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Advances in Nonlinear Analysis

Editor-in-Chief: Radulescu, Vicentiu / Squassina, Marco

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Volume 8, Issue 1

Well/ill-posedness for the dissipative Navier–Stokes system in generalized Carleson measure spaces

Yuzhao Wang
• Department of Mathematics and Physics, North China Electric Power University, Beijing 102206, P. R. China; and School of Mathematics, The University of Edinburgh, Edinburgh, EH9 3FD, United Kingdom
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• Other articles by this author:
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/ Jie Xiao
• Corresponding author
• Department of Mathematics and Statistics, Memorial University, St. John’s, NL A1C 5S7, Canada
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Published Online: 2017-01-12 | DOI: https://doi.org/10.1515/anona-2016-0042

Abstract

As an essential extension of the well known case $\beta \in \left(\frac{1}{2},1\right]$ to the hyper-dissipative case $\beta \in \left(1,\mathrm{\infty }\right)$, this paper establishes both well-posedness and ill-posedness (not only norm inflation but also indifferentiability of the solution map) for the mild solutions of the incompressible Navier–Stokes system with dissipation ${\left(-\mathrm{\Delta }\right)}^{\frac{1}{2}<\beta <\mathrm{\infty }}$ through the generalized Carleson measure spaces of initial data that unify many diverse spaces, including the Q space ${\left({Q}_{-s=-\alpha }\right)}^{n}$, the BMO-Sobolev space ${\left({\left(-\mathrm{\Delta }\right)}^{-\frac{s}{2}}\mathrm{BMO}\right)}^{n}$, the Lip-Sobolev space ${\left({\left(-\mathrm{\Delta }\right)}^{-\frac{s}{2}}\mathrm{Lip}\alpha \right)}^{n}$, and the Besov space ${\left({\stackrel{˙}{B}}_{\mathrm{\infty },\mathrm{\infty }}^{s}\right)}^{n}$.

MSC 2010: 30H25; 31C15; 35Q30; 42B37; 46E35

1 Introduction

Essentially, continuing from [47], we study the mild solutions (fluid velocities) of the so-called incompressible Navier–Stokes system with dissipation ${\left(-\mathrm{\Delta }\right)}^{\beta }$, under the assumption $\left(\beta ,x,t,\tau \right)\in \left(0,\mathrm{\infty }\right)×{ℝ}^{n}×\left(0,\tau \right)×\left(0,\mathrm{\infty }\right]$ (cf. e.g., [45, 54, 53, 39, 35, 52, 8, 28, 17]), given by

$\left\{\begin{array}{cc}& \left({\partial }_{t}𝐮+{\left(-\mathrm{\Delta }\right)}^{\beta }𝐮-𝐮\cdot \nabla 𝐮-\nabla p\right)\left(x,t\right)=0,\hfill \\ & \nabla \cdot 𝐮\left(x,t\right)=0,\hfill \\ & 𝐮\left(x,0\right)={𝐮}_{0}\left(x\right),\hfill \\ & \nabla \cdot {𝐮}_{0}\left(x\right)=0,\hfill \end{array}$(1.1)

with p being the pressure of a fluid, i.e., the solutions $𝐮$ that satisfy the following integral system arising from the initial data ${𝐮}_{0}$:

$𝐮\left(x,t\right)={e}^{-t{\left(-\mathrm{\Delta }\right)}^{\beta }}{𝐮}_{0}\left(x\right)-{\int }_{0}^{t}{e}^{-\left(t-s\right){\left(-\mathrm{\Delta }\right)}^{\beta }}ℙ\nabla \cdot \left(𝐮\left(x,s\right)\otimes 𝐮\left(x,s\right)\right)𝑑s,$(1.2)

where $ℙ={\left\{{ℙ}_{jk}\right\}}_{j,k=1,2,3}={\left\{{\delta }_{jk}+{R}_{j}{R}_{k}\right\}}_{j,k=1,2,\mathrm{\dots },n}$, ${\delta }_{jk}$ is the Kronecker symbol, and ${R}_{j}={\partial }_{j}{\left(-\mathrm{\Delta }\right)}^{-\frac{1}{2}}$ is the Riesz transform. In accordance with [38], the model (1.1) physically illustrates that the viscous stresses produce a dispersive momentum flow, which is determined by Darcy’s law of fractional order, while the dispersive flux divergence is the same as the change of momentum under Newton’s law and the assumption of fluid incompressibility. Furthermore, as explained in [45], in order to overcome some obstacles coming from numerical simulations of turbulent fluids induced by system (1.1) with $\beta =1$, we are suggested to handle system (1.1) with $\beta >1$, through replacing Δ (responsible for dissipating energy from the system) with a higher order dissipation mechanism $-{\left(-\mathrm{\Delta }\right)}^{\beta >1}$ (damping selectively the high wave numbers). Interestingly, upon taking the curl of the first equation of (1.1), setting $𝐰=\nabla \wedge 𝐮$, and using the computation on [40, p. 25], we find that the first equation in (1.1) can be rewritten as the following heat-type equation:

${\partial }_{t}𝐰+{\left(-\mathrm{\Delta }\right)}^{\beta }𝐰=\left(𝐰\cdot \nabla \right)𝐮-\left(𝐮\cdot \nabla \right)𝐰.$

Remarkably, the homogeneous form of the last system ${\partial }_{t}𝐰+{\left(-\mathrm{\Delta }\right)}^{\beta }𝐰=0$ (modelling anomalous diffusions) and its quasi-geophysical variant are of fundamental importance and interest in physics, probability and finance; see, e.g., [13, 29, 1, 24, 23, 37, 46, 18, 4, 12].

Here it is appropriate to mention three basic facts which reveal that the restriction $\frac{1}{2}<\beta <\mathrm{\infty }$ cannot be extended (to the challenging unsolved situation $0<\beta <\frac{1}{2}$) at least for our current casework regarding (1.1)–(1.2).

• System (1.1) with $\beta =1$ goes back to the classical incompressible Navier–Stokes system, see [7, 36] for more details.

• System (1.1) has a scaling property. If $\left(𝐮,p,{𝐮}_{0}\right)$ solves (1.1), then so does

• System (1.1) is more meaningful in a critical space which is invariant under the scaling

(1.3)

In fact, the solutions of (1.2) with $\beta =1$ in certain critical spaces have drawn a lot of attention since the pioneer work of Kato in [26], where he showed the global well-posedness with small data and the local well-posedness with large data in ${\left({L}^{n}\right)}^{n}$ (cf. [20] for an earlier work). Some similar well-posedness results can be found in [22, 44, 33] for certain Morrey spaces, in [30] for the space ${\left({\mathrm{BMO}}^{-1}\right)}^{n}$, and in [49] for the space ${\left({Q}_{\alpha }^{-1}\right)}^{n}$. Moreover, Li and Lin [31] showed global well-posedness in a subspace of ${\left({\mathrm{BMO}}^{-1}\right)}^{n}$ with large initial data, and Bourgain and Pavlovic̀ [3] found the norm inflation in ${\left({\stackrel{˙}{B}}_{\mathrm{\infty },\mathrm{\infty }}^{-1}\right)}^{n}$, which is the largest critical space with respect to (1.3) with $\beta =1$.

For $\frac{1}{2}<\beta \ne 1$, a study of (1.2) has been carried out partially. Wu [48] got a well-posed result for (1.1) with $1<\beta <\frac{5}{4}$ in the space ${\left({\stackrel{˙}{B}}_{1\le r\le \mathrm{\infty },2}^{\frac{5}{2}-2\beta }\right)}^{n}$. Li and Zhai [35] considered the fractional Navier–Stokes equation (1.1) with $\frac{1}{2}<\beta <1$, whence extending the above-mentioned well-posedness to Q-type spaces. Yu and Zhai [52] obtained a similar result in the largest critical space ${\left({\stackrel{˙}{B}}_{\mathrm{\infty },\mathrm{\infty }}^{1-2\beta }\right)}^{n}$. Cheskidov and Shvydkoy [11] discovered an ill-posed result in the largest critical space ${\left({\stackrel{˙}{B}}_{\mathrm{\infty },\mathrm{\infty }}^{1-2\alpha }\right)}^{n}$ under assumption (1.3). Deng and Yao [16, 15] obtained a similar ill-posedness in certain Triebel–Lizorkin spaces, providing a connection between the well-posedness in [30] and the ill-posedness in [3]. Li, Xiao and Yang [34] found a global well-posedness in some Besov-Q type spaces. Cheskidov and Dai [8] revealed a norm inflation phenomenon in the largest critical space ${\left({\stackrel{˙}{B}}_{\mathrm{\infty },\mathrm{\infty }}^{1-2\beta }\right)}^{n}$, with respect to (1.3) with $1\le \beta <\mathrm{\infty }$.

In this paper, partially motivated by [30, 49, 50, 25, 8, 35], under the natural constraint

$1<2\beta <\mathrm{\infty }\mathit{ }\text{and}\mathit{ }1-2\beta <\alpha <\mathrm{\infty },$

we develop a uniform framework to deal with a dichotomy of the well/ill-posed results in the generalized Carleson measure spaces ${\left({X}_{\beta }^{\alpha }\right)}^{n}$, which are critical with respect to (1.1) and, of course, contained in the homogeneous space ${\left({\stackrel{˙}{B}}_{\mathrm{\infty },\mathrm{\infty }}^{1-2\beta }\right)}^{n}$. In the above and below,

${X}_{\beta }^{\alpha }={X}_{\beta ,\mathrm{\infty }}^{\alpha },$

and for $0<\tau \le \mathrm{\infty }$, the space ${X}_{\beta ,\tau }^{\alpha }$ is defined by the norm

$\parallel f{\parallel }_{{X}_{\beta ,\tau }^{\alpha }}=\underset{\left(x,r\right)\in {ℝ}^{n}×\left(0,\tau \right)}{sup}\left({r}^{-\left(2\alpha +n\right)}{\int }_{0}^{{r}^{2\beta }}\left({\int }_{B\left(x,r\right)}|{e}^{-t{\left(-\mathrm{\Delta }\right)}^{\beta }}f\left(y\right){|}^{2}dy\right){t}^{-\frac{1-\alpha -\beta }{\beta }}dt\right){}^{\frac{1}{2}},$

where $B\left(x,r\right)$ is the ball centered at x with radius r. Meanwhile,

${\stackrel{˙}{B}}_{\mathrm{\infty },\mathrm{\infty }}^{1-2\beta }={\stackrel{˙}{B}}_{\mathrm{\infty },\mathrm{\infty },\mathrm{\infty }}^{1-2\beta },$

and for $0<\tau \le \mathrm{\infty }$, the space ${\stackrel{˙}{B}}_{\mathrm{\infty },\mathrm{\infty },\tau }^{1-2\beta }$ is determined by the norm (cf. [36])

${\parallel f\parallel }_{{\stackrel{˙}{B}}_{\mathrm{\infty },\mathrm{\infty },\tau }^{1-2\beta }}=\underset{\left(x,t\right)\in {ℝ}^{n}×\left(0,\tau \right)}{sup}{t}^{\frac{2\beta -1}{2\beta }}|{e}^{-t{\left(-\mathrm{\Delta }\right)}^{\beta }}f\left(x\right)|.$

Clearly, $\parallel \cdot {\parallel }_{{X}_{\beta ,\mathrm{\infty }}^{\alpha }}$ and ${\parallel f\parallel }_{{\stackrel{˙}{B}}_{\mathrm{\infty },\mathrm{\infty },\mathrm{\infty }}^{1-2\beta }}$ are invariant under the scaling transform (1.3). Moreover,

$\alpha \le {\alpha }^{\prime }⟹{X}_{\beta ,\mathrm{\infty }}^{\alpha }={X}_{\beta }^{\alpha }\subseteq {X}_{\beta ,\mathrm{\infty }}^{{\alpha }^{\prime }}={X}_{\beta }^{{\alpha }^{\prime }}\subseteq {\stackrel{˙}{B}}_{\mathrm{\infty },\mathrm{\infty },\mathrm{\infty }}^{1-2\beta }={\stackrel{˙}{B}}_{\mathrm{\infty },\mathrm{\infty }}^{1-2\beta },$

whose second inclusion becomes equality whenever ${\alpha }^{\prime }>0$. Accordingly,

${Y}_{\beta }^{\alpha }={Y}_{\beta ,\mathrm{\infty }}^{\alpha },$

and for $\tau \in \left(0,\mathrm{\infty }\right]$, the associated solution space ${Y}_{\beta ,\tau }^{\alpha }$ is decided by the norm

$\parallel u{\parallel }_{{Y}_{\beta ,\tau }^{\alpha }}=\underset{\left(x,r\right)\in {ℝ}^{n}×\left(0,\tau \right)}{sup}\left({r}^{-\left(2\alpha +n\right)}{\int }_{0}^{{r}^{2\beta }}\left({\int }_{B\left(x,r\right)}|u\left(y,t\right){|}^{2}dy\right){t}^{-\frac{1-\alpha -\beta }{\beta }}dt\right){}^{\frac{1}{2}}+sup{}_{\left(x,t\right)\in {ℝ}^{n}×\left(0,\tau \right)}t{}^{\frac{2\beta -1}{2\beta }}|u\left(x,t\right)|.$

The first theorem of this paper indicates that the well-posedness of (1.2) occurs only when α is relatively small.

Theorem 1.1.

Suppose

$\left\{\begin{array}{cc}& \beta \in \left(\frac{1}{2},1\right]\cup \left(1+\frac{n}{2},\mathrm{\infty }\right),\hfill \\ & 1-2\beta <\alpha \le 1-\beta ,\hfill \\ & 0<\tau \le \mathrm{\infty },\hfill \end{array}\mathit{ }\text{𝑜𝑟}\mathit{ }\left\{\begin{array}{cc}& \beta \in \left(\frac{1}{2},\mathrm{\infty }\right),\hfill \\ & 1-2\beta <\alpha \le 2-2\beta ,\hfill \\ & 0<\tau \le \mathrm{\infty }.\hfill \end{array}$

Then (1.2) is well-posed in ${\mathrm{\left(}{X}_{\beta \mathrm{,}\tau }^{\alpha }\mathrm{\right)}}^{n}$ with sufficiently small norm

$\parallel {𝐮}_{0}=\left({\left({u}_{0}\right)}_{1},\mathrm{\dots },{\left({u}_{0}\right)}_{n}\right){\parallel }_{{\left({X}_{\beta ,\tau }^{\alpha }\right)}^{n}}=\sum _{j=1}^{n}\parallel {\left({u}_{0}\right)}_{j}{\parallel }_{{X}_{\beta ,\tau }^{\alpha }}.$

Furthermore, the solution $\mathrm{u}\mathrm{\in }{\mathrm{\left(}{Y}_{\beta \mathrm{,}\tau }^{\alpha }\mathrm{\right)}}^{n}$, and the solution map $\mathcal{T}\mathrm{:}{\mathrm{u}}_{\mathrm{0}}\mathrm{\to }\mathrm{u}$ is analytic from a sufficient small neighborhood of origin of ${\mathrm{\left(}{X}_{\beta \mathrm{,}\tau }^{\alpha }\mathrm{\right)}}^{n}$ to ${\mathrm{\left(}{Y}_{\beta \mathrm{,}\tau }^{\alpha }\mathrm{\right)}}^{n}$.

Theorem 1.1 is essentially known for $\frac{1}{2}<\beta \le 1$ and $\alpha \in \left(1-2\beta ,0\right]$, see [30, 49, 50, 35, 52] and the relevant references therein. Needless to say that for the hyper-dissipative case $1<\beta <\mathrm{\infty }$, Theorem 1.1 is new. In order to prove Theorem 1.1, we follow the method originated from [30] (which was developed in [32, 49, 50, 35]), but we have to find a new idea to treat the singularity, appearing in ${\left(-\mathrm{\Delta }\right)}^{1-\beta }$, on the integrability of the kernel of

to meet the case $1<\beta <\mathrm{\infty }$. However, when $\beta \in \left(1,1+\frac{n}{2}\right)$, the singularity occurs both at the origin and at infinity, and so prevents us from getting the full range of α, see Lemma 2.1 for more details. Here, it should be pointed out that the well-posedness is understood under Kato’s sense as in [26, 30, 49, 50, 35, 52, 32, 33], i.e., both existence and uniqueness of a mild solution to (1.1) in the resolved space ${\left({Y}_{\beta }^{\alpha }\right)}^{n}$ are obtained by the standard fixed point theorem, which automatically ensures the analytic property of the solution map as stated above.

Remark 1.2.

