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

As an essential extension of the well known case β ∈ (1 2 , 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 (−∆) 2<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 ((−∆)− s 2BMO)n, the Lip-Sobolev space ((−∆)− s 2 Lipα)n, and the Besov space (Ḃs ∞,∞).

Here it is appropriate to mention three basic facts which reveal that the restriction 1 2 < β < ∞ cannot be extended (to the challenging unsolved situation 0 < β < 1 2 ) at least for our current casework regarding (1.1)-(1.2).
• System (1.1) is more meaningful in a critical space which is invariant under the scaling f λ (x) = λ 2β−1 f(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 wellposedness with large data in (L n ) n (cf. [20] for an earlier work). Some similar well-posedness results can be found in [22,33,44] for certain Morrey spaces, in [30] for the space (BMO −1 ) n , and in [49] for the space (Q −1 α ) n . Moreover, Li and Lin [31] showed global well-posedness in a subspace of (BMO −1 ) n with large initial data, and Bourgain and Pavlovic [3] found the norm inflation in (Ḃ −1 ∞,∞ ) n , which is the largest critical space with respect to (1.3) with β = 1.
Then (1.2) is well-posed in (X α β,τ ) n with sufficiently small norm Furthermore, the solution u ∈ (Y α β,τ ) n , and the solution map T : u 0 → u is analytic from a sufficient small neighborhood of origin of (X α β,τ ) n to (Y α β,τ ) n .
Theorem 1.1 is essentially known for 1 2 < β ≤ 1 and α ∈ (1 − 2β, 0], see [30,35,49,50,52] 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 [32,35,49,50]), but we have to find a new idea to treat the singularity, appearing in (−∆) 1−β , on the integrability of the kernel of (−∆) 1−β e −t(−∆) β for 1 2 < β ≤ 1, to meet the case 1 < β < ∞. However, when β ∈ (1, 1 + n 2 ), 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 [26,30,32,33,35,49,50,52], i.e., both existence and uniqueness of a mild solution to (1.1) in the resolved space (Y α β ) n are obtained by the standard fixed point theorem, which automatically ensures the analytic property of the solution map as stated above.
Upon taking into account 1 − β < α < ∞, the second theorem of this paper is concerned with the illposedness of (1.2), illustrating that Theorem 1.1 is optimal under certain circumstance.
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 (X α β ) n ) that becomes not only arbitrarily large in (X α β ) n for an arbitrarily small time, but also relatively large in the resolution space (Y α β ) n . Perhaps, it is appropriate to make two more comments on Theorems 1.1 and 1.4, and Corollary 1.3 as follows.
As described in Theorems 1.1 and 1.4, the well-posedness and the ill-posedness of (1.2) initialed in (X α β ) n can be summarized in Figure 1. The well-posedness is set up for all parameter (α, β) in the region between the polylineÂBC and polylineDEF but ∆PQB, while the ill-posed results are established for (α, β) above polylinê DEF. It is most likely that system (1.2) is well-posed when (α, β) in the triangle ∆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.
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 × (0, ∞). The symbol A ≲ B represents that there exists a positive constant C satisfying A ≤ CB, and thus A ≈ B represents the comparability of the quantities A and B, i.e., A ≲ B and B ≲ A.

Well-posedness in (X α β ) n
This section is devoted to a proof of Theorem 1.1 with τ = ∞. The argument for Theorem 1.1 with τ < ∞ is similar.

Estimation for some singular integrals
We need two technical results on some integrals of strong singularity.
(ii) Assume 1 < β < 1 + n 2 . Let ψ ∈ C ∞ 0 (ℝ n ) and ψ(ξ ) = 1 for |ξ | < 1, and denote by m the symbol of the operator Then, this symbol can be broken down into two terms: The first term m 1 is rewritten as For m 12 , 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 m 11 can be controlled similarly if s > 1 4 . So, without loss of generality, we may assume s ≪ 1 in the sequel. Write 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 , and so, if the multi-index α satisfies |α| = n + 1, then Thus, an integration by parts derives that the kernel K 113 of m 113 enjoys In order to prove (2.2), an improvement must be made when s It is easy to see that 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 + n 2 . In view of the argument in (ii), it is enough to handle K 113 . 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 + n 2 ) and 2 − 2β 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.

