Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Advances in Nonlinear Analysis

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


IMPACT FACTOR 2018: 6.636

CiteScore 2018: 5.03

SCImago Journal Rank (SJR) 2018: 3.215
Source Normalized Impact per Paper (SNIP) 2018: 3.225

Mathematical Citation Quotient (MCQ) 2017: 1.89

Open Access
Online
ISSN
2191-950X
See all formats and pricing
More options …

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
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Jie Xiao
  • Corresponding author
  • Department of Mathematics and Statistics, Memorial University, St. John’s, NL A1C 5S7, Canada
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2017-01-12 | DOI: https://doi.org/10.1515/anona-2016-0042

Abstract

As an essential extension of the well known case β(12,1] to the hyper-dissipative case β(1,), 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 (-Δ)12<β< through the generalized Carleson measure spaces of initial data that unify many diverse spaces, including the Q space (Q-s=-α)n, the BMO-Sobolev space ((-Δ)-s2BMO)n, the Lip-Sobolev space ((-Δ)-s2Lipα)n, and the Besov space (B˙,s)n.

Keywords: Incompressible Navier–Stokes system with dissipation; well/ill-posedness for mild solutions; generalized Carleson measure spaces

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 (-Δ)β, under the assumption (β,x,t,τ)(0,)×n×(0,τ)×(0,] (cf. e.g., [45, 54, 53, 39, 35, 52, 8, 28, 17]), given by

{(t𝐮+(-Δ)β𝐮-𝐮𝐮-p)(x,t)=0,𝐮(x,t)=0,𝐮(x,0)=𝐮0(x),𝐮0(x)=0,(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 𝐮0:

𝐮(x,t)=e-t(-Δ)β𝐮0(x)-0te-(t-s)(-Δ)β(𝐮(x,s)𝐮(x,s))𝑑s,(1.2)

where ={jk}j,k=1,2,3={δjk+RjRk}j,k=1,2,,n, δjk is the Kronecker symbol, and Rj=j(-Δ)-12 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 β=1, we are suggested to handle system (1.1) with β>1, through replacing Δ (responsible for dissipating energy from the system) with a higher order dissipation mechanism -(-Δ)β>1 (damping selectively the high wave numbers). Interestingly, upon taking the curl of the first equation of (1.1), setting 𝐰=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:

t𝐰+(-Δ)β𝐰=(𝐰)𝐮-(𝐮)𝐰.

Remarkably, the homogeneous form of the last system t𝐰+(-Δ)β𝐰=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 12<β< cannot be extended (to the challenging unsolved situation 0<β<12) at least for our current casework regarding (1.1)–(1.2).

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

  • System (1.1) has a scaling property. If (𝐮,p,𝐮0) solves (1.1), then so does

    (𝐮λ(x,t)=λ2β-1𝐮(λx,λ2βt),pλ(x,t)=λ4β-2p(λx,λ2βt),(𝐮0)λ=λ2β-1𝐮0(λx))for all λ>0.

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

    fλ(x)=λ2β-1f(λx)for all λ>0.(1.3)

In fact, the solutions of (1.2) with β=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 (Ln)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 (BMO-1)n, and in [49] for the space (Qα-1)n. Moreover, Li and Lin [31] showed global well-posedness in a subspace of (BMO-1)n with large initial data, and Bourgain and Pavlovic̀ [3] found the norm inflation in (B˙,-1)n, which is the largest critical space with respect to (1.3) with β=1.

For 12<β1, a study of (1.2) has been carried out partially. Wu [48] got a well-posed result for (1.1) with 1<β<54 in the space (B˙1r,252-2β)n. Li and Zhai [35] considered the fractional Navier–Stokes equation (1.1) with 12<β<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 (B˙,1-2β)n. Cheskidov and Shvydkoy [11] discovered an ill-posed result in the largest critical space (B˙,1-2α)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 (B˙,1-2β)n, with respect to (1.3) with 1β<.

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

1<2β<and1-2β<α<,

we develop a uniform framework to deal with a dichotomy of the well/ill-posed results in the generalized Carleson measure spaces (Xβα)n, which are critical with respect to (1.1) and, of course, contained in the homogeneous space (B˙,1-2β)n. In the above and below,

Xβα=Xβ,α,

and for 0<τ, the space Xβ,τα is defined by the norm

fXβ,τα=sup(x,r)n×(0,τ)(r-(2α+n)0r2β(B(x,r)|e-t(-Δ)βf(y)|2dy)t-1-α-ββdt)12,

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

B˙,1-2β=B˙,,1-2β,

and for 0<τ, the space B˙,,τ1-2β is determined by the norm (cf. [36])

fB˙,,τ1-2β=sup(x,t)n×(0,τ)t2β-12β|e-t(-Δ)βf(x)|.

Clearly, Xβ,α and fB˙,,1-2β are invariant under the scaling transform (1.3). Moreover,

ααXβ,α=XβαXβ,α=XβαB˙,,1-2β=B˙,1-2β,

whose second inclusion becomes equality whenever α>0. Accordingly,

Yβα=Yβ,α,

and for τ(0,], the associated solution space Yβ,τα is decided by the norm

uYβ,τα=sup(x,r)n×(0,τ)(r-(2α+n)0r2β(B(x,r)|u(y,t)|2dy)t-1-α-ββdt)12+sup(x,t)n×(0,τ)t2β-12β|u(x,t)|.

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

{β(12,1](1+n2,),1-2β<α1-β,0<τ,𝑜𝑟{β(12,),1-2β<α2-2β,0<τ.

Then (1.2) is well-posed in (Xβ,τα)n with sufficiently small norm

𝐮0=((u0)1,,(u0)n)(Xβ,τα)n=j=1n(u0)jXβ,τα.

Furthermore, the solution u(Yβ,τα)n, and the solution map T:u0u is analytic from a sufficient small neighborhood of origin of (Xβ,τα)n to (Yβ,τα)n.