Remarkably, the restriction $1-2\beta <\alpha$ in Theorem 1.1 is natural – this can be seen from the following assertion (cf. [45, 33, 50] for $\beta =1$). If ${L}_{2,n+2\alpha }$ (cf. [44]) stands for all real-valued ${L}_{\mathrm{loc}}^{2}\left({ℝ}^{n}\right)$-functions f obeying

$\parallel f{\parallel }_{{L}_{2,n+2\alpha }}=\underset{\left(x,r\right)\in {ℝ}^{n}×\left(0,\mathrm{\infty }\right)}{sup}\left({r}^{-\left(2\alpha +n\right)}{\int }_{B\left(x,r\right)}{|f\left(y\right){|}^{2}dy\right)}^{\frac{1}{2}}<\mathrm{\infty },$

then

and hence (1.2) is well-posed in ${\left({L}_{2,n+2\left(1-2\beta \right)}\right)}^{n}$ with sufficiently small norm

$\parallel {𝐮}_{0}={\left\{\left({𝐮}_{0}\right)\right\}}_{j=1}^{n}{\parallel }_{{\left({L}_{2,n+2\left(1-2\beta \right)}\right)}^{n}}\equiv \sum _{j=1}^{n}\parallel {\left({𝐮}_{0}\right)}_{j}{\parallel }_{{L}_{2,n+2\left(1-2\beta \right)}}.$

Corollary 1.3.

If $\frac{\mathrm{1}}{\mathrm{2}}\mathrm{<}\beta \mathrm{<}\mathrm{1}$ and $\mathrm{-}\mathrm{1}\mathrm{<}\alpha \mathrm{<}\frac{\mathrm{1}}{\mathrm{2}}$, then (1.2) is well-posed in ${\mathrm{\left(}{X}_{\beta }^{\alpha }\mathrm{\right)}}^{n}$ with sufficiently small norm

$\parallel {𝐮}_{0}={\left\{\left({𝐮}_{0}\right)\right\}}_{j=1}^{n}{\parallel }_{{\left({X}_{\beta }^{\alpha }\right)}^{n}}\equiv \sum _{j=1}^{n}\parallel {\left({𝐮}_{0}\right)}_{j}{\parallel }_{{X}_{\beta }^{\alpha }}.$

Furthermore, the solution $\mathrm{u}\mathrm{\in }{\mathrm{\left(}{Y}_{\beta }^{\alpha }\mathrm{\right)}}^{n}$, and the solution map $\mathcal{T}\mathrm{:}{\mathrm{u}}_{\mathrm{0}}\mathrm{\to }\mathrm{u}$ is analytic from a sufficient small neighborhood of the origin of ${\mathrm{\left(}{X}_{\beta }^{\alpha }\mathrm{\right)}}^{n}$ to ${\mathrm{\left(}{Y}_{\beta }^{\alpha }\mathrm{\right)}}^{n}$.

Note that

Thus, Corollary 1.3 extends and unifies partial well-posedness results in [35, 53, 52].

Upon taking into account $1-\beta <\alpha <\mathrm{\infty }$, the second theorem of this paper is concerned with the ill-posedness of (1.2), illustrating that Theorem 1.1 is optimal under certain circumstance.

Theorem 1.4.

Suppose

$1\le \beta <\mathrm{\infty }\mathit{ }\text{𝑎𝑛𝑑}\mathit{ }1-\beta <\alpha <\mathrm{\infty }.$

Then there exist a smooth space periodic solution $\mathrm{u}\mathit{}\mathrm{\left(}t\mathrm{\right)}$ of (1.2) with period $\mathrm{2}\mathit{}\pi$, and initial data ${\mathrm{u}}_{\mathrm{0}}$ such that the solution map $\mathcal{T}$ from ${\mathrm{\left(}{X}_{\beta \mathrm{,}\mathrm{1}}^{\alpha }\mathrm{\right)}}^{n}$ to ${\mathrm{\left(}{Y}_{\beta \mathrm{,}\mathrm{1}}^{\alpha }\mathrm{\right)}}^{n}$ is not differentiable at the origin of ${\mathrm{\left(}{X}_{\beta \mathrm{,}\mathrm{1}}^{\alpha }\mathrm{\right)}}^{n}$. Furthermore, for sufficiently small $ϵ\mathrm{\in }\mathrm{\left(}\mathrm{0}\mathrm{,}\mathrm{1}\mathrm{\right)}$, there exists a smooth space periodic solution $\mathrm{u}\mathit{}\mathrm{\left(}t\mathrm{\right)}$ of (1.2), with period $\mathrm{2}\mathit{}\pi$, such that

Additionally, the same assertion holds for ${\mathrm{\left(}{X}_{\beta }^{\alpha }\mathrm{\right)}}^{n}$ and ${\mathrm{\left(}{Y}_{\beta }^{\alpha }\mathrm{\right)}}^{n}$, provided that $\mathrm{0}\mathrm{\le }\alpha \mathrm{<}\mathrm{\infty }$.

In order to verify Theorem 1.4, we suitably employ the counter-example constructed in [3, 8] to get such a smooth space-periodic mild solution (with an arbitrarily small initial data in ${\left({X}_{\beta }^{\alpha }\right)}^{n}$) that becomes not only arbitrarily large in ${\left({X}_{\beta }^{\alpha }\right)}^{n}$ for an arbitrarily small time, but also relatively large in the resolution space ${\left({Y}_{\beta }^{\alpha }\right)}^{n}$.

Perhaps, it is appropriate to make two more comments on Theorems 1.1 and 1.4, and Corollary 1.3 as follows.

Figure 1

Here $P=\left(-n,\frac{n+2}{2}\right)$ and $Q=\left(-\frac{n}{2},\frac{n+2}{2}\right)$.

As described in Theorems 1.1 and 1.4, the well-posedness and the ill-posedness of (1.2) initialed in ${\left({X}_{\beta }^{\alpha }\right)}^{n}$ can be summarized in Figure 1. The well-posedness is set up for all parameter $\left(\alpha ,\beta \right)$ in the region between the polyline $\stackrel{^}{ABC}$ and polyline $\stackrel{^}{DEF}$ but $\mathrm{\Delta }PQB$, while the ill-posed results are established for $\left(\alpha ,\beta \right)$ above polyline $\stackrel{^}{DEF}$. It is most likely that system (1.2) is well-posed when $\left(\alpha ,\beta \right)$ in the triangle $\mathrm{\Delta }PQB$ – unfortunately, we have failed to show this possible well-posedness because of Lemma 2.1 (ii) (cf. Remark 2.2). It seems that a new method, such as the one in [2], is required to fill this unnatural gap.

As a direct consequence of Theorems 1.1 and 1.4, and Corollary 1.3 (whose argument ensures that ${X}_{\beta }^{0}={\mathrm{BMO}}^{1-2\beta }$ and ${X}_{\beta }^{\alpha }={\stackrel{˙}{B}}_{\mathrm{\infty },\mathrm{\infty }}^{1-2\beta }$ with $0<\alpha <\frac{1}{2}$), we assert that (1.2) is:

• well-posed when $\beta \in \left(\frac{1}{2},1\right]$, while ill-posed when $\beta >1$ initialed in ${\left({\mathrm{BMO}}^{1-2\beta }\right)}^{n}$,

• well-posed when $\beta \in \left(\frac{1}{2},1\right)$, while ill-posed when $\beta \ge 1$ initialed in ${\left({\stackrel{˙}{B}}_{\mathrm{\infty },\mathrm{\infty }}^{1-2\beta }\right)}^{n}$.

Although the well-posedness of this last assertion for $\beta =1$ and the ill-posedness for $\beta =1$ or $\beta >1$ reduce to the well-posedness in [30] for $\beta =1$ and the ill-posedness in [3, 8] (see, e.g., [9, 10, 51, 14] for more details) for $\beta =1$ or $\beta >1$, respectively, our ill-posedness in Theorem 1.4 cannot be implied by the results in [3, 8] at least because our space ${X}_{\beta }^{\alpha }$ with $\beta >1\ge 1-\alpha$ behaves differently from their space ${\stackrel{˙}{B}}_{\mathrm{\infty },p}^{-\gamma }$ with $\left(\gamma ,p\right)\in \left[1,\mathrm{\infty }\right)×\left(2,\mathrm{\infty }\right]$, and yet includes non-differentiability of the solution map as an extra property.

The preceding theorems can be straightforwardly applied to (1.2) initialed in the Campanato–Sobolev (CS) spaces explored in [47]:

where ${\left(-\mathrm{\Delta }\right)}^{-\frac{s}{2}}$ is determined by the Fourier transforms

$\stackrel{^}{f}\left(\xi \right)={\int }_{{ℝ}^{n}}{e}^{-i\xi \cdot x}f\left(x\right)𝑑x\mathit{ }\text{and}\mathit{ }\stackrel{^}{{\left(-\mathrm{\Delta }\right)}^{-\frac{s}{2}}f}\left(\xi \right)={|\xi |}^{-s}\stackrel{^}{f}\left(\xi \right),$

and ${\mathcal{ℒ}}_{2,n+2\alpha }$ denotes the square Campanato space (cf. [5, 6, 41]) on ${ℝ}^{n}$ of all real-valued ${L}_{\mathrm{loc}}^{2}\left({ℝ}^{n}\right)$-functions f satisfying

${\parallel f\parallel }_{{\mathcal{ℒ}}_{2,n+2\alpha }}=\underset{\left(x,r\right)\in {ℝ}^{n}×\left(0,\mathrm{\infty }\right)}{sup}{\left({r}^{-\left(2\alpha +n\right)}{\int }_{B\left(x,r\right)}{|f\left(y\right)-{r}^{-n}{\int }_{B\left(x,r\right)}f\left(z\right)𝑑z|}^{2}𝑑y\right)}^{\frac{1}{2}}<\mathrm{\infty }$

and

Interestingly, we have Table 1.

Table 1

Spaces.

Even more interestingly, we discover

$-1<\alpha <1\mathit{ }\text{and}\mathit{ }\frac{1}{2}<\beta <\frac{3}{2}⟹{X}_{\beta }^{\alpha }={\mathcal{ℒ}}_{2,n+2\alpha }^{1-\alpha -2\beta }.$

This fundamental identification, along with Theorems 1.1 and 1.4, produces the following assertion of relatively independent interest.

Corollary 1.5.

Suppose $\frac{\mathrm{1}}{\mathrm{2}}\mathrm{<}\beta \mathrm{<}\frac{\mathrm{3}}{\mathrm{2}}$ and $\mathrm{max}\mathit{}\mathrm{\left\{}\mathrm{-}\mathrm{1}\mathrm{,}\mathrm{1}\mathrm{-}\mathrm{2}\mathit{}\beta \mathrm{\right\}}\mathrm{<}\alpha \mathrm{<}\mathrm{1}\mathrm{.}$

• (i)

If $\frac{1}{2}<\beta <1$ , then ( 1.2 ) is well posed in ${\left({\mathcal{ℒ}}_{2,n+2\alpha }^{1-\alpha -2\beta }\right)}^{n}$ with sufficiently small norm

$\parallel {𝐮}_{0}={\left\{{\left({𝐮}_{0}\right)}_{j}\right\}}_{j=1}^{n}{\parallel }_{{\left({\mathcal{ℒ}}_{2,n+2\alpha }^{1-\alpha -2\beta }\right)}^{n}}=\sum _{j=1}^{n}\parallel {\left({𝐮}_{0}\right)}_{j}{\parallel }_{{\mathcal{ℒ}}_{2,n+2\alpha }^{1-\alpha -2\beta }}.$

• (ii)

If $1\le \beta <\frac{3}{2}$ , then ( 1.2 ) in ${\left({\mathcal{ℒ}}_{2,n+2\alpha }^{1-\alpha -2\beta }\right)}^{n}$ , with sufficiently small norm ${\parallel {𝐮}_{0}\parallel }_{{\left({\mathcal{ℒ}}_{2,n+2\alpha }^{1-\alpha -2\beta }\right)}^{n}}$ , is well-posed when $-1<\alpha \le 2-2\beta$ but ill-posed, in the sense of Theorem 1.4 , when $1-\beta <\alpha <1$.

The rest of the paper is organized as follows. In Section 2, we give an exposition of the details of the proofs of Theorem 1.1 and Remark 1.2. Section 3 provides a complete demonstration of Theorem 1.4. In Section 4, we check Corollary 1.5, using Theorems 1.1 and 1.4.

Notation.

From now on, ${ℝ}_{+}^{n+1}={ℝ}^{n}×\left(0,\mathrm{\infty }\right)$. The symbol $A\lesssim B$ represents that there exists a positive constant C satisfying $A\le CB$, and thus $A\approx B$ represents the comparability of the quantities A and B, i.e., $A\lesssim B$ and $B\lesssim A$.

2 Well-posedness in ${\left({X}_{\beta }^{\alpha }\right)}^{n}$

This section is devoted to a proof of Theorem 1.1 with $\tau =\mathrm{\infty }$. The argument for Theorem 1.1 with $\tau <\mathrm{\infty }$ is similar.

2.1 Estimation for some singular integrals

We need two technical results on some integrals of strong singularity.

Lemma 2.1.

Let $s\mathrm{\in }\mathrm{\left(}\mathrm{0}\mathrm{,}\mathrm{1}\mathrm{\right)}$ and ${K}_{s}^{\beta }\mathit{}\mathrm{\left(}x\mathrm{\right)}$ be the kernel of ${\mathrm{\left(}\mathrm{-}\mathrm{\Delta }\mathrm{\right)}}^{\mathrm{1}\mathrm{-}\beta }\mathit{}\mathrm{\left(}{e}^{\mathrm{-}{\mathrm{\left(}\mathrm{-}\mathrm{\Delta }\mathrm{\right)}}^{\beta }}\mathrm{-}{e}^{\mathrm{-}s\mathit{}{\mathrm{\left(}\mathrm{-}\mathrm{\Delta }\mathrm{\right)}}^{\beta }}\mathrm{\right)}$.

• (i)

If $\beta \in \left(0,1\right)$ , then

$|{K}_{s}^{\beta }\left(x\right)|\lesssim {\left(1+|x|\right)}^{-n-2+2\beta }+{s}^{-\frac{2+n-2\beta }{2\beta }}{\left(1+{s}^{-\frac{1}{2\beta }}|x|\right)}^{-n-2+2\beta }.$(2.1)

• (ii)

If $\beta \in \left(1,1+\frac{n}{2}\right)$ , then

$|{K}_{s}^{\beta }\left(x\right)|\lesssim {\left(1+|x|\right)}^{-n-1}+{s}^{-\frac{2+n-2\beta }{2\beta }}{\left(1+{s}^{-\frac{1}{2\beta }}|x|\right)}^{-n-2+2\beta }.$(2.2)

• (iii)

If $\beta =1$ or $\beta \in \left(1+\frac{n}{2},\mathrm{\infty }\right)$ , then

$|{K}_{s}^{\beta }\left(x\right)|\lesssim {\left(1+|x|\right)}^{-n-1}+{s}^{-\frac{2+n-2\beta }{2\beta }}{\left(1+{s}^{-\frac{1}{2\beta }}|x|\right)}^{-n-1}.$(2.3)

Proof.

(i)  Suppose $\beta \in \left(0,1\right)$, thus $1-\beta >0$. The kernel $K\left(x\right)$ of ${\left(-\mathrm{\Delta }\right)}^{1-\beta }{e}^{-{\left(-\mathrm{\Delta }\right)}^{\beta }}$ has the decay estimate

$|K\left(x\right)|\lesssim {\left(1+|x|\right)}^{\beta -n-1},$

see [39]. Then (2.1) follows by a scaling argument.

