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Advances in Nonlinear Analysis

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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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/ 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


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


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:


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:


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


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,


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


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


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


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


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


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


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

Theorem 1.1.



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


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



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


Corollary 1.3.

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


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.



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


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+2αs=(-Δ)s2f2,n+2αfor all f2,n+2αs.

Interestingly, we have Table 1.

Table 1


Even more interestingly, we discover


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


  • (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.


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


  • (ii)

    If β(1,1+n2) , then


  • (iii)

    If β=1 or β(1+n2,) , then



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


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:


The first term 𝗆1 is rewritten as


For 𝗆12, by scaling, we only need to show


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




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


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


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


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


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


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


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


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


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




  • (ii)

    If β>1+n2 or β=1, 1-2β<α1-β and




  • (iii)

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


(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


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


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


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


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


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


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


By a change of variables, we have



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


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


where B(,) is the following bilinear form:


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


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


and the L-bound




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:




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


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


In view of the definition of Yβα, we conclude


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


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


On the other hand, we similarly have


Consequently, we conclude


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


Then, by Hölder’s inequality, we get


Since α>1-2β, we have


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


and hence, for β>12, we have


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


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


Meanwhile, utilizing Hölder’s inequality, we derive




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:


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


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


and the bilinear operator


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


to derive


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


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




and hence


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


and hence


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:


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


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.


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


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).


i=1l|ki|βe-|ki|2βtt-12for all t1,

we have


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




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


Then 𝐮1 can be further decomposed according to


This in turn gives


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


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:



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.


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


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






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


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


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


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:


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<.



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


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:


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


It is not hard to check the following implication:


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



Assume f2,n+2αs. Since


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




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


On the other hand, noticing that


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


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


The above two-fold treatment yields


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


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


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


as desired. ∎


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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.

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