Theorem 1.1 is essentially known for 12<β1 and α(1-2β,0], see [30, 49, 50, 35, 52] and the relevant references therein. Needless to say that for the hyper-dissipative case 1<β<, 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 (-Δ)1-β, on the integrability of the kernel of

(-Δ)1-βe-t(-Δ)βfor 12<β1,

to meet the case 1<β<. However, when β(1,1+n2), 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 (Yβα)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 1-2β<α in Theorem 1.1 is natural – this can be seen from the following assertion (cf. [45, 33, 50] for β=1). If L2,n+2α (cf. [44]) stands for all real-valued Lloc2(n)-functions f obeying

fL2,n+2α=sup(x,r)n×(0,)(r-(2α+n)B(x,r)|f(y)|2dy)12<,

then

L2,n+2(1-2β)Xβαwhen 1-2β<α,

and hence (1.2) is well-posed in (L2,n+2(1-2β))n with sufficiently small norm

𝐮0={(𝐮0)}j=1n(L2,n+2(1-2β))nj=1n(𝐮0)jL2,n+2(1-2β).

Corollary 1.3.

If 12<β<1 and -1<α<12, then (1.2) is well-posed in (Xβα)n with sufficiently small norm

𝐮0={(𝐮0)}j=1n(Xβα)nj=1n(𝐮0)jXβα.

Furthermore, the solution u(Yβα)n, and the solution map T:u0u is analytic from a sufficient small neighborhood of the origin of (Xβα)n to (Yβα)n.

Note that

β(12,1)Xβα={Q1-α-β,β,-1if α(-1,0),BMO1-2βif α=0,B˙,1-2βif α(0,12).

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

Upon taking into account 1-β<α<, 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

1β<𝑎𝑛𝑑1-β<α<.

Then there exist a smooth space periodic solution u(t) of (1.2) with period 2π, and initial data u0 such that the solution map T from (Xβ,1α)n to (Yβ,1α)n is not differentiable at the origin of (Xβ,1α)n. Furthermore, for sufficiently small ϵ(0,1), there exists a smooth space periodic solution u(t) of (1.2), with period 2π, such that

𝐮(0)(Xβ,1α)nϵ  𝑎𝑛𝑑  𝐮(T)(Xβ,1α)nϵ-1for some T(0,ϵ).

Additionally, the same assertion holds for (Xβα)n and (Yβα)n, provided that 0α<.

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 (Xβα)n) that becomes not only arbitrarily large in (Xβα)n for an arbitrarily small time, but also relatively large in the resolution space (Yβα)n.

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

Here P=(-n,n+22){P=(-n,\frac{n+2}{2})} and Q=(-n2,n+22){Q=(-\frac{n}{2},\frac{n+2}{2})}.
Figure 1

Here P=(-n,n+22) and Q=(-n2,n+22).

As described in Theorems 1.1 and 1.4, the well-posedness and the ill-posedness of (1.2) initialed in (Xβα)n can be summarized in Figure 1. The well-posedness is set up for all parameter (α,β) in the region between the polyline ABC^ and polyline DEF^ but ΔPQB, while the ill-posed results are established for (α,β) above polyline DEF^. It is most likely that system (1.2) is well-posed when (α,β) in the triangle ΔPQB – 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 Xβ0=BMO1-2β and Xβα=B˙,1-2β with 0<α<12), we assert that (1.2) is:

  • well-posed when β(12,1], while ill-posed when β>1 initialed in (BMO1-2β)n,

  • well-posed when β(12,1), while ill-posed when β1 initialed in (B˙,1-2β)n.

Although the well-posedness of this last assertion for β=1 and the ill-posedness for β=1 or β>1 reduce to the well-posedness in [30] for β=1 and the ill-posedness in [3, 8] (see, e.g., [9, 10, 51, 14] for more details) for β=1 or β>1, respectively, our ill-posedness in Theorem 1.4 cannot be implied by the results in [3, 8] at least because our space Xβα with β>11-α behaves differently from their space B˙,p-γ with (γ,p)[1,)×(2,], 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]:

2,n+2αs=(-Δ)-s22,n+2αfor -1<α<1 and -<s<,

where (-Δ)-s2 is determined by the Fourier transforms

f^(ξ)=ne-iξxf(x)𝑑xand(-Δ)-s2f^(ξ)=|ξ|-sf^(ξ),

and 2,n+2α denotes the square Campanato space (cf. [5, 6, 41]) on n of all real-valued Lloc2(n)-functions f satisfying

f2,n+2α=sup(x,r)n×(0,)(r-(2α+n)B(x,r)|f(y)-r-nB(x,r)f(z)𝑑z|2𝑑y)12<

and

f2,n+2αs=(-Δ)s2f2,n+2αfor all f2,n+2αs.

Interestingly, we have Table 1.

Table 1

Spaces.

Even more interestingly, we discover

-1<α<1and12<β<32Xβα=2,n+2α1-α-2β.

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

Corollary 1.5.

Suppose 12<β<32 and max{-1,1-2β}<α<1.

  • (i)

    If 12<β<1 , then ( 1.2 ) is well posed in (2,n+2α1-α-2β)n with sufficiently small norm

    𝐮0={(𝐮0)j}j=1n(2,n+2α1-α-2β)n=j=1n(𝐮0)j2,n+2α1-α-2β.

  • (ii)

    If 1β<32 , then ( 1.2 ) in (2,n+2α1-α-2β)n , with sufficiently small norm 𝐮0(2,n+2α1-α-2β)n , is well-posed when -1<α2-2β but ill-posed, in the sense of Theorem 1.4 , when 1-β<α<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, +n+1=n×(0,). The symbol AB represents that there exists a positive constant C satisfying ACB, and thus AB represents the comparability of the quantities A and B, i.e., AB and BA.

2 Well-posedness in (Xβα)n

This section is devoted to a proof of Theorem 1.1 with τ=. The argument for Theorem 1.1 with τ< 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(0,1) and Ksβ(x) be the kernel of (-Δ)1-β(e-(-Δ)β-e-s(-Δ)β).

  • (i)

    If β(0,1) , then

    |Ksβ(x)|(1+|x|)-n-2+2β+s-2+n-2β2β(1+s-12β|x|)-n-2+2β.(2.1)

  • (ii)

    If β(1,1+n2) , then

    |Ksβ(x)|(1+|x|)-n-1+s-2+n-2β2β(1+s-12β|x|)-n-2+2β.(2.2)

  • (iii)

    If β=1 or β(1+n2,) , then

    |Ksβ(x)|(1+|x|)-n-1+s-2+n-2β2β(1+s-12β|x|)-n-1.(2.3)

Proof.