(ii)  Assume $1<\beta <1+\frac{n}{2}$. Let $\psi \in {C}_{0}^{\mathrm{\infty }}\left({ℝ}^{n}\right)$ and $\psi \left(\xi \right)=1$ for $|\xi |<1$, and denote by $𝗆$ the symbol of the operator ${\left(-\mathrm{\Delta }\right)}^{1-\beta }\left({e}^{-{\left(-\mathrm{\Delta }\right)}^{\beta }}-{e}^{-s{\left(-\mathrm{\Delta }\right)}^{\beta }}\right)$. Then, this symbol can be broken down into two terms:

$𝗆\left(\xi \right)={|\xi |}^{2-2\beta }\left({e}^{-{|\xi |}^{2\beta }}-{e}^{-s{|\xi |}^{2\beta }}\right)\psi \left({s}^{\frac{1}{2\beta }}\xi \right)+{|\xi |}^{2-2\beta }\left({e}^{-{|\xi |}^{2\beta }}-{e}^{-s{|\xi |}^{2\beta }}\right)\left(1-\psi \left({s}^{\frac{1}{2\beta }}\xi \right)\right)\equiv {𝗆}_{1}\left(\xi \right)+{𝗆}_{2}\left(\xi \right).$

The first term ${𝗆}_{1}$ is rewritten as

${𝗆}_{1}\left(\xi \right)={|\xi |}^{2-2\beta }\left({e}^{-{|\xi |}^{2\beta }}-1\right)\psi \left({s}^{\frac{1}{2\beta }}\xi \right)+{|\xi |}^{2-2\beta }\left(1-{e}^{-s{|\xi |}^{2\beta }}\right)\psi \left({s}^{\frac{1}{2\beta }}\xi \right)\equiv {𝗆}_{11}\left(\xi \right)+{𝗆}_{12}\left(\xi \right).$

For ${𝗆}_{12}$, by scaling, we only need to show

$|{\int }_{{ℝ}^{n}}{e}^{ix\cdot \xi }{|\xi |}^{2-2\beta }\left({e}^{-{|\xi |}^{2\beta }}-1\right)\psi \left(\xi \right)𝑑\xi |\lesssim {\left(1+|x|\right)}^{-n-1},$

which is obvious since the symbol ${|\xi |}^{2-2\beta }\left({e}^{-{|\xi |}^{2\beta }}-1\right)\psi \left(\xi \right)$ is compactly supported and has no singularity at the origin (cf. [39]). Note that the kernel of ${𝗆}_{11}$ can be controlled similarly if $s>\frac{1}{4}$. So, without loss of generality, we may assume $s\ll 1$ in the sequel. Write

${𝗆}_{11}\left(\xi \right)={𝗆}_{111}\left(\xi \right)+{𝗆}_{112}\left(\xi \right)+{𝗆}_{113}\left(\xi \right),$

where

${𝗆}_{111}\left(\xi \right)\equiv {|\xi |}^{2-2\beta }\left({e}^{-{|\xi |}^{2\beta }}-1\right)\psi \left(\xi \right),$${𝗆}_{112}\left(\xi \right)\equiv {|\xi |}^{2-2\beta }{e}^{-{|\xi |}^{2\beta }}\left(1-\psi \left(\xi \right)\right)\psi \left({s}^{\frac{1}{2\beta }}\xi \right),$${𝗆}_{111}\left(\xi \right)\equiv {|\xi |}^{2-2\beta }\left(1-\psi \left(\xi \right)\right)\psi \left({s}^{\frac{1}{2\beta }}\xi \right).$

In view of the previous argument, only the kernel of the last term, denoted by ${K}_{113}$, needs a control. By a simple calculation, we get that

${\parallel {𝗆}_{113}\parallel }_{{L}^{1}}\lesssim {s}^{-\frac{n+2-2\beta }{2\beta }},$

and so, if the multi-index α satisfies $|\alpha |=n+1$, then

${\parallel {\partial }_{\xi }^{\alpha }{𝗆}_{113}\parallel }_{{L}^{1}}\lesssim 1.$

Thus, an integration by parts derives that the kernel ${K}_{113}$ of ${𝗆}_{113}$ enjoys

$|{K}_{113}\left(x\right)|\lesssim \mathrm{min}\left\{{s}^{-\frac{n+2-2\beta }{2\beta }},{|x|}^{-n-1}\right\}.$

In order to prove (2.2), an improvement must be made when ${s}^{\frac{1}{2\beta }}\lesssim |x|\lesssim 1$. Now let $\delta \in \left(2,{s}^{-\frac{1}{2\beta }}/2\right)$. Then

${K}_{113}\left(x\right)={\int }_{{ℝ}^{n}}{e}^{ix\cdot \xi }{|\xi |}^{2-2\beta }\left(1-\psi \left(\xi \right)\right)\psi \left({s}^{\frac{1}{2\beta }}\xi \right)𝑑\xi$$={\int }_{\frac{1}{2}<|\xi |\le \delta }{e}^{ix\cdot \xi }{|\xi |}^{2-2\beta }\left(1-\psi \left(\xi \right)\right)𝑑\xi +{\int }_{\delta <|\xi |\le {s}^{-\frac{1}{2\beta }}}{e}^{ix\cdot \xi }{|\xi |}^{2-2\beta }\psi \left({s}^{\frac{1}{2\beta }}\xi \right)𝑑\xi$$\equiv 𝖠\left(\delta \right)+𝖡\left(\delta \right).$

It is easy to see that

Repeatedly using integration by parts, we obtain

$|𝖡\left(\delta \right)|\lesssim |x{|}^{-n}{\int }_{\delta <|\xi |\le {s}^{-\frac{1}{2\beta }}}|\xi {|}^{2-2\beta -n}d\xi \lesssim {\delta }^{2-2\beta }|x{|}^{-n},$

reaching the desired estimate (2.2) upon choosing $\delta =\frac{1}{|x|}$.

(iii)  For $\beta =1$, estimate (2.3) is obvious. So, it remains to treat $\beta >1+\frac{n}{2}$. In view of the argument in (ii), it is enough to handle ${K}_{113}$. Since

$|{K}_{113}\left(x\right)|\lesssim {\int }_{{ℝ}^{n}}|\xi {|}^{2-2\beta }\left(1-\psi \left(\xi \right)\right)\psi \left({s}^{\frac{1}{2\beta }}\xi \right)d\xi \lesssim {\int }_{|\xi |>\frac{1}{2}}|\xi {|}^{2-2\beta }d\xi \lesssim 1,$

an integration by parts gives (as estimated in (ii))

$|{K}_{113}\left(x\right)|\lesssim |x{|}^{-n-1}\sum _{|\alpha |=n+1}\parallel {\partial }_{\xi }^{\alpha }{𝗆}_{113}{\parallel }_{{L}^{1}}\lesssim |x{|}^{-n-1},$

and the desired result (2.3) follows. ∎

Remark 2.2.

It turns out that Lemma 2.1 (ii) is not sufficient for our purpose, since the decay in the second term of the right-hand side of (2.2) is not strong enough in small scale $|x|\le 1$. This is the main reason why our well-posed results fail to cover the case $\beta \in \left(1,1+\frac{n}{2}\right)$ and $2-2\beta <\alpha \le 1-\beta$ (the triangle $\mathrm{\Delta }PQB$ in Figure 1). Note that ${K}_{s}^{\beta }\left(x\right)$ can be rewritten as $M\left(x\right)+E\left(x\right)$, where $E\left(x\right)$ is well-behaved as an error term, and $M\left(x\right)$ behaves like

${\int }_{1\le |\xi |\le {s}^{-\frac{1}{2\beta }}}{e}^{ix\cdot \xi }{|\xi |}^{2-2\beta }𝑑\xi .$

So, in view of the identity (for a dimensional constant ${c}_{n}$)

it seems that (2.2) is the best expected decay in small scale as s tends to zero.

As one of our new-discovered tools, Lemma 2.1 will be used to prove the following lemma.

Lemma 2.3.

• (i)

If $\beta >\frac{1}{2}$, $1-2\beta <\alpha \le 1-\beta$ and

${\mathrm{C}}_{\beta }\left(f,t,x\right)={\int }_{0}^{t}{e}^{-\left(t-s\right){\left(-\mathrm{\Delta }\right)}^{\beta }}{\left(-\mathrm{\Delta }\right)}^{\beta }f\left(s,x\right)𝑑s,$

then

${\int }_{0}^{\mathrm{\infty }}\parallel {\mathrm{C}}_{\beta }\left(f,t,\cdot \right){\parallel }_{{L}^{2}}^{2}{t}^{-\frac{1-\alpha -\beta }{\beta }}dt\lesssim {\int }_{0}^{\mathrm{\infty }}\parallel f\left(t,\cdot \right){\parallel }_{{L}^{2}}^{2}{t}^{-\frac{1-\alpha -\beta }{\beta }}dt.$

• (ii)

If $\beta >1+\frac{n}{2}$ or $\beta =1$, $1-2\beta <\alpha \le 1-\beta$ and

${\mathrm{D}}_{\alpha ,\beta }\left(g\right)=\underset{\left(x,r\right)\in {ℝ}^{n}×\left(0,1\right)}{sup}{r}^{-\left(n+2\alpha \right)}{\int }_{0}^{{t}^{2\beta }}{\int }_{B\left(x,r\right)}|g\left(h,y\right)|dy\frac{dh}{{h}^{\frac{1-\alpha -\beta }{\beta }}},$

then

${\int }_{0}^{1}\parallel {\left(-\mathrm{\Delta }\right)}^{\frac{1}{2}}{e}^{-\frac{t}{2}{\left(-\mathrm{\Delta }\right)}^{\beta }}{\int }_{0}^{t}g\left(s,\cdot \right)ds\parallel {}_{{L}^{2}}{}^{2}\frac{dt}{{t}^{\frac{1-\alpha -\beta }{\beta }}}\lesssim \mathrm{D}{}_{\alpha ,\beta }\left(g\right)\int {}^{1}{}_{0}\parallel g\left(s,\cdot \right)\parallel {}_{{L}^{1}}\frac{ds}{{s}^{\frac{1-\alpha -\beta }{\beta }}}.$(2.4)

• (iii)

If $\beta >\frac{1}{2}$ and $1-2\beta <\alpha \le 2-2\beta$ , then ( 2.4 ) still holds.

Proof.

(i)  Suppose $\beta >\frac{1}{2}$ and $1-2\beta <\alpha \le 1-\beta$. An application of the definition of ${e}^{-\left(t-s\right){\left(-\mathrm{\Delta }\right)}^{\beta }}$, Plancehrel’s formula and Hölder’s inequality gives

${\int }_{0}^{\mathrm{\infty }}\parallel {\mathrm{C}}_{\beta }\left(f,t,\cdot \right){\parallel }_{{L}^{2}}^{2}{t}^{\frac{\alpha +\beta -1}{\beta }}dt\approx {\int }_{0}^{\mathrm{\infty }}\parallel {\int }_{0}^{t}{|\xi {|}^{2\beta }{e}^{-\left(t-s\right){|\xi |}^{2\beta }}\stackrel{^}{f\left(s,\xi \right)}ds\parallel }_{{L}^{2}}^{2}{t}^{\frac{\alpha +\beta -1}{\beta }}dt$$\lesssim {\int }_{0}^{\mathrm{\infty }}{\int }_{{ℝ}^{n}}{\int }_{0}^{t}|\xi {|}^{2\beta }{e}^{-\left(t-s\right){|\xi |}^{2\beta }}|\stackrel{^}{f\left(s,\xi \right)}{|}^{2}dsd\xi {t}^{\frac{\alpha +\beta -1}{\beta }}dt$$\lesssim {\int }_{{ℝ}^{n}}{\int }_{0}^{\mathrm{\infty }}\left(\int {}_{s}{}^{\mathrm{\infty }}|\xi {|}^{2\beta }{e}^{-\left(t-s\right){|\xi |}^{2\beta }}dt\right)|\stackrel{^}{f\left(s,\xi \right)}|{}^{2}s{}^{\frac{\alpha +\beta -1}{\beta }}dsd\xi$$\lesssim {\int }_{{ℝ}^{n}}{\int }_{0}^{\mathrm{\infty }}|\stackrel{^}{f\left(s,\xi \right)}{|}^{2}{s}^{\frac{\alpha +\beta -1}{\beta }}dsd\xi$$\approx {\int }_{0}^{\mathrm{\infty }}\parallel f\left(t,\cdot \right){\parallel }_{{L}^{2}}^{2}{t}^{\frac{\alpha +\beta -1}{\beta }}dt,$

as desired.

(ii)  Suppose $\beta >1+\frac{n}{2}$ or $\beta =1$ and $1-2\beta <\alpha \le 1-\beta$. Using the inner-product ${〈\cdot ,\cdot 〉}_{{L}^{2}}$ in ${L}^{2}$ with respect to the spatial variable $x\in {ℝ}^{n}$, we obtain

${\int }_{0}^{1}\parallel {\left(-\mathrm{\Delta }\right)}^{\frac{1}{2}}{e}^{-\frac{t}{2}{\left(-\mathrm{\Delta }\right)}^{\beta }}{\int }_{0}^{t}g\left(s,\cdot \right)ds{\parallel }_{{L}^{2}}^{2}{t}^{\frac{\alpha +\beta -1}{\beta }}dt$$={\int }_{0}^{1}〈{\int }_{0}^{t}{\left(-\mathrm{\Delta }\right)}^{\frac{1}{2}}{e}^{-\frac{t}{2}{\left(-\mathrm{\Delta }\right)}^{\beta }}g\left(s,\cdot \right)ds,{\int }_{0}^{t}{\left(-\mathrm{\Delta }\right)}^{\frac{1}{2}}{e}^{-\frac{t}{2}{\left(-\mathrm{\Delta }\right)}^{\beta }}g\left(h,\cdot \right)dh〉{}_{{L}^{2}}t{}^{\frac{\alpha +\beta -1}{\beta }}dt$$=2\mathrm{\Re }\left(\int {\int }_{0$\lesssim {\int }_{0}^{1}|{〈g\left(s,\cdot \right),{\int }_{0}^{s}{\left(-\mathrm{\Delta }\right)}^{1-\beta }\left({e}^{-{\left(-\mathrm{\Delta }\right)}^{\beta }}-{e}^{-s{\left(-\mathrm{\Delta }\right)}^{\beta }}\right)g\left(h,\cdot \right)𝑑h〉}_{{L}^{2}}|{s}^{\frac{\alpha +\beta -1}{\beta }}𝑑s.$

If ${K}_{s}^{\beta }\left(x\right)$ is the kernel of ${\left(-\mathrm{\Delta }\right)}^{1-\beta }\left({e}^{-{\left(-\mathrm{\Delta }\right)}^{\beta }}-{e}^{-s{\left(-\mathrm{\Delta }\right)}^{\beta }}\right)$, then an application of Lemma 2.1 and Hölder’s inequality derives

$|{\int }_{0}^{s}{\left(-\mathrm{\Delta }\right)}^{1-\beta }\left({e}^{-{\left(-\mathrm{\Delta }\right)}^{\beta }}-{e}^{-h{\left(-\mathrm{\Delta }\right)}^{\beta }}\right)g\left(h,x\right)dh|\lesssim sup{}_{0$\lesssim \underset{0$\lesssim \underset{t\in \left(0,1\right),k\in {ℤ}^{n}}{sup}{t}^{\frac{2\beta -n-2}{2\beta }}{\int }_{0}^{t}{\int }_{|k{t}^{\frac{1}{2\beta }}-y|<{t}^{\frac{1}{2\beta }}}|g\left(h,y\right)|dydh$$\lesssim \underset{\left(x,t\right)\in {ℝ}^{n}×\left(0,1\right)}{sup}{t}^{-\left(2\alpha +n\right)}{\int }_{0}^{{t}^{2\beta }}{\int }_{B\left(x,t\right)}|g\left(h,y\right)|dy\frac{dh}{{h}^{\frac{1-\alpha -\beta }{\beta }}}.$

This, along with another application of Hölder’s inequality, implies

${\int }_{0}^{1}{\parallel {\left(-\mathrm{\Delta }\right)}^{\frac{1}{2}}{e}^{-\frac{t}{2}{\left(-\mathrm{\Delta }\right)}^{\beta }}{\int }_{0}^{t}g\left(s,\cdot \right)ds\parallel }_{{L}^{2}}^{2}{t}^{\frac{\alpha +\beta -1}{\beta }}dt\lesssim {\mathrm{D}}_{\alpha ,\beta }\left(g\right){\int }_{0}^{1}\parallel g\left(s,\cdot \right){\parallel }_{{L}^{1}}{s}^{\frac{-1+\alpha +\beta }{\beta }}ds.$

(iii)  Suppose $\beta >\frac{1}{2}$ and $1-2\beta <\alpha \le 2\left(1-\beta \right)$. In view of the argument used in (ii), we obtain

${\int }_{0}^{1}\parallel {\left(-\mathrm{\Delta }\right)}^{\frac{1}{2}}{e}^{-\frac{t}{2}{\left(-\mathrm{\Delta }\right)}^{\beta }}{\int }_{0}^{t}g\left(s,\cdot \right)ds{\parallel }_{{L}^{2}}^{2}{t}^{\frac{\alpha +\beta -1}{\beta }}dt=2\mathrm{\Re }\left(\int {\int }_{0$=2\beta \mathrm{\Re }\left(\int {\int }_{0${\lesssim {\int }_{0}^{1}|〈g\left(s,\cdot \right),{s}^{\frac{\beta -1}{\beta }}{\int }_{0}^{s}\left[{\int }_{{s}^{\frac{1}{\beta }}}^{1}\left(-\mathrm{\Delta }\right){e}^{-{\left(-t\mathrm{\Delta }\right)}^{\beta }}dt\right]g\left(h,\cdot \right)dh〉}_{{L}^{2}}|s{}^{\frac{\alpha +\beta -1}{\beta }}ds.$