Proof of Theorem 1.1
The proof follows the idea originated from [30], see also [32,Chapter 16]. We rewrite (1.2) (cf. [22,26,27,32,44]) 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 X α β . Thanks to the definition, we have Step 1: L 2 -bound. Letting 1 r,x (y) = χ B(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 B 1 (u, v), we use the boundedness of the Riesz transform and Lemma 2.3 (i) to derive . Thus, using the Cauchy-Schwarz inequality, we have In view of the definition of Y α β , we conclude (2.10) For B 2 (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 (2.11) For B 3 (u, v), by the decay property of the kernel of e −t(−∆) β ℙ∇ we get that if |x − y| < r and s < r 2β , then Then, by Hölder's inequality, we get (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.

Proof of Remark 1.2
The argument is divided into two steps.
Step 1. Noting the following Minkowski-inequality-based estimates: Step 2. The desired well-posedness may be viewed as an extension of Kato's L p -theory, developed in [22,26,27,33,44,45,50], to (1.2). In order to deal with a mild solution of (1.1) initialized in (L 2,n+1−2β ) n , we are required to control the boundedness of the initial data semi-group u 0 → e t(−∆) β u 0 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: |g(y)| 4 dy) 1 4 , to derive whence defining the solution space (X β ) n of all vector-valued functions u = {u j } n j=1 with the norm 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 This, along with the standard fixed-point argument, as in [33], completes the proof.

Proof of Corollary 1.3
In accordance with Theorem 1.1 and the well-posedness of (1.2) arising from u 0 ∈ (Ḃ −1<1−2β<0 ∞,∞ ) n , obtained in [52], we are only required to prove that X α β can be identified withḂ and hence On the other hand, noting the following two facts: ∞,∞ is the largest space among all the Banach spaces that are translation-invariant and share the scaling (1.3) (cf. [7]), • X α β is translation-invariant and satisfies the scaling (1.3), we achieve ‖f ‖Ḃ 1−2β for all f ∈ X α β . Thus, the desired identification follows.

Ill-posedness in (X α β ) n
This section is devoted to validating Theorem 1.4. The construction in the proof relies heavily on [3,8].

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, 1 2 ) and the vectors k i ∈ ℤ n 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 u 0 (first constructed in [8] by an idea in [3]), we have div u 0 = 0, The following lemma is our main new tool, which asserts that the initial data div u 0 constructed above is well behaved in our spaces X α β .
Proof. In view of the definition of (X α β,1 ) n , we have So, it remains to show that for any r ∈ (0, 1), provided that α > 1 − β, which is sufficient since β ≥ 1. Furthermore, if α ≥ 0, then the above estimate for r ∈ (0, 1) is still valid, and hence it remains to establish a similar estimate for 1 ≤ r < ∞. As a matter of fact, since we utilize 1 ≪ N ≤ |k i | and α ≥ 0, to obtain 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 u 1 . A straightforward calculation derives Then u 1 can be further decomposed according to This in turn gives y(x, t). (3.4) It turns out that only u 10 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 1 ≤ β < ∞ and 0 < θ < 1 2 . Then for all t ∈ (0, T] when T is sufficiently small and l is sufficiently large. Actually, one can choose . Lemma 3.3. Let u 10 be defined as in (3.3). Then Furthermore, the solution u given by (3.4) is relatively large even in the resolution space: Proof. From (3.2)-(3.3) and a straightforward calculation, it follows that Consequently, In a similar calculation done as above, we have 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 < 1 2 . Then Since β > 1, we have γ < 2 − 3θ, whence getting provided that l is sufficiently large.

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 {u 0l } l with solution {u l = T(u 0l )} l such that ‖u 0l ‖ (X α β,1 ) n ≲ l −θ for α > 1 − β and β ≥ 1. However, using (3.6), we have for 0 < θ < 1 2 and l sufficiently large. Moreover, if α ≥ 0, then, by Lemma 3.1, ‖u 0l ‖ (X α β ) n ≲ l −θ . Similarly, by applying (3.6), we obtain for 0 < θ < 1 2 and l sufficiently large. Thus, we finish the proof by letting l → ∞.
On the other hand, noticing that and that (δ − φ r ) * R j e − 1 2 (t√−∆) 2β is a convolution operator with its kernelK t (x) satisfying we get, by the argument used in the proof of [47, Lemma 3.1] and Hölder's inequality, .