(i)  Suppose β(0,1), thus 1-β>0. The kernel K(x) of (-Δ)1-βe-(-Δ)β has the decay estimate

|K(x)|(1+|x|)β-n-1,

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

(ii)  Assume 1<β<1+n2. Let ψC0(n) and ψ(ξ)=1 for |ξ|<1, and denote by 𝗆 the symbol of the operator (-Δ)1-β(e-(-Δ)β-e-s(-Δ)β). Then, this symbol can be broken down into two terms:

𝗆(ξ)=|ξ|2-2β(e-|ξ|2β-e-s|ξ|2β)ψ(s12βξ)+|ξ|2-2β(e-|ξ|2β-e-s|ξ|2β)(1-ψ(s12βξ))𝗆1(ξ)+𝗆2(ξ).

The first term 𝗆1 is rewritten as

𝗆1(ξ)=|ξ|2-2β(e-|ξ|2β-1)ψ(s12βξ)+|ξ|2-2β(1-e-s|ξ|2β)ψ(s12βξ)𝗆11(ξ)+𝗆12(ξ).

For 𝗆12, by scaling, we only need to show

|neixξ|ξ|2-2β(e-|ξ|2β-1)ψ(ξ)𝑑ξ|(1+|x|)-n-1,

which is obvious since the symbol |ξ|2-2β(e-|ξ|2β-1)ψ(ξ) is compactly supported and has no singularity at the origin (cf. [39]). Note that the kernel of 𝗆11 can be controlled similarly if s>14. So, without loss of generality, we may assume s1 in the sequel. Write

𝗆11(ξ)=𝗆111(ξ)+𝗆112(ξ)+𝗆113(ξ),

where

𝗆111(ξ)|ξ|2-2β(e-|ξ|2β-1)ψ(ξ),𝗆112(ξ)|ξ|2-2βe-|ξ|2β(1-ψ(ξ))ψ(s12βξ),𝗆111(ξ)|ξ|2-2β(1-ψ(ξ))ψ(s12βξ).

In view of the previous argument, only the kernel of the last term, denoted by K113, needs a control. By a simple calculation, we get that

𝗆113L1s-n+2-2β2β,

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

ξα𝗆113L11.

Thus, an integration by parts derives that the kernel K113 of 𝗆113 enjoys

|K113(x)|min{s-n+2-2β2β,|x|-n-1}.

In order to prove (2.2), an improvement must be made when s12β|x|1. Now let δ(2,s-12β/2). Then

K113(x)=neixξ|ξ|2-2β(1-ψ(ξ))ψ(s12βξ)𝑑ξ=12<|ξ|δeixξ|ξ|2-2β(1-ψ(ξ))𝑑ξ+δ<|ξ|s-12βeixξ|ξ|2-2βψ(s12βξ)𝑑ξ𝖠(δ)+𝖡(δ).

It is easy to see that

|𝖠(δ)|12<|ξ|δ|ξ|2-2βdξδn+2-2βif β<1+n2.

Repeatedly using integration by parts, we obtain

|𝖡(δ)||x|-nδ<|ξ|s-12β|ξ|2-2β-ndξδ2-2β|x|-n,

reaching the desired estimate (2.2) upon choosing δ=1|x|.

(iii)  For β=1, estimate (2.3) is obvious. So, it remains to treat β>1+n2. In view of the argument in (ii), it is enough to handle K113. Since

|K113(x)|n|ξ|2-2β(1-ψ(ξ))ψ(s12βξ)dξ|ξ|>12|ξ|2-2βdξ1,

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

|K113(x)||x|-n-1|α|=n+1ξα𝗆113L1|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|1. This is the main reason why our well-posed results fail to cover the case β(1,1+n2) and 2-2β<α1-β (the triangle ΔPQB in Figure 1). Note that Ksβ(x) can be rewritten as M(x)+E(x), where E(x) is well-behaved as an error term, and M(x) behaves like

1|ξ|s-12βeixξ|ξ|2-2β𝑑ξ.

So, in view of the identity (for a dimensional constant cn)

|neixξ|ξ|2-2βdξ|=cn|x|2β-n-2for all β(1,1+n2),

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 β>12, 1-2β<α1-β and

    Cβ(f,t,x)=0te-(t-s)(-Δ)β(-Δ)βf(s,x)𝑑s,

    then

    0Cβ(f,t,)L22t-1-α-ββdt0f(t,)L22t-1-α-ββdt.

  • (ii)

    If β>1+n2 or β=1, 1-2β<α1-β and

    Dα,β(g)=sup(x,r)n×(0,1)r-(n+2α)0t2βB(x,r)|g(h,y)|dydhh1-α-ββ,

    then

    01(-Δ)12e-t2(-Δ)β0tg(s,)dsL22dtt1-α-ββDα,β(g)10g(s,)L1dss1-α-ββ.(2.4)

  • (iii)

    If β>12 and 1-2β<α2-2β , then ( 2.4 ) still holds.

Proof.

(i)  Suppose β>12 and 1-2β<α1-β. An application of the definition of e-(t-s)(-Δ)β, Plancehrel’s formula and Hölder’s inequality gives

0Cβ(f,t,)L22tα+β-1βdt00t|ξ|2βe-(t-s)|ξ|2βf(s,ξ)^dsL22tα+β-1βdt0n0t|ξ|2βe-(t-s)|ξ|2β|f(s,ξ)^|2dsdξtα+β-1βdtn0(s|ξ|2βe-(t-s)|ξ|2βdt)|f(s,ξ)^|2sα+β-1βdsdξn0|f(s,ξ)^|2sα+β-1βdsdξ0f(t,)L22tα+β-1βdt,

as desired.

(ii)  Suppose β>1+n2 or β=1 and 1-2β<α1-β. Using the inner-product ,L2 in L2 with respect to the spatial variable xn, we obtain

01(-Δ)12e-t2(-Δ)β0tg(s,)dsL22tα+β-1βdt=010t(-Δ)12e-t2(-Δ)βg(s,)ds,0t(-Δ)12e-t2(-Δ)βg(h,)dhL2tα+β-1βdt=2(0<h<s<1g(s,),s1(-Δ)e-t(-Δ)βg(h,)tα+β-1βdtL2dsdh)01|g(s,),0s(-Δ)1-β(e-(-Δ)β-e-s(-Δ)β)g(h,)𝑑hL2|sα+β-1β𝑑s.