Denote by $\stackrel{~}{K}$ and $\stackrel{~}{𝗆}$ the kernel and symbol of the differential operator

${\int }_{{s}^{\frac{1}{\beta }}}^{1}\left(-\mathrm{\Delta }\right){e}^{-{\left(-t\mathrm{\Delta }\right)}^{\beta }}𝑑t.$

In view of the argument used in (ii), it suffices to prove

$|\stackrel{~}{K}\left(x\right)|\lesssim {\left(1+|x|\right)}^{-n-1}+{s}^{-\frac{n}{2\beta }}{\left(1+{s}^{-\frac{1}{2\beta }}|x|\right)}^{-n-1}.$(2.5)

By a change of variables, we have

$\begin{array}{cc}\hfill \stackrel{~}{𝗆}\left(\xi \right)& ={\int }_{{s}^{\frac{1}{\beta }}}^{1}|\xi {|}^{2}{e}^{-{\left(t{|\xi |}^{2}\right)}^{\beta }}dt={\int }_{{s}^{\frac{1}{\beta }}|\xi {|}^{2}}^{{|\xi |}^{2}}{e}^{-{t}^{\beta }}dt\hfill \\ & =F\left({s}^{\frac{1}{\beta }}{|\xi |}^{2}\right)-F\left({|\xi |}^{2}\right),\hfill \end{array}$

where

It is clear that $F\left({|\xi |}^{2}\right)\in {L}^{1}$ and ${\partial }_{\xi }^{\gamma }F\left({|\xi |}^{2}\right)\in {L}^{1}$ with $|\gamma |=n+1$. So, an integration by parts shows

$|{\int }_{{ℝ}^{n}}{e}^{ix\cdot \xi }F\left(|\xi {|}^{2}\right)d\xi |\lesssim \left(1+|x|\right){}^{-n-1}.$

Thus, (2.5) follows by a scaling argument thanks to $F\left({s}^{\frac{1}{\beta }}{|\xi |}^{2}\right)=F\left({|{s}^{\frac{1}{2\beta }}\xi |}^{2}\right).$

2.2 Proof of Theorem 1.1

The proof follows the idea originated from [30], see also [32, Chapter 16]. We rewrite (1.2) (cf. [26, 27, 22, 44, 32]) as

$𝐮\left(x,t\right)={e}^{-t{\left(-\mathrm{\Delta }\right)}^{\beta }}{𝐮}_{0}\left(x\right)-B\left(𝐮,𝐮\right),$(2.6)

where $B\left(\cdot ,\cdot \right)$ is the following bilinear form:

$B\left(𝐮,𝐯\right)={\int }_{0}^{t}{e}^{-\left(t-s\right){\left(-\mathrm{\Delta }\right)}^{\beta }}ℙ\nabla \cdot \left(𝐮\otimes 𝐯\right)𝑑s.$(2.7)

Let α, β satisfy the conditions in Theorem 1.1. According to the standard fixed point argument, it suffices to prove that the integral equation (2.6) is solvable in a small neighborhood of the origin in ${X}_{\beta }^{\alpha }$. Thanks to the definition, we have

${\parallel {e}^{-t{\left(-\mathrm{\Delta }\right)}^{\beta }}{𝐮}_{0}\parallel }_{{\left({Y}_{\beta }^{\alpha }\right)}^{n}}\lesssim {\parallel {𝐮}_{0}\parallel }_{{\left({X}_{\beta }^{\alpha }\right)}^{n}},$

thus it remains to verify that (2.7) is bounded from ${\left({Y}_{\beta }^{\alpha }\right)}^{n}×{\left({Y}_{\beta }^{\alpha }\right)}^{n}$ to ${\left({Y}_{\beta }^{\alpha }\right)}^{n}$. Of course, it suffices to show both the ${L}^{2}$-bound

${r}^{-\left(2\alpha +n\right)}{\int }_{0}^{{r}^{2\beta }}{\int }_{|y-x|(2.8)

and the ${L}^{\mathrm{\infty }}$-bound

${\parallel B\left(𝐮,𝐯\right)\parallel }_{{L}^{\mathrm{\infty }}}\lesssim {t}^{\frac{1}{2\beta }-1}{\parallel 𝐮\parallel }_{{\left({Y}_{\beta }^{\alpha }\right)}^{n}}{\parallel 𝐯\parallel }_{{\left({Y}_{\beta }^{\alpha }\right)}^{n}},$(2.9)

where

$𝐮=\left({u}_{1},{u}_{2},\mathrm{\dots },{u}_{n}\right),$$\parallel 𝐮{\parallel }_{{\left({Y}_{\beta }^{\alpha }\right)}^{n}}=\sum _{j=1}^{n}\parallel {u}_{j}{\parallel }_{{Y}_{\beta }^{\alpha }},$$𝐯=\left({v}_{1},{v}_{2},\mathrm{\dots },{v}_{n}\right),$$\parallel 𝐯{\parallel }_{{\left({Y}_{\beta }^{\alpha }\right)}^{n}}=\sum _{j=1}^{n}\parallel {v}_{j}{\parallel }_{{Y}_{\beta }^{\alpha }}.$

Step 1: ${L}^{\mathrm{2}}$-bound. Letting ${1}_{r,x}\left(y\right)={\chi }_{B\left(x,10r\right)}\left(y\right)$ be the characteristic function of $B\left(x,10r\right)$ and I the identity map, we divide $B\left(𝐮,𝐯\right)$ into three parts:

$B\left(𝐮,𝐯\right)={B}_{1}\left(𝐮,𝐯\right)+{B}_{2}\left(𝐮,𝐯\right)+{B}_{3}\left(𝐮,𝐯\right),$

where

${B}_{1}\left(𝐮,𝐯\right)={\left(-\mathrm{\Delta }\right)}^{-\frac{1}{2}}ℙ\nabla \cdot {\int }_{0}^{s}{e}^{-\left(s-h\right){\left(-\mathrm{\Delta }\right)}^{\beta }}{\left(-\mathrm{\Delta }\right)}^{\frac{1}{2}}\left(I-{e}^{-h{\left(-\mathrm{\Delta }\right)}^{\beta }}\right)\left({1}_{r,x}𝐮\otimes 𝐯\right)𝑑h,$${B}_{2}\left(𝐮,𝐯\right)={\left(-\mathrm{\Delta }\right)}^{-\frac{1}{2}}ℙ\nabla \cdot {\left(-\mathrm{\Delta }\right)}^{\frac{1}{2}}{e}^{-s{\left(-\mathrm{\Delta }\right)}^{\beta }}{\int }_{0}^{s}\left({1}_{r,x}𝐮\otimes 𝐯\right)𝑑h,$${B}_{3}\left(𝐮,𝐯\right)={\int }_{0}^{s}{e}^{-\left(s-h\right){\left(-\mathrm{\Delta }\right)}^{\beta }}ℙ\nabla \cdot \left(\left(1-{1}_{r,x}\right)𝐮\otimes 𝐯\right)𝑑h.$

For ${B}_{1}\left(𝐮,𝐯\right)$, we use the boundedness of the Riesz transform and Lemma 2.3 (i) to derive

${\int }_{0}^{{r}^{2\beta }}\parallel {B}_{1}\left(𝐮,𝐯\right){\parallel }_{{L}^{2}}^{2}\frac{dt}{{t}^{\frac{1-\alpha -\beta }{\beta }}}\lesssim {\int }_{0}^{{r}^{2\beta }}\parallel {\int }_{0}^{s}{e}^{-\left(s-h\right){\left(-\mathrm{\Delta }\right)}^{\beta }}{\left(-\mathrm{\Delta }\right)}^{\frac{1}{2}}\left(I-{e}^{-h{\left(-\mathrm{\Delta }\right)}^{\beta }}\right)\left({1}_{r,x}𝐮\otimes 𝐯\right)dh{\parallel }_{{L}^{2}}^{2}\frac{dt}{{t}^{\frac{1-\alpha -\beta }{\beta }}}$$\lesssim {\int }_{0}^{{r}^{2\beta }}\parallel {\left(-\mathrm{\Delta }\right)}^{\frac{1}{2}-\beta }\left(I-{e}^{-t{\left(-\mathrm{\Delta }\right)}^{\beta }}\right)\left({1}_{r,x}𝐮\otimes 𝐯\right){\parallel }_{{L}^{2}}^{2}\frac{dt}{{t}^{\frac{1-\alpha -\beta }{\beta }}}.$

Notice that ${\left(-\mathrm{\Delta }\right)}^{\frac{1}{2}-\beta }\left(I-{e}^{-t{\left(-\mathrm{\Delta }\right)}^{\beta }}\right)$ is bounded on ${L}^{2}$, provided $\frac{1}{2}<\beta <\mathrm{\infty }$, with its operator norm $\lesssim {t}^{1-\frac{1}{2\beta }}$. Thus, using the Cauchy–Schwarz inequality, we have

${\int }_{0}^{{r}^{2\beta }}\parallel {B}_{1}\left(𝐮,𝐯\right){\parallel }_{{L}^{2}}^{2}\frac{dt}{{t}^{\frac{1-\alpha -\beta }{\beta }}}\lesssim {\int }_{0}^{{r}^{2\beta }}{t}^{2-\frac{1}{\beta }}\parallel {1}_{r,x}𝐮\otimes 𝐯{\parallel }_{{L}^{2}}^{2}\frac{dt}{{t}^{\frac{1-\alpha -\beta }{\beta }}}$$\lesssim {\int }_{0}^{{r}^{2\beta }}{t}^{2-\frac{1}{\beta }}{\int }_{|y-x|$\lesssim \left(\underset{t\in \left(0,T\right)}{sup}{t}^{\frac{2\beta -1}{2\beta }}\parallel 𝐮\left(y,t\right){\parallel }_{{L}^{\mathrm{\infty }}}\right)\left(\underset{t\in \left(0,T\right)}{sup}{t}^{\frac{2\beta -1}{2\beta }}\parallel 𝐯\left(y,t\right){\parallel }_{{L}^{\mathrm{\infty }}}\right)$$×\left({\int }_{0}^{{r}^{2\beta }}{\int }_{|y-x|

In view of the definition of ${Y}_{\beta }^{\alpha }$, we conclude

${\int }_{0}^{{r}^{2\beta }}\parallel {B}_{1}\left(𝐮,𝐯\right){\parallel }_{{L}^{2}}^{2}\frac{dt}{{t}^{\frac{1-\alpha -\beta }{\beta }}}\lesssim {r}^{n+2\alpha }\parallel 𝐮{\parallel }_{{\left({Y}_{\beta }^{\alpha }\right)}^{n}}^{2}\parallel 𝐯{\parallel }_{{\left({Y}_{\beta }^{\alpha }\right)}^{n}}^{2}.$(2.10)

For ${B}_{2}\left(𝐮,𝐯\right)$, by the boundedness of the Riesz transform and Lemma 2.3 (ii), we have

${\int }_{0}^{{r}^{2\beta }}\parallel {B}_{2}\left(𝐮,𝐯\right){\parallel }_{{L}^{2}}^{2}\frac{dt}{{t}^{\frac{1-\alpha -\beta }{\beta }}}\lesssim {\int }_{0}^{{r}^{2\beta }}\parallel {\left(-\mathrm{\Delta }\right)}^{\frac{1}{2}}{e}^{-t{\left(-\mathrm{\Delta }\right)}^{\beta }}{\int }_{0}^{t}\left({1}_{r,x}𝐮\otimes 𝐯\right)dh\parallel {}_{{L}^{2}}{}^{2}\frac{dt}{{t}^{\frac{1-\alpha -\beta }{\beta }}}$$\lesssim {\mathrm{D}}_{\alpha ,\beta }\left({1}_{r,x}𝐮\otimes 𝐯\right){\int }_{0}^{{r}^{2\beta }}{\int }_{{ℝ}^{n}}|{1}_{r,x}𝐮\otimes 𝐯\left(x,s\right)|\frac{dxds}{{s}^{\frac{1-\alpha -\beta }{\beta }}}.$

On the one hand, we employ Hölder’s inequality to derive

$\begin{array}{cc}\hfill {\mathrm{D}}_{\alpha ,\beta }\left({1}_{r,x}𝐮\otimes 𝐯\right)& \lesssim \underset{\left(x,r\right)\in {ℝ}_{+}^{n+1}}{sup}{r}^{-2\alpha -n}{\int }_{0}^{{r}^{2\beta }}{\int }_{{ℝ}^{n}}{1}_{r,x}\left(s\right)|𝐮\otimes 𝐯\left(x,s\right)|\frac{dxds}{{s}^{\frac{1-\alpha -\beta }{\beta }}}\hfill \\ & \lesssim {\parallel 𝐮\parallel }_{{\left({Y}_{\beta }^{\alpha }\right)}^{n}}{\parallel 𝐯\parallel }_{{\left({Y}_{\beta }^{\alpha }\right)}^{n}}.\hfill \end{array}$

On the other hand, we similarly have

${\int }_{0}^{{r}^{2\beta }}{\int }_{{ℝ}^{n}}|{1}_{r,x}𝐮\otimes 𝐯\left(x,s\right)|\frac{dxds}{{s}^{\frac{1-\alpha -\beta }{\beta }}}\lesssim {r}^{n+2\alpha }\parallel 𝐮{\parallel }_{{\left({Y}_{\beta }^{\alpha }\right)}^{n}}\parallel 𝐯{\parallel }_{{\left({Y}_{\beta }^{\alpha }\right)}^{n}}.$

Consequently, we conclude

${\int }_{0}^{{r}^{2\beta }}\parallel {B}_{2}\left(𝐮,𝐯\right){\parallel }_{{L}^{2}}^{2}\frac{dt}{{t}^{\frac{1-\alpha -\beta }{\beta }}}\lesssim {r}^{n+2\alpha }\parallel 𝐮{\parallel }_{{\left({Y}_{\beta }^{\alpha }\right)}^{n}}^{2}\parallel 𝐯{\parallel }_{{\left({Y}_{\beta }^{\alpha }\right)}^{n}}^{2}.$(2.11)

For ${B}_{3}\left(𝐮,𝐯\right)$, by the decay property of the kernel of ${e}^{-t{\left(-\mathrm{\Delta }\right)}^{\beta }}ℙ\nabla$ we get that if $|x-y| and $s<{r}^{2\beta }$, then

$|{B}_{3}\left(𝐮,𝐯\right)|\le |{\int }_{0}^{s}{e}^{-\left(s-h\right){\left(-\mathrm{\Delta }\right)}^{\beta }}ℙ\nabla \cdot \left(\left(1-{1}_{r,x}\right)𝐮\otimes 𝐯\right)dh|$$\lesssim {\int }_{0}^{s}{\int }_{|z-x|\ge 10r}\frac{|𝐮\left(h,z\right)||𝐯\left(h,z\right)|}{{\left({\left(s-h\right)}^{\frac{1}{2\beta }}+|z-y|\right)}^{n+1}}𝑑z𝑑h$$\lesssim {\int }_{0}^{{r}^{2\beta }}{\int }_{|z-x|\ge 10r}\frac{|𝐮\left(h,z\right)||𝐯\left(h,z\right)|}{{|x-z|}^{n+1}}𝑑z𝑑h$$\lesssim \sum _{j=3}^{\mathrm{\infty }}{\left({2}^{j}r\right)}^{-n-1}{\int }_{0}^{{r}^{2\beta }}{\int }_{B\left(x,{2}^{j+1}r\right)\setminus B\left(x,{2}^{j}r\right)}|𝐮\left(h,z\right)||𝐯\left(h,z\right)|dzdh$$\lesssim \sum _{j=3}^{\mathrm{\infty }}{\left({2}^{j}r\right)}^{-n-1}{r}^{2-2\alpha -2\beta }{\int }_{0}^{{r}^{2\beta }}\left(\int {}_{B\left(x,{2}^{j+1}r\right)}|𝐮\left(h,z\right)||𝐯\left(h,z\right)|dz\right)h{}^{\frac{\alpha +\beta -1}{\beta }}dh$$\lesssim {r}^{1-2\beta -n-2\alpha }{\int }_{0}^{{r}^{2\beta }}\left({\int }_{B\left(x,{2}^{j+1}r\right)}|𝐮\left(h,z\right)||𝐯\left(h,z\right)|dz\right){h}^{\frac{\alpha +\beta -1}{\beta }}dh.$

Then, by Hölder’s inequality, we get

$|{B}_{3}\left(𝐮,𝐯\right)|\lesssim {r}^{1-2\beta }{\parallel 𝐮\parallel }_{{\left({Y}_{\beta }^{\alpha }\right)}^{n}}{\parallel 𝐯\parallel }_{{\left({Y}_{\beta }^{\alpha }\right)}^{n}}.$

Since $\alpha >1-2\beta$, we have

${\int }_{0}^{{r}^{2\beta }}{\int }_{|y-x|(2.12)

Putting the estimates (2.10), (2.11) and (2.12) together, we reach (2.8). Step 2: ${L}^{\mathrm{\infty }}$-bound. Two situations are handled in the sequel.