If Ksβ(x) is the kernel of (-Δ)1-β(e-(-Δ)β-e-s(-Δ)β), then an application of Lemma 2.1 and Hölder’s inequality derives

|0s(-Δ)1-β(e-(-Δ)β-e-h(-Δ)β)g(h,x)dh|sup0<s<1s2β-n-22βs0n|g(h,y)|dydh(1+h-12β|x-y|)n+1sup0<s<1s2β-n-22βkn0ss-12β(x-y)k+[0,1]n|g(h,y)|dydh(1+h-12β|x-y|)n+1supt(0,1),knt2β-n-22β0t|kt12β-y|<t12β|g(h,y)|dydhsup(x,t)n×(0,1)t-(2α+n)0t2βB(x,t)|g(h,y)|dydhh1-α-ββ.

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

01(-Δ)12e-t2(-Δ)β0tg(s,)dsL22tα+β-1βdtDα,β(g)01g(s,)L1s-1+α+ββds.

(iii)  Suppose β>12 and 1-2β<α2(1-β). In view of the argument used in (ii), we obtain

01(-Δ)12e-t2(-Δ)β0tg(s,)dsL22tα+β-1βdt=2(0<h<s<1g(s,),s1(-Δ)e-t(-Δ)βg(h,)tα+β-1βdtL2dsdh)=2β(0<h<s<1g(s,),s1β1(-Δ)e-(-tΔ)βtα+2β-2β𝑑tg(h,)L2𝑑s𝑑h)01|g(s,),sβ-1β0s[s1β1(-Δ)e-(-tΔ)βdt]g(h,)dhL2|sα+β-1βds.

Denote by K~ and 𝗆~ the kernel and symbol of the differential operator

s1β1(-Δ)e-(-tΔ)β𝑑t.

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

|K~(x)|(1+|x|)-n-1+s-n2β(1+s-12β|x|)-n-1.(2.5)

By a change of variables, we have

𝗆~(ξ)=s1β1|ξ|2e-(t|ξ|2)βdt=s1β|ξ|2|ξ|2e-tβdt=F(s1β|ξ|2)-F(|ξ|2),

where

F(τ)τe-tβ𝑑tfor all τ>0.

It is clear that F(|ξ|2)L1 and ξγF(|ξ|2)L1 with |γ|=n+1. So, an integration by parts shows

|neixξF(|ξ|2)dξ|(1+|x|)-n-1.

Thus, (2.5) follows by a scaling argument thanks to F(s1β|ξ|2)=F(|s12βξ|2).

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(x,t)=e-t(-Δ)β𝐮0(x)-B(𝐮,𝐮),(2.6)

where B(,) is the following bilinear form:

B(𝐮,𝐯)=0te-(t-s)(-Δ)β(𝐮𝐯)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 Xβα. Thanks to the definition, we have

e-t(-Δ)β𝐮0(Yβα)n𝐮0(Xβα)n,

thus it remains to verify that (2.7) is bounded from (Yβα)n×(Yβα)n to (Yβα)n. Of course, it suffices to show both the L2-bound

r-(2α+n)0r2β|y-x|<r|B(𝐮,𝐯)|2dydtt1-α-ββ𝐮(Yβα)n2𝐯(Yβα)n2(2.8)

and the L-bound

B(𝐮,𝐯)Lt12β-1𝐮(Yβα)n𝐯(Yβα)n,(2.9)

where

𝐮=(u1,u2,,un),𝐮(Yβα)n=j=1nujYβα,𝐯=(v1,v2,,vn),𝐯(Yβα)n=j=1nvjYβα.

Step 1: L2-bound. Letting 1r,x(y)=χB(x,10r)(y) be the characteristic function of B(x,10r) and I the identity map, we divide B(𝐮,𝐯) into three parts:

B(𝐮,𝐯)=B1(𝐮,𝐯)+B2(𝐮,𝐯)+B3(𝐮,𝐯),

where

B1(𝐮,𝐯)=(-Δ)-120se-(s-h)(-Δ)β(-Δ)12(I-e-h(-Δ)β)(1r,x𝐮𝐯)𝑑h,B2(𝐮,𝐯)=(-Δ)-12(-Δ)12e-s(-Δ)β0s(1r,x𝐮𝐯)𝑑h,B3(𝐮,𝐯)=0se-(s-h)(-Δ)β((1-1r,x)𝐮𝐯)𝑑h.

For B1(𝐮,𝐯), we use the boundedness of the Riesz transform and Lemma 2.3 (i) to derive

0r2βB1(𝐮,𝐯)L22dtt1-α-ββ0r2β0se-(s-h)(-Δ)β(-Δ)12(I-e-h(-Δ)β)(1r,x𝐮𝐯)dhL22dtt1-α-ββ0r2β(-Δ)12-β(I-e-t(-Δ)β)(1r,x𝐮𝐯)L22dtt1-α-ββ.

Notice that (-Δ)12-β(I-e-t(-Δ)β) is bounded on L2, provided 12<β<, with its operator norm t1-12β. Thus, using the Cauchy–Schwarz inequality, we have

0r2βB1(𝐮,𝐯)L22dtt1-α-ββ0r2βt2-1β1r,x𝐮𝐯L22dtt1-α-ββ0r2βt2-1β|y-x|<r|𝐮(y,t)|2|𝐯(y,t)|2dydtt1-α-ββ(supt(0,T)t2β-12β𝐮(y,t)L)(supt(0,T)t2β-12β𝐯(y,t)L)×(0r2β|y-x|<r|𝐮(y,t)|2dydtt1-α-ββ)12(0r2β|y-x|<r|𝐯(y,t)|2dydtt1-α-ββ)12.

In view of the definition of Yβα, we conclude

0r2βB1(𝐮,𝐯)L22dtt1-α-ββrn+2α𝐮(Yβα)n2𝐯(Yβα)n2.(2.10)

For B2(𝐮,𝐯), by the boundedness of the Riesz transform and Lemma 2.3 (ii), we have

0r2βB2(𝐮,𝐯)L22dtt1-α-ββ0r2β(-Δ)12e-t(-Δ)β0t(1r,x𝐮𝐯)dhL22dtt1-α-ββDα,β(1r,x𝐮𝐯)0r2βn|1r,x𝐮𝐯(x,s)|dxdss1-α-ββ.