If $\frac{t}{2}\le s, then

${\parallel {e}^{-\left(t-s\right){\left(-\mathrm{\Delta }\right)}^{\beta }}ℙ\nabla \cdot \left(𝐮\otimes 𝐯\right)\parallel }_{{L}^{\mathrm{\infty }}}\lesssim {\left(t-s\right)}^{-\frac{1}{2\beta }}{\parallel 𝐮\parallel }_{{L}^{\mathrm{\infty }}}{\parallel 𝐯\parallel }_{{L}^{\mathrm{\infty }}}\lesssim {\left(t-s\right)}^{-\frac{1}{2\beta }}{s}^{\frac{1}{\beta }-2}{\parallel 𝐮\parallel }_{{\left({Y}_{\beta }^{\alpha }\right)}^{n}}{\parallel 𝐯\parallel }_{{\left({Y}_{\beta }^{\alpha }\right)}^{n}},$

and hence, for $\beta >\frac{1}{2}$, we have

$|{\int }_{\frac{t}{2}}^{t}{e}^{-\left(t-s\right){\left(-\mathrm{\Delta }\right)}^{\beta }}ℙ\nabla \cdot \left(𝐮\otimes 𝐯\right)ds|\lesssim \int {}_{\frac{t}{2}}{}^{t}\left(t-s\right){}^{-\frac{1}{2\beta }}s{}^{\frac{1}{\beta }-2}ds\parallel 𝐮\parallel {}_{{\left({Y}_{\beta }^{\alpha }\right)}^{n}}\parallel 𝐯\parallel {}_{{\left({Y}_{\beta }^{\alpha }\right)}^{n}}\lesssim t{}^{\frac{1-2\beta }{2\beta }}\parallel 𝐮\parallel {}_{{\left({Y}_{\beta }^{\alpha }\right)}^{n}}\parallel 𝐯\parallel {}_{{\left({Y}_{\beta }^{\alpha }\right)}^{n}}.$(2.13)

If $0, then $t-s\approx t$, and hence

$|{e}^{-\left(t-s\right){\left(-\mathrm{\Delta }\right)}^{\beta }}ℙ\nabla \cdot \left(𝐮\otimes 𝐯\right)|\lesssim {\int }_{{ℝ}^{n}}\frac{|𝐮\left(y,s\right)||𝐯\left(y,s\right)|}{{\left({\left(t-s\right)}^{\frac{1}{2\beta }}+|x-y|\right)}^{n+1}}𝑑y$$\lesssim {\int }_{{ℝ}^{n}}\frac{|𝐮\left(y,s\right)||𝐯\left(y,s\right)|}{{\left({t}^{\frac{1}{2\beta }}+|x-y|\right)}^{n+1}}𝑑y$$\lesssim {t}^{-\frac{n+1}{2\beta }}{\int }_{|x-y|\le 10{t}^{\frac{1}{2\beta }}}|𝐮\left(y,s\right)||𝐯\left(y,s\right)|dy+{\int }_{|x-y|>10{t}^{\frac{1}{2\beta }}}\frac{|𝐮\left(y,s\right)||𝐯\left(y,s\right)|}{{|x-y|}^{n+1}}dy.$

Using the same calculation as in ${B}_{3}\left(𝐮,𝐯\right)$ with $r={t}^{\frac{1}{2\beta }}$, we obtain

${\int }_{0}^{\frac{t}{2}}{\int }_{|x-y|>10{t}^{\frac{1}{2\beta }}}\frac{|𝐮\left(y,s\right)||𝐯\left(y,s\right)|}{{|x-y|}^{n+1}}dyds\lesssim {t}^{\frac{1-2\beta }{2\beta }}\parallel 𝐮{\parallel }_{{\left({Y}_{\beta }^{\alpha }\right)}^{n}}\parallel 𝐯{\parallel }_{{\left({Y}_{\beta }^{\alpha }\right)}^{n}}.$

Meanwhile, utilizing Hölder’s inequality, we derive

${t}^{-\frac{n+1}{2\beta }}{\int }_{0}^{\frac{t}{2}}{\int }_{|x-y|\le 10{t}^{\frac{1}{2\beta }}}|𝐮\left(y,s\right)||𝐯\left(y,s\right)|dyds\lesssim {t}^{-\frac{2\beta -1}{2\beta }}{t}^{-\frac{n+2\alpha }{2\beta }}{\int }_{0}^{\frac{t}{2}}\left({\int }_{B\left(x,10{t}^{\frac{1}{2\beta }}\right)}|𝐮\left(z,s\right)||𝐯\left(z,s\right)|dz\right){s}^{\frac{\alpha +\beta -1}{\beta }}ds$$\lesssim {t}^{\frac{1-2\beta }{2\beta }}\parallel 𝐮{\parallel }_{{\left({Y}_{\beta }^{\alpha }\right)}^{n}}\parallel 𝐯{\parallel }_{{\left({Y}_{\beta }^{\alpha }\right)}^{n}}.$

Consequently,

$|{\int }_{0}^{\frac{t}{2}}{e}^{-\left(t-s\right){\left(-\mathrm{\Delta }\right)}^{\beta }}ℙ\nabla \cdot \left(𝐮\otimes 𝐯\right)ds|\lesssim t{}^{\frac{1-2\beta }{2\beta }}\parallel 𝐮{\parallel }_{{\left({Y}_{\beta }^{\alpha }\right)}^{n}}\parallel 𝐯{\parallel }_{{\left({Y}_{\beta }^{\alpha }\right)}^{n}}.$(2.14)

Now, putting estimates (2.13) and (2.14) together yields the ${L}^{\mathrm{\infty }}$-bound (2.9).

2.3 Proof of Remark 1.2

The argument is divided into two steps. Step 1. Noting the following Minkowski-inequality-based estimates:

$\underset{t>0}{sup}{t}^{\frac{2\beta -1}{2\beta }}\parallel {e}^{-t{\left(-\mathrm{\Delta }\right)}^{\beta }}f{\parallel }_{{L}^{\mathrm{\infty }}}\lesssim \parallel f{\parallel }_{{L}_{2,n+1-2\beta }},\underset{t>0}{sup}\parallel {e}^{-t{\left(-\mathrm{\Delta }\right)}^{\beta }}f{\parallel }_{{L}_{2,n+1-2\beta }}\lesssim \parallel f{\parallel }_{{L}_{2,n+1-2\beta }},$

we get that if $\alpha >1-2\beta$ and $\left(x,r\right)\in {ℝ}_{+}^{n+1}$, then

${r}^{-\left(2\alpha +n\right)}{\int }_{0}^{{r}^{2\beta }}\left(\int {}_{B\left(x,r\right)}|{e}^{-t{\left(-\mathrm{\Delta }\right)}^{\beta }}f\left(y\right){|}^{2}dy\right)t{}^{-\frac{1-\alpha -\beta }{\beta }}dt\lesssim \int {}_{0}{}^{{r}^{2\beta }}\parallel e{}^{-t{\left(-\mathrm{\Delta }\right)}^{\beta }}f\parallel {}^{2}{}_{{L}_{2,n+1-2\beta }}\frac{{r}^{2\left(1-2\beta -\alpha \right)}dt}{{t}^{\frac{1-\alpha -\beta }{\beta }}}\lesssim \parallel f\parallel {}^{2}{}_{{L}_{2,n+1-2\beta }},$

whence deriving

Step 2. The desired well-posedness may be viewed as an extension of Kato’s ${L}^{p}$-theory, developed in [26, 27, 45, 22, 44, 33, 50], to (1.2). In order to deal with a mild solution of (1.1) initialized in ${\left({L}_{2,n+1-2\beta }\right)}^{n}$, we are required to control the boundedness of the initial data semi-group

${𝐮}_{0}↦{e}^{t{\left(-\mathrm{\Delta }\right)}^{\beta }}{𝐮}_{0}$

and the bilinear operator

$\left(𝐮,𝐯\right)↦B\left(𝐮,𝐯\right)={\int }_{0}^{t}{e}^{-\left(t-s\right){\left(-\mathrm{\Delta }\right)}^{\beta }}ℙ\nabla \cdot \left(𝐮\otimes 𝐯\right)𝑑s,$

acting on a suitable solution space. To see this, let us use the foregoing Minkowski-inequality-based estimates and the following Morrey norm:

$\parallel g{\parallel }_{{L}_{4,2\left(n+1-2\beta \right)}}=\underset{\left(x,r\right)\in {ℝ}_{+}^{n+1}}{sup}\left({r}^{-2\left(n+1-2\beta \right)}{\int }_{B\left(x,r\right)}{|g\left(y\right){|}^{4}dy\right)}^{\frac{1}{4}},$

to derive

${\parallel {e}^{-t{\left(-\mathrm{\Delta }\right)}^{\beta }}f\parallel }_{{L}_{4,2\left(n+1-2\beta \right)}}\lesssim \sqrt{{\parallel {e}^{-t{\left(-\mathrm{\Delta }\right)}^{\beta }}f\parallel }_{{L}_{2,n+1-2\beta }}{\parallel {e}^{-t{\left(-\mathrm{\Delta }\right)}^{\beta }}f\parallel }_{{L}^{\mathrm{\infty }}}}\lesssim {t}^{\frac{2\beta -1}{4\beta }}{\parallel f\parallel }_{{L}_{2,n+1-2\beta }},$

whence defining the solution space ${\left({X}_{\beta }\right)}^{n}$ of all vector-valued functions $𝐮={\left\{{u}_{j}\right\}}_{j=1}^{n}$ with the norm

$\parallel 𝐮{\parallel }_{{\left({X}_{\beta }\right)}^{n}}=\sum _{j=1}^{n}\parallel {u}_{j}{\parallel }_{{X}_{\beta }}=\sum _{j=1}^{n}\left(\underset{t>0}{sup}{t}^{\frac{2\beta -1}{4\beta }}\parallel {u}_{j}\left(\cdot ,t\right){\parallel }_{{L}_{4,2\left(n+1-2\beta \right)}}+\underset{t>0}{sup}{t}^{\frac{2\beta -1}{2\beta }}\parallel {u}_{j}\left(\cdot ,t\right){\parallel }_{{L}^{\mathrm{\infty }}}\right).$

On the one hand, for the initial data ${𝐮}_{0}$ in (1.1), we have

On the other hand, for the corresponding bilinear part, a direct computation as in [33] shows that if $t>s$, then

${\parallel {e}^{-\left(t-s\right){\left(-\mathrm{\Delta }\right)}^{\beta }}ℙ\nabla \cdot \left(𝐮\otimes 𝐯\right)\parallel }_{{\left({L}_{4,2\left(n+1-2\beta \right)}\right)}^{n}}\lesssim {\left({\left(t-s\right)}^{\frac{1}{2\beta }}{s}^{\frac{3\left(2\beta -1\right)}{4\beta }}\right)}^{-1}{s}^{\frac{2\beta -1}{4\beta }}{\parallel 𝐮\parallel }_{{\left({L}_{4,2\left(n+1-2\beta \right)}\right)}^{n}}{s}^{\frac{2\beta -1}{2\beta }}{\parallel 𝐯\parallel }_{{\left({L}^{\mathrm{\infty }}\right)}^{n}}$$\lesssim {\left({\left(t-s\right)}^{\frac{1}{2\beta }}{s}^{\frac{3\left(2\beta -1\right)}{4\beta }}\right)}^{-1}{\parallel 𝐮\parallel }_{{\left({X}_{\beta }\right)}^{n}}{\parallel 𝐯\parallel }_{{\left({X}_{\beta }\right)}^{n}}$

and

${\left(t-s\right)}^{\frac{1}{2\beta }}{\parallel {e}^{-\left(t-s\right){\left(-\mathrm{\Delta }\right)}^{\beta }}ℙ\nabla \cdot \left(𝐮\otimes 𝐯\right)\parallel }_{{\left({L}^{\mathrm{\infty }}\right)}^{n}}$$\lesssim \mathrm{min}\left\{\frac{{s}^{\frac{2\beta -1}{4\beta }}\parallel 𝐮{\parallel }_{{\left({L}_{4,2\left(n+1-2\beta \right)}\right)}^{n}}{s}^{\frac{2\beta -1}{4\beta }}\parallel 𝐯{\parallel }_{{\left({L}_{4,2\left(n+1-2\beta \right)}\right)}^{n}}}{{\left(t-s\right)}^{\frac{1}{2\beta }}{s}^{\frac{2\beta -1}{2\beta }}},\frac{{s}^{\frac{2\beta -1}{2\beta }}{\parallel 𝐮\parallel }_{{\left({L}^{\mathrm{\infty }}\right)}^{n}}{s}^{\frac{2\beta -1}{2\beta }}{\parallel 𝐯\parallel }_{{\left({L}^{\mathrm{\infty }}\right)}^{n}}}{{s}^{\frac{2\beta -1}{\beta }}}\right\}$$\lesssim \mathrm{min}\left\{{\left(t-s\right)}^{-\frac{1}{2\beta }}{s}^{-\frac{2\beta -1}{2\beta }},{s}^{-\frac{2\beta -1}{\beta }}\right\}{\parallel 𝐮\parallel }_{{\left({X}_{\beta }\right)}^{n}}{\parallel 𝐯\parallel }_{{\left({X}_{\beta }\right)}^{n}},$

and hence

$\parallel {\int }_{0}^{t}{e}^{-\left(t-s\right){\left(-\mathrm{\Delta }\right)}^{2\beta }}ℙ\nabla \cdot \left(𝐮\otimes 𝐯\right)ds\parallel {}_{{\left({X}_{\beta }\right)}^{n}}\lesssim \parallel 𝐮\parallel {}_{{\left({X}_{\beta }\right)}^{n}}\parallel 𝐯\parallel {}_{{\left({X}_{\beta }\right)}^{n}}.$

This, along with the standard fixed-point argument, as in [33], completes the proof.

2.4 Proof of Corollary 1.3

In accordance with Theorem 1.1 and the well-posedness of (1.2) arising from ${𝐮}_{0}\in {\left({\stackrel{˙}{B}}_{\mathrm{\infty },\mathrm{\infty }}^{-1<1-2\beta <0}\right)}^{n}$, obtained in [52], we are only required to prove that ${X}_{\beta }^{\alpha }$ can be identified with ${\stackrel{˙}{B}}_{\mathrm{\infty },\mathrm{\infty }}^{1-2\beta }$ for $\frac{1}{2}>\alpha >1-\beta >0$. On the one hand, if $f\in {\stackrel{˙}{B}}_{\mathrm{\infty },\mathrm{\infty }}^{1-2\beta }$, then

${\parallel f\parallel }_{{\stackrel{˙}{B}}_{\mathrm{\infty },\mathrm{\infty }}^{1-2\beta }}\approx \underset{\left(x,t\right)\in {ℝ}_{+}^{n+1}}{sup}{t}^{\frac{2\beta -1}{2\beta }}|{e}^{-t{\left(-\mathrm{\Delta }\right)}^{2\beta }}f\left(x\right)|<\mathrm{\infty },$

and hence

$\parallel f{\parallel }_{{X}_{\beta }^{\alpha }}\approx \underset{\left({x}_{0},r\right)\in {ℝ}_{+}^{n+1}}{sup}\left({r}^{-\left(2\alpha +n\right)}{\int }_{B\left({x}_{0},r\right)}{\int }_{0}^{r}{|{e}^{-t{\left(-\mathrm{\Delta }\right)}^{2\beta }}f\left(x\right){|}^{2}\frac{dtdx}{{t}^{\frac{1-\alpha -\beta }{\beta }}}\right)}^{\frac{1}{2}}$$\lesssim \parallel f{\parallel }_{{\stackrel{˙}{B}}_{\mathrm{\infty },\mathrm{\infty }}^{1-2\beta }}\underset{\left({x}_{0},r\right)\in {ℝ}_{+}^{n+1}}{sup}\left({r}^{-\left(2\alpha +n\right)}{\int }_{B\left({x}_{0},r\right)}{\int }_{0}^{r}{|{t}^{-\frac{2\beta -1}{2\beta }}{|}^{2}\frac{dtdx}{{t}^{\frac{1-\alpha -\beta }{\beta }}}\right)}^{\frac{1}{2}}$$\lesssim {\parallel f\parallel }_{{\stackrel{˙}{B}}_{\mathrm{\infty },\mathrm{\infty }}^{1-2\beta }},$

thanks to $\alpha >1-\beta >0$. On the other hand, noting the following two facts:

• ${\stackrel{˙}{B}}_{\mathrm{\infty },\mathrm{\infty }}^{1-2\beta }$ is the largest space among all the Banach spaces that are translation-invariant and share the scaling (1.3) (cf. [7]),

• ${X}_{\beta }^{\alpha }$ is translation-invariant and satisfies the scaling (1.3),

we achieve

Thus, the desired identification follows.

3 Ill-posedness in ${\left({X}_{\beta }^{\alpha }\right)}^{n}$

This section is devoted to validating Theorem 1.4. The construction in the proof relies heavily on [3, 8].

3.1 Proof of Theorem 1.4 – Construction

To validate Theorem 1.4, we are required to find the initial data and its associated solution. Clearly, it is enough to handle the situation for $n=3$. Referring to [3, 8], for a large integer $l>0$, we choose the following initial data:

${𝐮}_{0}\left(x\right)={l}^{-\theta }\sum _{i=1}^{l}|{k}_{i}{|}^{\beta }\left(v\mathrm{cos}\left({k}_{i}\cdot x\right)+{v}^{\prime }\mathrm{cos}\left({k}_{i}^{\prime }\cdot x\right)\right),$(3.1)

where $\theta \in \left(0,\frac{1}{2}\right)$ and the vectors ${k}_{i}\in {ℤ}^{n}$ are parallel to $\zeta =\left(1,0,0\right)$. For $i=1,2,\mathrm{\dots },l$ and a large integer N dependent on l, let

$|{k}_{i}|={2}^{i-1}N,{k}_{i}^{\prime }={k}_{i}+\eta \in {ℤ}^{n},v=\left(0,0,1\right),{v}^{\prime }=\left(0,1,0\right).$

For the initial data ${𝐮}_{0}$ (first constructed in [8] by an idea in [3]), we have

$\mathrm{div}{𝐮}_{0}=0,$${e}^{-t{\left(-\mathrm{\Delta }\right)}^{\beta }}{𝐮}_{0}={l}^{-\theta }\sum _{i=1}^{l}|{k}_{i}{|}^{\beta }\left(v\mathrm{cos}\left({k}_{i}\cdot x\right){e}^{-{|{k}_{i}|}^{2\beta }t}+{v}^{\prime }\mathrm{cos}\left({k}_{i}^{\prime }\cdot x\right){e}^{-{|{k}_{i}^{\prime }|}^{2\beta }t}\right),$

The following lemma is our main new tool, which asserts that the initial data $\mathrm{div}{𝐮}_{0}$ constructed above is well behaved in our spaces ${X}_{\beta }^{\alpha }$.

Lemma 3.1.

Suppose $\mathrm{1}\mathrm{\le }\beta \mathrm{<}\mathrm{\infty }$. If ${\mathrm{u}}_{\mathrm{0}}$ is given in (3.1), then

Proof.

In view of the definition of ${\left({X}_{\beta ,1}^{\alpha }\right)}^{n}$, we have

$\parallel {𝐮}_{0}{\parallel }_{{\left({X}_{\beta ,1}^{\alpha }\right)}^{n}}^{2}=\underset{0$\lesssim {l}^{-2\theta }\underset{0

So, it remains to show that

Since

we have

${{r}^{-2\alpha }{\int }_{0}^{{r}^{2\beta }}\left(\sum _{i=1}^{l}|{k}_{i}{|}^{\beta }\left({e}^{-{|{k}_{i}|}^{2\beta }t}+{e}^{-{|{k}_{i}^{\prime }|}^{2\beta }t}\right)\right)}^{2}{t}^{-\frac{1-\alpha -\beta }{\beta }}dt\lesssim {r}^{-2\alpha }{\int }_{0}^{{r}^{2\beta }}\left(\sum {}_{i=1}{}^{l}|{k}_{i}{|}^{\beta }{e}^{-{|{k}_{i}|}^{2\beta }t}\right){}^{2}t{}^{-\frac{1-\alpha -\beta }{\beta }}dt\lesssim r{}^{2\beta -2}$

for any $r\in \left(0,1\right)$, provided that $\alpha >1-\beta$, which is sufficient since $\beta \ge 1$.

Furthermore, if $\alpha \ge 0$, then the above estimate for $r\in \left(0,1\right)$ is still valid, and hence it remains to establish a similar estimate for $1\le r<\mathrm{\infty }$. As a matter of fact, since

we utilize $1\ll N\le |{k}_{i}|$ and $\alpha \ge 0$, to obtain

${r}^{-2\alpha }{\int }_{0}^{{r}^{2\beta }}\left(\sum _{i=1}^{l}{|{k}_{i}{|}^{\beta }\left({e}^{-{|{k}_{i}|}^{2\beta }t}+{e}^{-{|{k}_{i}^{\prime }|}^{2\beta }t}\right)\right)}^{2}{t}^{-\frac{1-\alpha -\beta }{\beta }}dt\lesssim {r}^{-2\alpha }{\int }_{0}^{1}+{\int }_{1}^{{r}^{2\beta }}\left(\sum _{i=1}^{l}|{k}_{i}{|}^{\beta }{e}^{-{|{k}_{i}|}^{2\beta }t}\right){}^{2}t{}^{-\frac{1-\alpha -\beta }{\beta }}dt$$\lesssim {r}^{-2\alpha }{\int }_{0}^{2}{t}^{-1}{t}^{-\frac{1-\alpha -\beta }{\beta }}𝑑t+{r}^{-2\alpha }{\int }_{2}^{{r}^{2\beta }}{e}^{-{N}^{2\beta }t}{t}^{-\frac{1-\alpha -\beta }{\beta }}𝑑t$

The proof is completed. ∎

Next, as in [8], we write

$𝐮\left(t\right)={e}^{-t{\left(-\mathrm{\Delta }\right)}^{\beta }}{𝐮}_{0}-{𝐮}_{1}\left(t\right)+𝐲\left(t\right),$${𝐮}_{1}\left(t\right)=B\left({e}^{-t{\left(-\mathrm{\Delta }\right)}^{\beta }}{𝐮}_{0},{e}^{-t{\left(-\mathrm{\Delta }\right)}^{\beta }}{𝐮}_{0}\right),$$𝐲\left(t\right)=-{\int }_{0}^{t}{e}^{-\left(t-\tau \right){\left(-\mathrm{\Delta }\right)}^{\beta }}\left[{G}_{0}\left(\tau \right)+{G}_{1}\left(\tau \right)+{G}_{2}\left(\tau \right)\right]𝑑\tau ,$

where

${G}_{0}=ℙ\left[\left({e}^{-t{\left(-\mathrm{\Delta }\right)}^{\beta }}{𝐮}_{0}\cdot \nabla \right){𝐮}_{1}+\left({𝐮}_{1}\cdot \nabla \right){e}^{-t{\left(-\mathrm{\Delta }\right)}^{\beta }}{𝐮}_{0}+\left({𝐮}_{1}\cdot \nabla \right){𝐮}_{1}\right],$${G}_{1}=ℙ\left[\left({e}^{-t{\left(-\mathrm{\Delta }\right)}^{\beta }}{𝐮}_{0}\cdot \nabla \right)𝐲+\left({𝐮}_{1}\cdot \nabla \right)𝐲+\left(𝐲\cdot \nabla \right){e}^{-t{\left(-\mathrm{\Delta }\right)}^{\beta }}{𝐮}_{0}+\left(𝐲\cdot \nabla \right){𝐮}_{1}\right],$${G}_{2}=ℙ\left[\left(𝐲\cdot \nabla \right)𝐲\right].$

It turns out that $𝐲$ gives no trouble as an error term. So, the main contribution comes from the bilinear term ${𝐮}_{1}$. A straightforward calculation derives

$\left({e}^{-t{\left(-\mathrm{\Delta }\right)}^{\beta }}{𝐮}_{0}\cdot \nabla \right){e}^{-t{\left(-\mathrm{\Delta }\right)}^{\beta }}{𝐮}_{0}=-{l}^{-2\theta }\sum _{i=1}^{l}\sum _{i=1}^{l}|{k}_{i}{|}^{\beta }|{k}_{j}{|}^{\beta }{e}^{-\left({|{k}_{i}|}^{2\alpha }+{|{k}_{j}^{\prime }|}^{2\beta }\right)t}{v}^{\prime }\mathrm{cos}\left({k}_{i}\cdot x\right)\mathrm{sin}\left({k}_{j}^{\prime }\cdot x\right)$$=-\frac{{l}^{-2\theta }}{2}\sum _{i=1}^{l}|{k}_{i}{|}^{2\beta }{e}^{-\left({|{k}_{i}^{\prime }|}^{2\beta }+{|{k}_{i}|}^{2\beta }\right)t}\mathrm{sin}\left(\eta \cdot x\right){v}^{\prime }$$-\frac{{l}^{-2\theta }}{2}\sum _{i\ne j}^{l}{|{k}_{i}|}^{\beta }{|{k}_{j}|}^{\beta }{e}^{-\left({|{k}_{i}|}^{2\beta }+{|{k}_{j}^{\prime }|}^{2\beta }\right)t}\mathrm{sin}\left(\left({k}_{j}^{\prime }-{k}_{i}\right)\cdot x\right){v}^{\prime }$$-\frac{{l}^{-2\theta }}{2}\sum _{i=1}^{l}\sum _{i=1}^{l}|{k}_{i}{|}^{\beta }|{k}_{j}{|}^{\beta }{e}^{-\left({|{k}_{i}|}^{2\beta }+{|{k}_{j}^{\prime }|}^{2\beta }\right)t}\mathrm{sin}\left(\left({k}_{j}^{\prime }+{k}_{i}\right)\cdot x\right){v}^{\prime }$$\equiv {E}_{0}+{E}_{1}+{E}_{2}.$(3.2)

Then ${𝐮}_{1}$ can be further decomposed according to

${𝐮}_{1}={\int }_{0}^{t}{e}^{-\left(t-\tau \right){\left(-\mathrm{\Delta }\right)}^{\beta }}{E}_{0}𝑑\tau +{\int }_{0}^{t}{e}^{-\left(t-\tau \right){\left(-\mathrm{\Delta }\right)}^{\beta }}{E}_{1}𝑑\tau +{\int }_{0}^{t}{e}^{-\left(t-\tau \right){\left(-\mathrm{\Delta }\right)}^{\beta }}{E}_{2}𝑑\tau$$\equiv {𝐮}_{10}+{𝐮}_{11}+{𝐮}_{12}.$(3.3)

This in turn gives

$𝐮\left(x,t\right)={e}^{-t{\left(-\mathrm{\Delta }\right)}^{\beta }}{𝐮}_{0}\left(x\right)-{𝐮}_{10}\left(x,t\right)-{𝐮}_{11}\left(x,t\right)-{𝐮}_{12}\left(t\right)-𝐲\left(x,t\right).$(3.4)

It turns out that only ${𝐮}_{10}$ matters, while other terms can be controlled easily under the ${L}^{\mathrm{\infty }}$-norm. More precisely, we have the following two lemmas.

Lemma 3.2 (${L}^{\mathrm{\infty }}$-estimates from [8]).

Let $\mathrm{1}\mathrm{\le }\beta \mathrm{<}\mathrm{\infty }$ and $\mathrm{0}\mathrm{<}\theta \mathrm{<}\frac{\mathrm{1}}{\mathrm{2}}$. Then

${\parallel {e}^{-t{\left(-\mathrm{\Delta }\right)}^{\beta }}{𝐮}_{0}\left(\cdot \right)\parallel }_{{\left({L}^{\mathrm{\infty }}\right)}^{n}}\lesssim {l}^{-\theta }{t}^{-\frac{1}{2}},$${\parallel {𝐮}_{10}\left(\cdot ,t\right)\parallel }_{{\left({L}^{\mathrm{\infty }}\right)}^{n}}\lesssim {l}^{1-2\theta },$${\parallel {𝐮}_{11}\left(\cdot ,t\right)\parallel }_{{\left({L}^{\mathrm{\infty }}\right)}^{n}}\lesssim {l}^{-2\theta },$${\parallel {𝐮}_{12}\left(\cdot ,t\right)\parallel }_{{\left({L}^{\mathrm{\infty }}\right)}^{n}}\lesssim {l}^{-2\theta },$${\parallel 𝐲\left(\cdot ,t\right)\parallel }_{{\left({L}^{\mathrm{\infty }}\right)}^{n}}\lesssim {l}^{1-3\theta }{t}^{\frac{1}{2}-\frac{1}{2\beta }}+{l}^{2-4\theta }{t}^{1-\frac{1}{2\beta }}$

for all $t\mathrm{\in }\mathrm{\left(}\mathrm{0}\mathrm{,}T\mathrm{\right]}$ when T is sufficiently small and l is sufficiently large. Actually, one can choose

Lemma 3.3.

Let ${\mathrm{u}}_{\mathrm{10}}$ be defined as in (3.3). Then

(3.5)

Furthermore, the solution $\mathrm{u}$ given by (3.4) is relatively large even in the resolution space:

${\parallel 𝐮\parallel }_{{\left({Y}_{\beta }^{\alpha }\right)}^{n}}\gtrsim {\parallel 𝐮\parallel }_{{\left({Y}_{\beta ,1}^{\alpha }\right)}^{n}}\gtrsim {l}^{-\frac{\theta }{2}}.$(3.6)

Proof.

From (3.2)–(3.3) and a straightforward calculation, it follows that

${e}^{-{t}^{\prime }{\left(-\mathrm{\Delta }\right)}^{\beta }}{𝐮}_{10}=-\frac{{l}^{-2\theta }{e}^{-{t}^{\prime }{|\eta |}^{2\beta }}}{2}{\int }_{0}^{t}\sum _{i=1}^{l}|{k}_{i}{|}^{2\beta }{e}^{-\left({|{k}_{i}^{\prime }|}^{2\beta }+{|{k}_{i}|}^{2\beta }\right)\tau }{e}^{-{|\eta |}^{2\beta }\left(t-\tau \right)}\mathrm{sin}\left(\eta \cdot x\right){v}^{\prime }d\tau$$=-\frac{{l}^{-2\theta }{e}^{-{t}^{\prime }{|\eta |}^{2\beta }}}{2}\mathrm{sin}\left(\eta \cdot x\right){v}^{\prime }\sum _{i=1}^{l}|{k}_{i}{|}^{2\beta }{e}^{-t}\frac{1-{e}^{-\left({|{k}_{i}^{\prime }|}^{2\beta }+{|{k}_{i}|}^{2\beta }-1\right)t}}{{|{k}_{i}^{\prime }|}^{2\beta }+{|{k}_{i}|}^{2\beta }-1}$$\approx -{l}^{-2\theta }{e}^{-{t}^{\prime }{|\eta |}^{2\beta }}\mathrm{sin}\left(\eta \cdot x\right){v}^{\prime }\sum _{i=1}^{l}{e}^{-t}\left(1-{e}^{-{|{k}_{i}|}^{2\beta }t}\right)$

Consequently,

$\parallel {𝐮}_{10}\left(\cdot ,t\right){\parallel }_{{\left({X}_{\beta }^{\alpha }\right)}^{n}}^{2}\gtrsim \underset{0$\gtrsim {l}^{2-4\theta }\underset{0$\gtrsim {l}^{2-4\theta }\underset{0

Next we estimate ${𝐮}_{10}$ in ${\left({Y}_{\beta }^{\alpha }\right)}^{n}$. In a similar calculation done as above, we have

whence, in view of (3.4) and Lemma 3.2, getting

${\parallel 𝐮\parallel }_{{\left({Y}_{\beta }^{\alpha }\right)}^{n}}\gtrsim {\parallel 𝐮\parallel }_{{\left({Y}_{\beta ,1}^{\alpha }\right)}^{n}}\gtrsim \underset{\left(x,t\right)\in {ℝ}^{n}×\left(0,1\right)}{sup}{t}^{\frac{2\beta -1}{2\beta }}|u\left(x,t\right)|$$\gtrsim \underset{\left(x,t\right)\in {ℝ}^{n}×\left[{N}^{-2\beta },T\right]}{sup}{t}^{\frac{2\beta -1}{2\beta }}\left(|{𝐮}_{10}\left(x,t\right)|-|{e}^{-t{\left(-\mathrm{\Delta }\right)}^{\beta }}{𝐮}_{0}\left(x\right)|-|{𝐮}_{11}\left(x,t\right)|-|{𝐮}_{12}\left(x,t\right)|-|𝐲\left(x,t\right)|\right)$$\gtrsim \underset{\left(x,t\right)\in {ℝ}^{n}×\left[{N}^{-2\beta },T\right]}{sup}{t}^{\frac{2\beta -1}{2\beta }}\left({l}^{1-2\theta }-{l}^{-\theta }{t}^{-\frac{1}{2}}-{l}^{-2\theta }-{l}^{1-3\theta }{t}^{\frac{1}{2}-\frac{1}{2\beta }}+{l}^{2-4\theta }{t}^{1-\frac{1}{2\beta }}\right)$$\gtrsim {l}^{1-2\theta }{T}^{1-\frac{1}{2\beta }}-{l}^{-\theta }{T}^{\frac{1}{2}-\frac{1}{2\beta }}-{l}^{-2\theta }{T}^{1-\frac{1}{2\beta }}-{l}^{1-3\theta }{T}^{\frac{3}{2}-\frac{1}{\beta }}-{l}^{2-4\theta }{T}^{2-\frac{1}{\beta }}.$