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

Dα,β(1r,x𝐮𝐯)sup(x,r)+n+1r-2α-n0r2βn1r,x(s)|𝐮𝐯(x,s)|dxdss1-α-ββ𝐮(Yβα)n𝐯(Yβα)n.

On the other hand, we similarly have

0r2βn|1r,x𝐮𝐯(x,s)|dxdss1-α-ββrn+2α𝐮(Yβα)n𝐯(Yβα)n.

Consequently, we conclude

0r2βB2(𝐮,𝐯)L22dtt1-α-ββrn+2α𝐮(Yβα)n2𝐯(Yβα)n2.(2.11)

For B3(𝐮,𝐯), by the decay property of the kernel of e-t(-Δ)β we get that if |x-y|<r and s<r2β, then

|B3(𝐮,𝐯)||0se-(s-h)(-Δ)β((1-1r,x)𝐮𝐯)dh|0s|z-x|10r|𝐮(h,z)||𝐯(h,z)|((s-h)12β+|z-y|)n+1𝑑z𝑑h0r2β|z-x|10r|𝐮(h,z)||𝐯(h,z)||x-z|n+1𝑑z𝑑hj=3(2jr)-n-10r2βB(x,2j+1r)B(x,2jr)|𝐮(h,z)||𝐯(h,z)|dzdhj=3(2jr)-n-1r2-2α-2β0r2β(B(x,2j+1r)|𝐮(h,z)||𝐯(h,z)|dz)hα+β-1βdhr1-2β-n-2α0r2β(B(x,2j+1r)|𝐮(h,z)||𝐯(h,z)|dz)hα+β-1βdh.

Then, by Hölder’s inequality, we get

|B3(𝐮,𝐯)|r1-2β𝐮(Yβα)n𝐯(Yβα)n.

Since α>1-2β, we have

0r2β|y-x|<r|B3(𝐮,𝐯)|2dydtt1-α-ββrn+2-4β0r2βdtt1-α-ββ𝐮(Yβα)n2𝐯(Yβα)n2rn-2α𝐮(Yβα)n2𝐯(Yβα)n2.(2.12)

Putting the estimates (2.10), (2.11) and (2.12) together, we reach (2.8). Step 2: L-bound. Two situations are handled in the sequel.

If t2s<t, then

e-(t-s)(-Δ)β(𝐮𝐯)L(t-s)-12β𝐮L𝐯L(t-s)-12βs1β-2𝐮(Yβα)n𝐯(Yβα)n,

and hence, for β>12, we have

|t2te-(t-s)(-Δ)β(𝐮𝐯)ds|t2t(t-s)-12βs1β-2ds𝐮(Yβα)n𝐯(Yβα)nt1-2β2β𝐮(Yβα)n𝐯(Yβα)n.(2.13)

If 0<s<t2, then t-st, and hence

|e-(t-s)(-Δ)β(𝐮𝐯)|n|𝐮(y,s)||𝐯(y,s)|((t-s)12β+|x-y|)n+1𝑑yn|𝐮(y,s)||𝐯(y,s)|(t12β+|x-y|)n+1𝑑yt-n+12β|x-y|10t12β|𝐮(y,s)||𝐯(y,s)|dy+|x-y|>10t12β|𝐮(y,s)||𝐯(y,s)||x-y|n+1dy.

Using the same calculation as in B3(𝐮,𝐯) with r=t12β, we obtain

0t2|x-y|>10t12β|𝐮(y,s)||𝐯(y,s)||x-y|n+1dydst1-2β2β𝐮(Yβα)n𝐯(Yβα)n.

Meanwhile, utilizing Hölder’s inequality, we derive

t-n+12β0t2|x-y|10t12β|𝐮(y,s)||𝐯(y,s)|dydst-2β-12βt-n+2α2β0t2(B(x,10t12β)|𝐮(z,s)||𝐯(z,s)|dz)sα+β-1βdst1-2β2β𝐮(Yβα)n𝐯(Yβα)n.

Consequently,

|0t2e-(t-s)(-Δ)β(𝐮𝐯)ds|t1-2β2β𝐮(Yβα)n𝐯(Yβα)n.(2.14)

Now, putting estimates (2.13) and (2.14) together yields the L-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:

supt>0t2β-12βe-t(-Δ)βfLfL2,n+1-2β,supt>0e-t(-Δ)βfL2,n+1-2βfL2,n+1-2β,

we get that if α>1-2β and (x,r)+n+1, then

r-(2α+n)0r2β(B(x,r)|e-t(-Δ)βf(y)|2dy)t-1-α-ββdt0r2βe-t(-Δ)βf2L2,n+1-2βr2(1-2β-α)dtt1-α-ββf2L2,n+1-2β,

whence deriving

L2,n+1-2βXβαwhen 1-2β<α.

Step 2. The desired well-posedness may be viewed as an extension of Kato’s Lp-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 (L2,n+1-2β)n, we are required to control the boundedness of the initial data semi-group

𝐮0et(-Δ)β𝐮0

and the bilinear operator

(𝐮,𝐯)B(𝐮,𝐯)=0te-(t-s)(-Δ)β(𝐮𝐯)𝑑s,

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

gL4,2(n+1-2β)=sup(x,r)+n+1(r-2(n+1-2β)B(x,r)|g(y)|4dy)14,

to derive

e-t(-Δ)βfL4,2(n+1-2β)e-t(-Δ)βfL2,n+1-2βe-t(-Δ)βfLt2β-14βfL2,n+1-2β,

whence defining the solution space (Xβ)n of all vector-valued functions 𝐮={uj}j=1n with the norm

𝐮(Xβ)n=j=1nujXβ=j=1n(supt>0t2β-14βuj(,t)L4,2(n+1-2β)+supt>0t2β-12βuj(,t)L).

On the one hand, for the initial data 𝐮0 in (1.1), we have

e-t(-Δ)β𝐮0(Xβ)nwith e-t(-Δ)β𝐮0(Xβ)n𝐮0(L2,n+1-2β)n.