Recall that $T={l}^{-\gamma }$ is as in Lemma 3.2 and $0<\theta <\frac{1}{2}$. Then

${\parallel 𝐮\parallel }_{{\left({Y}_{\beta }^{\alpha }\right)}^{n}}\gtrsim {l}^{1-2\theta }{l}^{-\gamma \left(1-\frac{1}{2\beta }\right)}-{l}^{-\theta }{l}^{-\gamma \left(\frac{1}{2}-\frac{1}{2\beta }\right)}-{l}^{-2\theta }{l}^{-\gamma \left(1-\frac{1}{2\beta }\right)}-{l}^{1-3\theta }{l}^{-\gamma \left(\frac{3}{2}-\frac{1}{\beta }\right)}-{l}^{2-4\theta }{l}^{-\gamma \left(2-\frac{1}{\beta }\right)}.$

If

$\gamma =\frac{1-\frac{3\theta }{2}}{1-\frac{1}{2\beta }}>\frac{1-2\theta }{1-\frac{1}{2\beta }}>0,$

then

${\parallel 𝐮\parallel }_{{\left({Y}_{\beta }^{\alpha }\right)}^{n}}\gtrsim {l}^{-\frac{\theta }{2}}-{l}^{-\theta }-{l}^{-2\theta }-{l}^{-1}{l}^{\frac{\gamma }{2}}-{l}^{-\theta }.$

Since $\beta >1$, we have $\gamma <2-3\theta$, whence getting

${\parallel 𝐮\parallel }_{{\left({Y}_{\beta }^{\alpha }\right)}^{n}}\gtrsim {l}^{-\frac{\theta }{2}}-{l}^{-\theta }-{l}^{-2\theta }-{l}^{-\frac{3\theta }{2}}-{l}^{-\theta }\gtrsim {l}^{-\frac{\theta }{2}},$

provided that l is sufficiently large. ∎

3.2 Proof of Theorem 1.4 – Conclusion

The desired norm inflation part of Theorem 1.4 follows from Lemma 3.2 and (3.5) by a similar argument as that used in [8, Section 4.4]. It is only needed to disprove the differentiability of the associated solution map. In view of Lemma 3.1, we conclude that there exists a sequence ${\left\{{𝐮}_{0l}\right\}}_{l}$ with solution ${\left\{{𝐮}_{l}=\mathcal{𝒯}\left({𝐮}_{0l}\right)\right\}}_{l}$ such that

However, using (3.6), we have

$\frac{{\parallel \mathcal{𝒯}\left({𝐮}_{0l}\right)\parallel }_{{\left({Y}_{\beta ,1}^{\alpha }\right)}^{n}}}{{\parallel {𝐮}_{0l}\parallel }_{{\left({X}_{\beta ,1}^{\alpha }\right)}^{n}}}=\frac{{\parallel {𝐮}_{l}\parallel }_{{\left({Y}_{\beta ,1}^{\alpha }\right)}^{n}}}{{\parallel {𝐮}_{0l}\parallel }_{{\left({X}_{\beta ,1}^{\alpha }\right)}^{n}}}\gtrsim {l}^{\frac{\theta }{2}}$

for $0<\theta <\frac{1}{2}$ and l sufficiently large. Moreover, if $\alpha \ge 0$, then, by Lemma 3.1, ${\parallel {𝐮}_{0l}\parallel }_{{\left({X}_{\beta }^{\alpha }\right)}^{n}}\lesssim {l}^{-\theta }.$ Similarly, by applying (3.6), we obtain

$\frac{{\parallel \mathcal{𝒯}\left({𝐮}_{0l}\right)\parallel }_{{\left({Y}_{\beta }^{\alpha }\right)}^{n}}}{{\parallel {𝐮}_{0l}\parallel }_{{\left({X}_{\beta }^{\alpha }\right)}^{n}}}\approx \frac{{\parallel {𝐮}_{l}\parallel }_{{\left({Y}_{\beta }^{\alpha }\right)}^{n}}}{{\parallel {𝐮}_{0l}\parallel }_{{\left({X}_{\beta }^{\alpha }\right)}^{n}}}\gtrsim {l}^{\frac{\theta }{2}}$

for $0<\theta <\frac{1}{2}$ and l sufficiently large. Thus, we finish the proof by letting $l\to \mathrm{\infty }$.

4 Application to ${\left({\mathcal{ℒ}}_{2,n+2\alpha }^{1-\alpha -2\beta }\right)}^{n}$

In this section, we demonstrate Corollary 1.5.

4.1 Characterization of CS functions

Given $\alpha \in \left(-1,1\right)$. According to [49, Lemma 2.1] and [42, Theorem 2.5], each $f\in {\mathcal{ℒ}}_{2,n+2\alpha }$ has an equivalent norm:

$\parallel f{\parallel }_{{\mathcal{ℒ}}_{2,n+2\alpha }}\approx \underset{\left(x,r\right)\in {ℝ}_{+}^{n+1}}{sup}\left({r}^{-\left(2\alpha +n\right)}{\int }_{B\left(x,r\right)}{\int }_{0}^{r}{|{\varphi }_{t}*f\left(y\right){|}^{2}\frac{dtdy}{t}\right)}^{\frac{1}{2}},$

where ϕ is a radial function on ${ℝ}^{n}$ such that

$\varphi \in {L}^{1},{\varphi }_{t}\left(x\right)={t}^{-\frac{n}{2}}\varphi \left(\frac{x}{t}\right),$${\int }_{{ℝ}^{n}}\varphi \left(x\right)dx=0,0<{\int }_{0}^{\mathrm{\infty }}|\stackrel{^}{\varphi }\left(t\xi \right){|}^{2}\frac{dt}{t}<\mathrm{\infty }.$

Since

${\psi }_{t}*\left[{\left(\sqrt{-\mathrm{\Delta }}\right)}^{s}f\right]\left(x\right)={\left(\sqrt{-\mathrm{\Delta }}\right)}^{s}{\psi }_{t}*f\left(x\right)={t}^{-s}{\left[{\left(\sqrt{-\mathrm{\Delta }}\right)}^{s}\psi \right]}_{t}*f\left(x\right),$

we discover an equivalent norm for ${\mathcal{ℒ}}_{2,n+2\alpha }^{s}$:

${\parallel f\parallel }_{{\mathcal{ℒ}}_{2,n+2\alpha }^{s}}\approx \underset{\left(x,r\right)\in {ℝ}_{+}^{n+1}}{sup}{\left({r}^{-\left(2\alpha +n\right)}{\int }_{B\left(x,r\right)}{\int }_{0}^{r}{|{\left[{\left(\sqrt{-\mathrm{\Delta }}\right)}^{s}\psi \right]}_{t}*f\left(y\right)|}^{2}\frac{dtdy}{{t}^{1+2s}}\right)}^{\frac{1}{2}}$$\approx \underset{\left(x,r\right)\in {ℝ}_{+}^{n+1}}{sup}\left({r}^{-\left(2\alpha +n\right)}{\int }_{B\left(x,r\right)}{\int }_{0}^{r}{|{\varphi }_{t}*f\left(y\right){|}^{2}\frac{dtdy}{{t}^{1+2s}}\right)}^{\frac{1}{2}},$

provided that ψ satisfies the above conditions on ϕ, where ${\left(\sqrt{-\mathrm{\Delta }}\right)}^{s}\psi =\varphi$.

Now, set ϕ be the inverse Fourier transform of $t|\xi |{e}^{-{\left(t|\xi |\right)}^{2\beta }}$ and $-\mathrm{\infty }. In view of the above analysis, we have a semi-group characterization for each Campanato–Sobolev (CS) function $f\in {\mathcal{ℒ}}_{2,n+2\alpha }^{s}$:

$\parallel f{\parallel }_{{\mathcal{ℒ}}_{2,n+2\alpha }^{s}}\approx \underset{\left(x,r\right)\in {ℝ}_{+}^{n+1}}{sup}\left({r}^{-\left(2\alpha +n\right)}{\int }_{B\left(x,r\right)}{\int }_{0}^{r}{|t\nabla {e}^{{t}^{2\beta }{\left(-\mathrm{\Delta }\right)}^{\beta }}f\left(y\right){|}^{2}\frac{dtdx}{{t}^{1+2s}}\right)}^{\frac{1}{2}}$$\approx \underset{\left(x,r\right)\in {ℝ}_{+}^{n+1}}{sup}\left({r}^{-\left(2\alpha +n\right)}{\int }_{B\left(x,r\right)}{\int }_{0}^{{r}^{2\beta }}{|\nabla {e}^{-t{\left(-\mathrm{\Delta }\right)}^{\beta }}f\left(y\right){|}^{2}{t}^{\frac{1-s-\beta }{\beta }}dtdy\right)}^{\frac{1}{2}},$

where $\nabla$ stands for the spatial gradient.

4.2 Proof of Corollary 1.5

The preceding characterization leads to introducing the space ${\left({\mathcal{ℒ}}_{2,n+2\alpha }^{s}\right)}^{-1}$ of all functions $f\in {L}_{\mathrm{loc}}^{2}$ on ${ℝ}^{n}$ with the norm

$\parallel f{\parallel }_{{\left({\mathcal{ℒ}}_{2,n+2\alpha }^{s}\right)}^{-1}}=\underset{\left(x,r\right)\in {ℝ}_{+}^{n+1}}{sup}\left({r}^{-\left(2\alpha +n\right)}{\int }_{B\left(x,r\right)}{\int }_{0}^{{r}^{2\beta }}{|{e}^{-t{\left(-\mathrm{\Delta }\right)}^{\beta }}f\left(y\right){|}^{2}{t}^{\frac{1-s-\beta }{\beta }}dtdy\right)}^{\frac{1}{2}}.$

It is not hard to check the following implication:

$\left(\alpha ,\beta ,s\right)\in \left(-1,1\right)×\left[\frac{1}{2},\mathrm{\infty }\right)×\left(-\mathrm{\infty },1\right)⟹{X}_{\beta }^{\alpha }={\left({\mathcal{ℒ}}_{2,n+2\alpha }^{2-\alpha -2\beta }\right)}^{-1}.$(4.1)

Therefore, the assertions in Corollary 1.5 follow immediately from (4.1), Theorems 1.1 and 1.4, and the following lemma.

Lemma 4.1.

Suppose $\mathrm{\left(}\alpha \mathrm{,}\beta \mathrm{,}s\mathrm{\right)}\mathrm{\in }\mathrm{\left(}\mathrm{-}\mathrm{1}\mathrm{,}\mathrm{1}\mathrm{\right)}\mathrm{×}\mathrm{\left[}\frac{\mathrm{1}}{\mathrm{2}}\mathrm{,}\frac{\mathrm{3}}{\mathrm{2}}\mathrm{\right)}\mathrm{×}\mathrm{\left(}\mathrm{-}\mathrm{\infty }\mathrm{,}\mathrm{1}\mathrm{\right)}$ and $j\mathrm{=}\mathrm{1}\mathrm{,}\mathrm{2}\mathrm{,}\mathrm{\dots }\mathrm{,}n$. If ${R}_{j}\mathrm{=}\frac{{\mathrm{\partial }}_{{x}_{j}}}{\sqrt{\mathrm{-}\mathrm{\Delta }}}$ is the j-th Riesz transform, then

and hence

${\mathcal{ℒ}}_{2,n+2\alpha }^{s-1}={\left({\mathcal{ℒ}}_{2,n+2\alpha }^{s}\right)}^{-1}=\nabla \cdot {\left({\mathcal{ℒ}}_{2,n+2\alpha }^{s}\right)}^{n}.$

Proof.

Assume $f\in {\mathcal{ℒ}}_{2,n+2\alpha }^{s}$. Since

$\parallel {R}_{j}f{\parallel }_{{\mathcal{ℒ}}_{2,n+2\alpha }^{s}}^{2}=\underset{\left(x,r\right)\in {ℝ}_{+}^{n+1}}{sup}{r}^{-\left(2\alpha +n\right)}{\int }_{B\left(x,r\right)}{\int }_{0}^{{r}^{2\beta }}|\nabla {e}^{-t{\left(-\mathrm{\Delta }\right)}^{\beta }}{R}_{j}f\left(y\right){|}^{2}{t}^{\frac{1-s-\beta }{\beta }}dtdy,$

we split ${R}_{j}f$ into two pieces via the point-mass function δ:

${R}_{j}f={\phi }_{r}*\left({R}_{j}f\right)+\left(\delta -{\phi }_{r}\right)*\left({R}_{j}f\right),$

where

$\phi \in {C}_{0}^{\mathrm{\infty }}\left({ℝ}^{n}\right),\mathrm{supp}\phi \subset B\left(0,1\right),{\int }_{{ℝ}^{n}}\phi \left(x\right)𝑑x=1,{\phi }_{r}\left(x\right)={r}^{-n}\phi \left(\frac{x}{r}\right).$

On the one hand, using the fact that the predual of ${\stackrel{˙}{B}}_{\mathrm{\infty },\mathrm{\infty }}^{s+\alpha -1}$ is the homogeneous Besov space ${\stackrel{˙}{B}}_{1,1}^{1-s-\alpha }$, we estimate

${\int }_{B\left(x,r\right)}{\int }_{0}^{{r}^{2\beta }}|{\phi }_{r}*\nabla {e}^{-t{\left(-\mathrm{\Delta }\right)}^{\beta }}{R}_{j}f\left(y\right){|}^{2}{t}^{\frac{1-s-\beta }{\beta }}dtdy\lesssim {r}^{n-2s+2}\parallel {\phi }_{r}*\nabla {e}^{-t{\left(-\mathrm{\Delta }\right)}^{\beta }}{R}_{j}f\left(y\right){\parallel }_{{L}^{\mathrm{\infty }}\left({ℝ}_{+}^{n+1}\right)}^{2}$$\lesssim {r}^{n+2-2s}{\parallel {\phi }_{r}\parallel }_{{\stackrel{˙}{B}}_{1,1}^{1-s-\alpha }}^{2}{\parallel \nabla {e}^{-{\left(t\sqrt{-\mathrm{\Delta }}\right)}^{2\beta }}{R}_{j}f\parallel }_{{\stackrel{˙}{B}}_{\mathrm{\infty },\mathrm{\infty }}^{s+\alpha -1}}^{2}$$\lesssim {r}^{n+2\alpha }{\parallel \nabla f\parallel }_{{\stackrel{˙}{B}}_{\mathrm{\infty },\mathrm{\infty }}^{s+\alpha -1}}^{2}$$\lesssim {r}^{n+2\alpha }{\parallel \nabla f\parallel }_{{\left({\mathcal{ℒ}}_{2,n+2\alpha }^{s}\right)}^{-1}}^{2}$$\lesssim {r}^{n+2\alpha }{\parallel f\parallel }_{{\mathcal{ℒ}}_{2,n+2\alpha }^{s}}^{2}.$

On the other hand, noticing that

$\left(\delta -{\phi }_{r}\right)*{R}_{j}\nabla {e}^{-{\left(t\sqrt{-\mathrm{\Delta }}\right)}^{\beta }}f\left(x\right)=\left(\delta -{\phi }_{r}\right)*{R}_{j}{e}^{-\frac{1}{2}{\left(t\sqrt{-\mathrm{\Delta }}\right)}^{\beta }}\nabla {e}^{-\frac{1}{2}{\left(t\sqrt{-\mathrm{\Delta }}\right)}^{\beta }}f\left(x\right)$

and that $\left(\delta -{\phi }_{r}\right)*{R}_{j}{e}^{-\frac{1}{2}{\left(t\sqrt{-\mathrm{\Delta }}\right)}^{2\beta }}$ is a convolution operator with its kernel ${\stackrel{~}{K}}_{t}\left(x\right)$ satisfying

$\underset{t>0}{sup}{\int }_{{ℝ}^{n}}|{\stackrel{~}{K}}_{t}\left(x\right)|dx\lesssim 1,$

we get, by the argument used in the proof of [47, Lemma 3.1] and Hölder’s inequality,

${r}^{-\left(2\alpha +n\right)}{\int }_{B\left({x}_{0},r\right)}{\int }_{0}^{{r}^{2\beta }}{|\left(\delta -{\phi }_{r}\right)*\nabla {e}^{-t{\left(-\mathrm{\Delta }\right)}^{\beta }}{R}_{j}f\left(x\right)|}^{2}{t}^{\frac{1-s-\beta }{\beta }}𝑑t𝑑x$$\lesssim {r}^{-\left(2\alpha +n\right)}{\int }_{B\left({x}_{0},r\right)}{\int }_{0}^{r}|{\int }_{{ℝ}^{n}}{\stackrel{~}{K}}_{t}\left(x-y\right)\left({\nabla }_{y}{e}^{-\frac{1}{2}{\left(t\sqrt{-\mathrm{\Delta }}\right)}^{2\beta }}f\left(y\right)\right)dy|{}^{2}t{}^{1-2s}dtdx$$\lesssim \underset{t>0}{sup}\left(\int {}_{{ℝ}^{n}}|{\stackrel{~}{K}}_{t}\left(x\right)|dx\right){}^{2}\parallel f\parallel {}^{2}{}_{{\mathcal{ℒ}}_{2,n+2\alpha }^{s}}$$\lesssim {\parallel f\parallel }_{{\mathcal{ℒ}}_{2,n+2\alpha }^{s}}^{2}.$

The above two-fold treatment yields

${\parallel {R}_{j}f\parallel }_{{\mathcal{ℒ}}_{2,n+2\alpha }^{s}}\lesssim {\parallel f\parallel }_{{\mathcal{ℒ}}_{2,n+2\alpha }^{s}}.$

To check the identification between those three spaces, we consider two inclusions.