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

e-(t-s)(-Δ)β(𝐮𝐯)(L4,2(n+1-2β))n((t-s)12βs3(2β-1)4β)-1s2β-14β𝐮(L4,2(n+1-2β))ns2β-12β𝐯(L)n((t-s)12βs3(2β-1)4β)-1𝐮(Xβ)n𝐯(Xβ)n

and

(t-s)12βe-(t-s)(-Δ)β(𝐮𝐯)(L)nmin{s2β-14β𝐮(L4,2(n+1-2β))ns2β-14β𝐯(L4,2(n+1-2β))n(t-s)12βs2β-12β,s2β-12β𝐮(L)ns2β-12β𝐯(L)ns2β-1β}min{(t-s)-12βs-2β-12β,s-2β-1β}𝐮(Xβ)n𝐯(Xβ)n,

and hence

0te-(t-s)(-Δ)2β(𝐮𝐯)ds(Xβ)n𝐮(Xβ)n𝐯(Xβ)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 𝐮0(B˙,-1<1-2β<0)n, obtained in [52], we are only required to prove that Xβα can be identified with B˙,1-2β for 12>α>1-β>0. On the one hand, if fB˙,1-2β, then

fB˙,1-2βsup(x,t)+n+1t2β-12β|e-t(-Δ)2βf(x)|<,

and hence

fXβαsup(x0,r)+n+1(r-(2α+n)B(x0,r)0r|e-t(-Δ)2βf(x)|2dtdxt1-α-ββ)12fB˙,1-2βsup(x0,r)+n+1(r-(2α+n)B(x0,r)0r|t-2β-12β|2dtdxt1-α-ββ)12fB˙,1-2β,

thanks to α>1-β>0. On the other hand, noting the following two facts:

  • B˙,1-2β is the largest space among all the Banach spaces that are translation-invariant and share the scaling (1.3) (cf. [7]),

  • Xβα is translation-invariant and satisfies the scaling (1.3),

we achieve

fB˙,1-2βfXβαfor all fXβα.

Thus, the desired identification follows.

3 Ill-posedness in (Xβα)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:

𝐮0(x)=l-θi=1l|ki|β(vcos(kix)+vcos(kix)),(3.1)

where θ(0,12) and the vectors kin are parallel to ζ=(1,0,0). For i=1,2,,l and a large integer N dependent on l, let

|ki|=2i-1N,ki=ki+ηn,v=(0,0,1),v=(0,1,0).

For the initial data 𝐮0 (first constructed in [8] by an idea in [3]), we have

div𝐮0=0,e-t(-Δ)β𝐮0=l-θi=1l|ki|β(vcos(kix)e-|ki|2βt+vcos(kix)e-|ki|2βt),e-t(-Δ)β𝐮0Ll-θt-12for all t>0.

The following lemma is our main new tool, which asserts that the initial data div𝐮0 constructed above is well behaved in our spaces Xβα.

Lemma 3.1.

Suppose 1β<. If u0 is given in (3.1), then

l-θ{𝐮0(Xβ,1α)nif α>1-β,𝐮0(Xβα)nif α0 and β1.

Proof.

In view of the definition of (Xβ,1α)n, we have

𝐮0(Xβ,1α)n2=sup0<r<1r-(n+2α)0r2βB(x0,r)|e-t(-Δ)βu0|2t-1-α-ββdtdyl-2θsup0<r<1r-2α0r2β(i=1l|ki|β(e-|ki|2βt+e-|ki|2βt))2t-1-α-ββdt.

So, it remains to show that

r-2α0r2β(i=1l|ki|β(e-|ki|2βt+e-|ki|2βt))2t-1-α-ββdt1for all r(0,1).

Since

i=1l|ki|βe-|ki|2βtt-12for all t1,

we have

r-2α0r2β(i=1l|ki|β(e-|ki|2βt+e-|ki|2βt))2t-1-α-ββdtr-2α0r2β(i=1l|ki|βe-|ki|2βt)2t-1-α-ββdtr2β-2

for any r(0,1), provided that α>1-β, which is sufficient since β1.

Furthermore, if α0, then the above estimate for r(0,1) is still valid, and hence it remains to establish a similar estimate for 1r<. As a matter of fact, since

i=1l|ki|βe-|ki|2βte-N2βtfor all t2,

we utilize 1N|ki| and α0, to obtain

r-2α0r2β(i=1l|ki|β(e-|ki|2βt+e-|ki|2βt))2t-1-α-ββdtr-2α01+1r2β(i=1l|ki|βe-|ki|2βt)2t-1-α-ββdtr-2α02t-1t-1-α-ββ𝑑t+r-2α2r2βe-N2βtt-1-α-ββ𝑑tr-2α1for all r[1,).

The proof is completed. ∎

Next, as in [8], we write

𝐮(t)=e-t(-Δ)β𝐮0-𝐮1(t)+𝐲(t),𝐮1(t)=B(e-t(-Δ)β𝐮0,e-t(-Δ)β𝐮0),𝐲(t)=-0te-(t-τ)(-Δ)β[G0(τ)+G1(τ)+G2(τ)]𝑑τ,

where

G0=[(e-t(-Δ)β𝐮0)𝐮1+(𝐮1)e-t(-Δ)β𝐮0+(𝐮1)𝐮1],G1=[(e-t(-Δ)β𝐮0)𝐲+(𝐮1)𝐲+(𝐲)e-t(-Δ)β𝐮0+(𝐲)𝐮1],G2=[(𝐲)𝐲].

It turns out that 𝐲 gives no trouble as an error term. So, the main contribution comes from the bilinear term 𝐮1. A straightforward calculation derives

(e-t(-Δ)β𝐮0)e-t(-Δ)β𝐮0=-l-2θi=1li=1l|ki|β|kj|βe-(|ki|2α+|kj|2β)tvcos(kix)sin(kjx)=-l-2θ2i=1l|ki|2βe-(|ki|2β+|ki|2β)tsin(ηx)v-l-2θ2ijl|ki|β|kj|βe-(|ki|2β+|kj|2β)tsin((kj-ki)x)v-l-2θ2i=1li=1l|ki|β|kj|βe-(|ki|2β+|kj|2β)tsin((kj+ki)x)vE0+E1+E2.(3.2)

Then 𝐮1 can be further decomposed according to

𝐮1=0te-(t-τ)(-Δ)βE0𝑑τ+0te-(t-τ)(-Δ)βE1𝑑τ+0te-(t-τ)(-Δ)βE2𝑑τ𝐮10+𝐮11+𝐮12.(3.3)

This in turn gives

𝐮(x,t)=e-t(-Δ)β𝐮0(x)-𝐮10(x,t)-𝐮11(x,t)-𝐮12(t)-𝐲(x,t).(3.4)

It turns out that only 𝐮10 matters, while other terms can be controlled easily under the L-norm. More precisely, we have the following two lemmas.

Lemma 3.2 (L-estimates from [8]).

Let 1β< and 0<θ<12. Then

e-t(-Δ)β𝐮0()(L)nl-θt-12,𝐮10(,t)(L)nl1-2θ,𝐮11(,t)(L)nl-2θ,𝐮12(,t)(L)nl-2θ,𝐲(,t)(L)nl1-3θt12-12β+l2-4θt1-12β

for all t(0,T] when T is sufficiently small and l is sufficiently large. Actually, one can choose

T=l-γwith γ>1-2θ1-12β.

Lemma 3.3.

Let u10 be defined as in (3.3). Then

𝐮10(,t)(Xβα)nl1-2θfor all t[N-2β,1].(3.5)

Furthermore, the solution u given by (3.4) is relatively large even in the resolution space:

𝐮(Yβα)n𝐮(Yβ,1α)nl-θ2.(3.6)

Proof.

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

e-t(-Δ)β𝐮10=-l-2θe-t|η|2β20ti=1l|ki|2βe-(|ki|2β+|ki|2β)τe-|η|2β(t-τ)sin(ηx)vdτ=-l-2θe-t|η|2β2sin(ηx)vi=1l|ki|2βe-t1-e-(|ki|2β+|ki|2β-1)t|ki|2β+|ki|2β-1-l-2θe-t|η|2βsin(ηx)vi=1le-t(1-e-|ki|2βt)-l1-2θe-t|η|2βsin(ηx)vwhen N-2βt1.

Consequently,

𝐮10(,t)(Xβα)n2sup0<r<1r-(n+2α)0r2βB(x0,r)|e-t(-Δ)β𝐮10|2t-1-α-ββdtdyl2-4θsup0<r<1r-2α0r2β(e-t|η|2β)2t-1-α-ββ𝑑tl2-4θsup0<r<1r-2α0r2βt-1-α-ββ𝑑tl2-4θwhen 12β<.

Next we estimate 𝐮10 in (Yβα)n. In a similar calculation done as above, we have

u10-l1-2θsin(ηx)vwhen N-2βt1,

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

𝐮(Yβα)n𝐮(Yβ,1α)nsup(x,t)n×(0,1)t2β-12β|u(x,t)|sup(x,t)n×[N-2β,T]t2β-12β(|𝐮10(x,t)|-|e-t(-Δ)β𝐮0(x)|-|𝐮11(x,t)|-|𝐮12(x,t)|-|𝐲(x,t)|)sup(x,t)n×[N-2β,T]t2β-12β(l1-2θ-l-θt-12-l-2θ-l1-3θt12-12β+l2-4θt1-12β)l1-2θT1-12β-l-θT12-12β-l-2θT1-12β-l1-3θT32-1β-l2-4θT2-1β.

Recall that T=l-γ is as in Lemma 3.2 and 0<θ<12. Then

𝐮(Yβα)nl1-2θl-γ(1-12β)-l-θl-γ(12-12β)-l-2θl-γ(1-12β)-l1-3θl-γ(32-1β)-l2-4θl-γ(2-1β).

If

γ=1-3θ21-12β>1-2θ1-12β>0,

then

𝐮(Yβα)nl-θ2-l-θ-l-2θ-l-1lγ2-l-θ.

Since β>1, we have γ<2-3θ, whence getting

𝐮(Yβα)nl-θ2-l-θ-l-2θ-l-3θ2-l-θl-θ2,

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 {𝐮0l}l with solution {𝐮l=𝒯(𝐮0l)}l such that

𝐮0l(Xβ,1α)nl-θfor α>1-β and β1.

However, using (3.6), we have

𝒯(𝐮0l)(Yβ,1α)n𝐮0l(Xβ,1α)n=𝐮l(Yβ,1α)n𝐮0l(Xβ,1α)nlθ2

for 0<θ<12 and l sufficiently large. Moreover, if α0, then, by Lemma 3.1, 𝐮0l(Xβα)nl-θ. Similarly, by applying (3.6), we obtain

𝒯(𝐮0l)(Yβα)n𝐮0l(Xβα)n𝐮l(Yβα)n𝐮0l(Xβα)nlθ2

for 0<θ<12 and l sufficiently large. Thus, we finish the proof by letting l.

4 Application to (2,n+2α1-α-2β)n

In this section, we demonstrate Corollary 1.5.

4.1 Characterization of CS functions

Given α(-1,1). According to [49, Lemma 2.1] and [42, Theorem 2.5], each f2,n+2α has an equivalent norm:

f2,n+2αsup(x,r)+n+1(r-(2α+n)B(x,r)0r|ϕt*f(y)|2dtdyt)12,

where ϕ is a radial function on n such that

ϕL1,ϕt(x)=t-n2ϕ(xt),|ϕ(x)|(1+|x|)-ϵ-nfor some ϵ>0,nϕ(x)dx=0,0<0|ϕ^(tξ)|2dtt<.

Since

ψt*[(-Δ)sf](x)=(-Δ)sψt*f(x)=t-s[(-Δ)sψ]t*f(x),

we discover an equivalent norm for 2,n+2αs:

f2,n+2αssup(x,r)+n+1(r-(2α+n)B(x,r)0r|[(-Δ)sψ]t*f(y)|2dtdyt1+2s)12sup(x,r)+n+1(r-(2α+n)B(x,r)0r|ϕt*f(y)|2dtdyt1+2s)12,

provided that ψ satisfies the above conditions on ϕ, where (-Δ)sψ=ϕ.

Now, set ϕ be the inverse Fourier transform of t|ξ|e-(t|ξ|)2β and -<s<1. In view of the above analysis, we have a semi-group characterization for each Campanato–Sobolev (CS) function f2,n+2αs:

f2,n+2αssup(x,r)+n+1(r-(2α+n)B(x,r)0r|tet2β(-Δ)βf(y)|2dtdxt1+2s)12sup(x,r)+n+1(r-(2α+n)B(x,r)0r2β|e-t(-Δ)βf(y)|2t1-s-ββdtdy)12,

where stands for the spatial gradient.