On the one hand, if $f\in {\mathcal{ℒ}}_{2,n+2\alpha }^{s-1}$, then ${\left(\sqrt{-\mathrm{\Delta }}\right)}^{-1}f\in {\mathcal{ℒ}}_{2,n+2\alpha }^{s}$, by definition. An application of the estimate for the Riesz transform gives

$\frac{{\partial }_{{x}_{j}}}{-\mathrm{\Delta }}f=\left(\frac{{\partial }_{{x}_{j}}}{\sqrt{-\mathrm{\Delta }}}\right){\left(\sqrt{-\mathrm{\Delta }}\right)}^{-1}f\in {\mathcal{ℒ}}_{2,n+2\alpha }^{s},$

and consequently $f\in \nabla \cdot {\left({\mathcal{ℒ}}_{2,n+2\alpha }^{s}\right)}^{n}$. This in turn produces

$\left({f}_{1},\mathrm{\dots },{f}_{n}\right)\in {\left({\mathcal{ℒ}}_{2,n+2\alpha }^{s}\right)}^{n}\mathit{ }\text{such that}\mathit{ }f=\sum _{j=1}^{n}{\partial }_{{x}_{j}}{f}_{j}.$

An application of the triangle inequality implies $f\in {\left({\mathcal{ℒ}}_{2,n+2\alpha }^{s}\right)}^{-1}$.

On the other hand, if $f\in {\left({\mathcal{ℒ}}_{2,n+2\alpha }^{s}\right)}^{-1}$, then choosing

one has ${f}_{j,k}\in {\left({\mathcal{ℒ}}_{2,n+2\alpha }^{s}\right)}^{-1}$ due to the above-proved Riesz transform estimate. This further derives

${g}_{j}={\partial }_{{x}_{j}}{\left(-\mathrm{\Delta }\right)}^{-1}f\in {\mathcal{ℒ}}_{2,n+2\alpha }^{s}.$

So, there exist ${f}_{j}\in {\mathcal{ℒ}}_{2,n+2\alpha }^{s}$ for $j=1,\mathrm{\dots },n$ such that $f={\sum }_{j=1}^{n}{\partial }_{{x}_{j}}{f}_{j}$, and then

$\parallel f{\parallel }_{{\mathcal{ℒ}}_{2,n+2\alpha }^{s-1}}\le \sum _{j=1}^{n}\parallel {\partial }_{{x}_{j}}{f}_{j}{\parallel }_{{\mathcal{ℒ}}_{2,n+2\alpha }^{s-1}}\approx \sum _{j=1}^{n}\parallel {R}_{j}{f}_{j}{\parallel }_{{\mathcal{ℒ}}_{2,n+2\alpha }^{s}}\lesssim \sum _{j=1}^{n}\parallel {f}_{j}{\parallel }_{{\mathcal{ℒ}}_{2,n+2\alpha }^{s}},$

as desired. ∎

References

• [1]

S. Abe and S. Thurner, Anomalous diffusion in view of Einsteins 1905 theory of Brownian motion, Phys. A 365 (2005), 403–407.  Google Scholar

• [2]

P. Auscher and D. Frey, A new proof for Koch and Tataru’s result on the well-posedness of Navier–Stokes equations in ${\mathrm{BMO}}^{-1}$, preprint (2013), https://arxiv.org/abs/1310.3783.

• [3]

J. Bourgain and N. Pavlović, Ill-posedness of the Navier–Stokes equations in a critical space in 3D, J. Funct. Anal. 255 (2008), 2233–2247.

• [4]

L. A. Caffarelli and A. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Ann. of Math. (2) 171 (2010), 1903–1930.

• [5]

S. Campanato, Proprietá di hölderianitá di alcune classi di funzioni, Ann. Sc. Norm. Super. Pisa, Sci. Fis. Mat. III. Ser. 17 (1963), 175–188.  Google Scholar

• [6]

S. Campanato, Proprietá di una famiglia dispazi funzionali, Ann. Sc. Norm. Super. Pisa, Sci. Fis. Mat. III. Ser. 18 (1964), 137–160.  Google Scholar

• [7]

M. Cannone, Harmonic analysis tools for solving the incompressible Navier–Stokes equations, Handbook of Mathematical Fluid Dynamics. Vol. III, North-Holland, Amsterdam (2004), 161–244.  Google Scholar

• [8]

A. Cheskidov and M. Dai, Norm inflation for generalized Navier–Stokes equations, Indiana Univ. Math. J. 63 (2014), 869–884.

• [9]

A. Cheskidov and M. Dai, Norm inflation for generalized magneto-hydrodynamic system, Nonlinearity 28 (2015), 129–142.

• [10]

A. Cheskidov and R. Shvydkoy, Ill-posedness of basic equations of fluid dynamics in Besov spaces, Proc. Amer. Math. Soc. 138 (2010), 1059–1067.

• [11]

A. Cheskidov and R. Shvydkoy, Ill-posedness for subcritical hyper dissipative Navier–Stokes equations in the largest critical spaces, J. Math. Phys. 53 (2012), Article ID 115620.  Google Scholar

• [12]

P. Constantin, Euler equations, Navier–Stokes equations and turbulence, Mathematical Foundation of Turbulent Viscous Flows, Lecture Notes in Math. 1871, Springer, New York (2006), 1–43.  Google Scholar

• [13]

A. Cordoba and D. Cordoba, A maximum principle applied to quasi-geostrophic equations, Commun. Math. Phys. 249 (2004), 511–528.

• [14]

M. Dai, J. Qing and M. E. Schonbek, Norm inflation for incompressible magneto-hydrodynamic system in ${\stackrel{˙}{B}}_{\mathrm{\infty }}^{-1,\mathrm{\infty }}$, Adv. Differential Equations 16 (2011), 725–746.  Google Scholar

• [15]

C. Deng and X. Yao, Ill-posedness of the incompressible Navier–Stokes equations in ${\stackrel{˙}{F}}_{\mathrm{\infty }}^{-1,q}\left({ℝ}^{3}\right)$, preprint (2013), https://arxiv.org/abs/1302.7084v1.

• [16]

C. Deng and X. Yao, Well-posedness and ill-posedness for the generalized Navier–Stokes equations in ${\stackrel{˙}{F}}_{\frac{3}{\alpha -1}}^{-\alpha ,r}$, Discrete Contin. Dyn. Syst. 34 (2014), 437–459.  Google Scholar

• [17]

S. Dubois, What is a solution to the Navier–Stokes equations?, C.R. Acad. Sci. Paris Ser. I 335 (2002), 27–32.

• [18]

G. Duvaut and J.-L. Lions, Les inéquations en mécanique et en physique, Trav. Rech. Math. 21, Dunod, Paris, 1972.  Google Scholar

• [19]

M. Essén, S. Janson, L. Peng and J. Xiao, Q spaces of several real variables, Indiana Univ. Math. J. 49 (2000), 575–615.  Google Scholar

• [20]

H. Fujita and T. Kato, On the nonstationary Navier–Stokes system, Rend. Semin. Mat. Univ. Padova 32 (1962), 243–260.  Google Scholar

• [21]

J. Garnett, P. W. Jones, T. M. Le and L. A. Vese, Modeling oscillatory components with the homogeneous spaces $B\stackrel{˙}{M}{O}^{-\alpha }$ and ${\stackrel{˙}{W}}^{-\alpha ,p}$, Pure Appl. Math. Q. 7 (2011), 275–318.  Google Scholar

• [22]

Y. Giga and T. Miyakawa, Solutions in ${L}^{r}$ of the Navier–Stokes initial value problem, Arch. Ration. Mech. Anal. 89 (1985), 267–281.  Google Scholar

• [23]

M. Jara, Limit theorems for additive functionals of a Markov chain, Ann. Appl. Probab. 19 (2009), no. 6, 2270–2300.

• [24]

M. Jara, Nonequilibrium scaling limit for a tagged particle in the simple exclusion process with long jumps, Comm. Pure Appl. Math. 62 (2009), 198–214.

• [25]

R. Jiang, J. Xiao and D. Yang, Towards spaces of harmonic functions with traces in square Campanato space and its scaling invariant, Anal. Appl. (Singap.) 14 (2016), 10.1142/S0219530515500190.  Google Scholar

• [26]

T. Kato, Strong ${L}^{p}$-solutions of the Navier–Stokes equation in ${ℝ}^{m}$, with applications to weak solutions, Math. Z. 187 (1984), 471–480.  Google Scholar

• [27]

T. Kato, Strong solutions of the Navier–Stokes equations in Morrey spaces, Bol. Soc. Brasil. Math. 22 (1992), 127–155.

• [28]

N. H. Katz and N. Pavlović, A cheap Caffarelli–Kohn–Nirenberg inequality for the Navier–Stokes equation with hyper-dissipation, Geom. Funct. Anal. 12 (2002), 355–379.

• [29]

A. Kiselev, F. Nazarov and A. Volberg, Global well-posedness for the critical 2D dissipative quasi-geostrophic equation, Invent. Math. 167 (2007), 445–453.

• [30]

H. Koch and D. Tataru, Well-posedness for the Navier–Stokes equations, Adv. Math. 157 (2001), 22–35.

• [31]

Z. Lei and F. Lin, Global mild solutions of Navier–Stokes equations, Comm. Pure Appl. Math. 64 (2011), 1297–1304.

• [32]

P. G. Lemarié-Rieusset, Recent Developments in the Navier–Stokes Problem, Chapman & Hall/CRC Res. Notes Math. 431, Chapman & Hall/CRC, Boca Raton, 2002.  Google Scholar

• [33]

P. G. Lemariè-Rieusset, The Navier–Stokes equations in the critical Morrey–Campanato space, Rev. Mat. Iberoam. 23 (2002), 897–930.  Google Scholar

• [34]

P. Li, J. Xiao and Q. Yang, Global mild solutions of modified Naiver–Stokes equations with small initial data in critical Besov-Q spaces, Electron. J. Differential Equations 2014 (2014), no. 185, 1–37.  Google Scholar

• [35]

P. Li and Z. Zhai, Well-posedness and regularity of generalized Navier–Stokes equations in some critical Q-spaces, J. Funct. Anal. 259 (2010), 2457–2519.

• [36]

P.-L. Lions and N. Masmoudi, Uniqueness of mild solutions of the Navier–Stokes system in ${L}^{N}$, Comm. Partial Differential Equations 26 (2001), 2211–2226.  Google Scholar

• [37]

A. Mellet, S. Mischler and C. Mouhot, Fractional diffusion limit for collisional kinetic equations, Arch. Ration. Mech. Anal. 199 (2011), 493–525.

• [38]

J. R. Mercado-Escalante, P. Guido-Aldana, W. Ojeda-Bustamante and J. Sánchez-Sesma, The drag coefficient and the Navier–Stokes fractional equation, Experimental & Computational Fluid Mechanics, Springer, Cham (2014), 399–407.  Google Scholar

• [39]

C. Miao, B. Yuan and B. Zhang, Well-posedness of the Cauchy problem for the fractional power dissipative equations, Nonlinear Anal. 68 (2008), 461–484.

• [40]

J. C. Robinson, An introduction to the classical theory of the Navier–Stokes equations, preprint (2010), IMECC-Unicamp.

• [41]

G. Stampacchia, ${\mathcal{ℒ}}^{\left(p,\lambda \right)}$-spaces and interpolation, Comm. Pure Appl. Math. 17 (1964), 293–306.  Google Scholar

• [42]

R. Strichartz, Bounded mean oscillation and Sobolev spaces, Indiana Univ. Math. J. 29 (1980), 539–558.

• [43]

R. Strichartz, Traces of BMO-Sobolev spaces, Proc. Amer. Math. Soc. 83 (1981), 509–513.

• [44]

M. E. Taylor, Analysis on Morrey spaces and applications to Navier–Stokes and other evolution equations, Comm. Partial Differential Equations 17 (1992), 1407–1456.

• [45]

S. Tourville, Existence and uniqueness of solutions for a modified Navier–Stokes equation in ${ℝ}^{2}$, Comm. Partial Differential Equations 23 (1998), 97–121.  Google Scholar

• [46]

L. Vlahos, H. Isliker, Y. Kominis and K. Hizonidis, Normal and anomalous diffusion: A tutorial, preprint (2008), https://arxiv.org/abs/0805.0419.

• [47]

Y. Z. Wang and J. Xiao, Homogeneous Campanato–Sobolev classes, Appl. Comput. Harmon. Anal. 39 (2015), 214–247.

• [48]

J. Wu, Lower bounds for an integral involving fractional Laplacians and the generalized Navier–Stokes equations in Besov spaces, Comm. Math. Phys. 263 (2006), 803–831.

• [49]

J. Xiao, Homothetic variant of fractional Sobolev space with application to Navier–Stokes system, Dyn. Partial Differ. Equ. 4 (2007), 227–245.

• [50]

J. Xiao, Homothetic variant of fractional Sobolev space with application to Navier–Stokes system revisited, Dyn. Partial Differ. Equ. 11 (2014), 167–181.

• [51]

T. Yoneda, Ill-posedness of the 3D Navier–Stokes equations in a generalized Besov space near ${\mathrm{BMO}}^{-1}$, J. Funct. Anal. 258 (2010), 3376–3387.  Google Scholar

• [52]

X. Yu and Z. Zhai, Well-posedness for fractional Navier–Stokes equations in the largest critical spaces ${\stackrel{˙}{B}}_{\mathrm{\infty },\mathrm{\infty }}^{1-2\beta }$, Math. Methods Appl. Sci. 35 (2012), 676–683.  Google Scholar

• [53]

Z. Zhai, Well-posedness for fractional Navier–Stokes equations in critical spaces close to ${\stackrel{˙}{B}}_{\mathrm{\infty }}^{-2\beta +1,\mathrm{\infty }}\left({ℝ}^{n}\right)$, Dyn. Partial Differ. Equ. 7 (2010), 25–44.  Google Scholar

• [54]

L. Zhang, On the modified Navier–Stokes equations in n-dimensional spaces, Bull. Inst. Math. Acad. Sin. 32 (2004), 185–193.  Google Scholar

Revised: 2016-07-28

Accepted: 2016-11-08

Published Online: 2017-01-12

Funding Source: National Natural Science Foundation of China

Award identifier / Grant number: 11201143

Funding Source: Chinese Universities Scientific Fund

Award identifier / Grant number: 2014ZZD10

Funding Source: Natural Sciences and Engineering Research Council of Canada

Award identifier / Grant number: 202979463102000

The first author was supported by AARMS Postdoctoral Fellowship (2013.9-2015.8), NSFC (no. 11201143) and the Fundamental Research Funds for the Central Universities (2014ZZD10), respectively. The second author was supported by NSERC of Canada (FOAPAL # 202979463102000).

Citation Information: Advances in Nonlinear Analysis, Volume 8, Issue 1, Pages 203–224, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496,

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