4.2 Proof of Corollary 1.5

The preceding characterization leads to introducing the space (2,n+2αs)-1 of all functions fLloc2 on n with the norm

f(2,n+2αs)-1=sup(x,r)+n+1(r-(2α+n)B(x,r)0r2β|e-t(-Δ)βf(y)|2t1-s-ββdtdy)12.

It is not hard to check the following implication:

(α,β,s)(-1,1)×[12,)×(-,1)Xβα=(2,n+2α2-α-2β)-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 (α,β,s)(-1,1)×[12,32)×(-,1) and j=1,2,,n. If Rj=xj-Δ is the j-th Riesz transform, then

Rjf2,n+2αsf2,n+2αsfor all f2,n+2αs,

and hence

2,n+2αs-1=(2,n+2αs)-1=(2,n+2αs)n.

Proof.

Assume f2,n+2αs. Since

Rjf2,n+2αs2=sup(x,r)+n+1r-(2α+n)B(x,r)0r2β|e-t(-Δ)βRjf(y)|2t1-s-ββdtdy,

we split Rjf into two pieces via the point-mass function δ:

Rjf=φr*(Rjf)+(δ-φr)*(Rjf),

where

φC0(n),suppφB(0,1),nφ(x)𝑑x=1,φr(x)=r-nφ(xr).

On the one hand, using the fact that the predual of B˙,s+α-1 is the homogeneous Besov space B˙1,11-s-α, we estimate

B(x,r)0r2β|φr*e-t(-Δ)βRjf(y)|2t1-s-ββdtdyrn-2s+2φr*e-t(-Δ)βRjf(y)L(+n+1)2rn+2-2sφrB˙1,11-s-α2e-(t-Δ)2βRjfB˙,s+α-12rn+2αfB˙,s+α-12rn+2αf(2,n+2αs)-12rn+2αf2,n+2αs2.

On the other hand, noticing that

(δ-φr)*Rje-(t-Δ)βf(x)=(δ-φr)*Rje-12(t-Δ)βe-12(t-Δ)βf(x)

and that (δ-φr)*Rje-12(t-Δ)2β is a convolution operator with its kernel K~t(x) satisfying

supt>0n|K~t(x)|dx1,

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

r-(2α+n)B(x0,r)0r2β|(δ-φr)*e-t(-Δ)βRjf(x)|2t1-s-ββ𝑑t𝑑xr-(2α+n)B(x0,r)0r|nK~t(x-y)(ye-12(t-Δ)2βf(y))dy|2t1-2sdtdxsupt>0(n|K~t(x)|dx)2f22,n+2αsf2,n+2αs2.

The above two-fold treatment yields

Rjf2,n+2αsf2,n+2αs.

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

On the one hand, if f2,n+2αs-1, then (-Δ)-1f2,n+2αs, by definition. An application of the estimate for the Riesz transform gives

xj-Δf=(xj-Δ)(-Δ)-1f2,n+2αs,

and consequently f(2,n+2αs)n. This in turn produces

(f1,,fn)(2,n+2αs)nsuch thatf=j=1nxjfj.

An application of the triangle inequality implies f(2,n+2αs)-1.

On the other hand, if f(2,n+2αs)-1, then choosing

fj,k=xjxk(-Δ)-1ffor all j,k{1,2,,n},

one has fj,k(2,n+2αs)-1 due to the above-proved Riesz transform estimate. This further derives

gj=xj(-Δ)-1f2,n+2αs.

So, there exist fj2,n+2αs for j=1,,n such that f=j=1nxjfj, and then

f2,n+2αs-1j=1nxjfj2,n+2αs-1j=1nRjfj2,n+2αsj=1nfj2,n+2αs,

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 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.  CrossrefGoogle Scholar

  • [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.  CrossrefGoogle Scholar

  • [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.  CrossrefGoogle Scholar

  • [9]

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

  • [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.  CrossrefGoogle Scholar

  • [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.  CrossrefGoogle Scholar

  • [14]

    M. Dai, J. Qing and M. E. Schonbek, Norm inflation for incompressible magneto-hydrodynamic system in B˙-1,, Adv. Differential Equations 16 (2011), 725–746.  Google Scholar

  • [15]

    C. Deng and X. Yao, Ill-posedness of the incompressible Navier–Stokes equations in F˙-1,q(3), 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 F˙3α-1-α,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.  CrossrefGoogle Scholar

  • [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 BM˙O-α and W˙-α,p, Pure Appl. Math. Q. 7 (2011), 275–318.  Google Scholar

  • [22]

    Y. Giga and T. Miyakawa, Solutions in Lr 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.  CrossrefGoogle Scholar

  • [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.  CrossrefGoogle Scholar

  • [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 Lp-solutions of the Navier–Stokes equation in 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.  CrossrefGoogle Scholar

  • [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.  CrossrefGoogle Scholar

  • [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.  CrossrefGoogle Scholar

  • [30]

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

  • [31]

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

  • [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.  CrossrefGoogle Scholar

  • [36]

    P.-L. Lions and N. Masmoudi, Uniqueness of mild solutions of the Navier–Stokes system in LN, 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.  CrossrefGoogle Scholar

  • [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.  CrossrefGoogle Scholar

  • [40]

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

  • [41]

    G. Stampacchia, (p,λ)-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.  CrossrefGoogle Scholar

  • [43]

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

  • [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.  CrossrefGoogle Scholar

  • [45]

    S. Tourville, Existence and uniqueness of solutions for a modified Navier–Stokes equation in 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.  CrossrefGoogle Scholar

  • [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.  CrossrefGoogle Scholar

  • [49]

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

  • [50]

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

  • [51]

    T. Yoneda, Ill-posedness of the 3D Navier–Stokes equations in a generalized Besov space near 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 B˙,1-2β, 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 B˙-2β+1,(n), 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

About the article

Received: 2016-02-21

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, DOI: https://doi.org/10.1515/anona-2016-0042.

Export Citation

© 2019 Walter de Gruyter GmbH, Berlin/Boston. This work is licensed under the Creative Commons Attribution 4.0 Public License. BY 4.0

Comments (0)

Please log in or register to comment.
Log in