Essentially, continuing from , we study the mild solutions (fluid velocities) of the so-called incompressible Navier–Stokes system with dissipation , under the assumption (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 that satisfy the following integral system arising from the initial data :
where , is the Kronecker symbol, and is the Riesz transform. In accordance with , 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 , in order to overcome some obstacles coming from numerical simulations of turbulent fluids induced by system (1.1) with , we are suggested to handle system (1.1) with , through replacing Δ (responsible for dissipating energy from the system) with a higher order dissipation mechanism (damping selectively the high wave numbers). Interestingly, upon taking the curl of the first equation of (1.1), setting , 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 (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 cannot be extended (to the challenging unsolved situation ) 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
In fact, the solutions of (1.2) with in certain critical spaces have drawn a lot of attention since the pioneer work of Kato in , where he showed the global well-posedness with small data and the local well-posedness with large data in (cf.  for an earlier work). Some similar well-posedness results can be found in [22, 44, 33] for certain Morrey spaces, in  for the space , and in  for the space . Moreover, Li and Lin  showed global well-posedness in a subspace of with large initial data, and Bourgain and Pavlovic̀  found the norm inflation in , which is the largest critical space with respect to (1.3) with .
For , a study of (1.2) has been carried out partially. Wu  got a well-posed result for (1.1) with in the space . Li and Zhai  considered the fractional Navier–Stokes equation (1.1) with , whence extending the above-mentioned well-posedness to Q-type spaces. Yu and Zhai  obtained a similar result in the largest critical space . Cheskidov and Shvydkoy  discovered an ill-posed result in the largest critical space 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  and the ill-posedness in . Li, Xiao and Yang  found a global well-posedness in some Besov-Q type spaces. Cheskidov and Dai  revealed a norm inflation phenomenon in the largest critical space , with respect to (1.3) with .
we develop a uniform framework to deal with a dichotomy of the well/ill-posed results in the generalized Carleson measure spaces , which are critical with respect to (1.1) and, of course, contained in the homogeneous space . In the above and below,
and for , the space is defined by the norm
where is the ball centered at x with radius r. Meanwhile,
and for , the space is determined by the norm (cf. )
Clearly, and are invariant under the scaling transform (1.3). Moreover,
whose second inclusion becomes equality whenever . Accordingly,
and for , the associated solution space 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.
Then (1.2) is well-posed in with sufficiently small norm
Furthermore, the solution , and the solution map is analytic from a sufficient small neighborhood of origin of to .
Theorem 1.1 is essentially known for and , see [30, 49, 50, 35, 52] and the relevant references therein. Needless to say that for the hyper-dissipative case , Theorem 1.1 is new. In order to prove Theorem 1.1, we follow the method originated from  (which was developed in [32, 49, 50, 35]), but we have to find a new idea to treat the singularity, appearing in , on the integrability of the kernel of
to meet the case . However, when , 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 are obtained by the standard fixed point theorem, which automatically ensures the analytic property of the solution map as stated above.
and hence (1.2) is well-posed in with sufficiently small norm
If and , then (1.2) is well-posed in with sufficiently small norm
Furthermore, the solution , and the solution map is analytic from a sufficient small neighborhood of the origin of to .
Then there exist a smooth space periodic solution of (1.2) with period , and initial data such that the solution map from to is not differentiable at the origin of . Furthermore, for sufficiently small , there exists a smooth space periodic solution of (1.2), with period , such that
Additionally, the same assertion holds for and , provided that .
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 ) that becomes not only arbitrarily large in for an arbitrarily small time, but also relatively large in the resolution space .
As described in Theorems 1.1 and 1.4, the well-posedness and the ill-posedness of (1.2) initialed in can be summarized in Figure 1. The well-posedness is set up for all parameter in the region between the polyline and polyline but , while the ill-posed results are established for above polyline . It is most likely that system (1.2) is well-posed when in the triangle – 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 , is required to fill this unnatural gap.
well-posed when , while ill-posed when initialed in ,
well-posed when , while ill-posed when initialed in .
Although the well-posedness of this last assertion for and the ill-posedness for or reduce to the well-posedness in  for and the ill-posedness in [3, 8] (see, e.g., [9, 10, 51, 14] for more details) for or , respectively, our ill-posedness in Theorem 1.4 cannot be implied by the results in [3, 8] at least because our space with behaves differently from their space with , and yet includes non-differentiability of the solution map as an extra property.
where is determined by the Fourier transforms
Interestingly, we have Table 1.
Even more interestingly, we discover
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, . The symbol represents that there exists a positive constant C satisfying , and thus represents the comparability of the quantities A and B, i.e., and .
2 Well-posedness in
2.1 Estimation for some singular integrals
We need two technical results on some integrals of strong singularity.
Let and be the kernel of .
If , then
If , then
If or , then
(i) Suppose , thus . The kernel of has the decay estimate
(ii) Assume . Let and for , and denote by the symbol of the operator . Then, this symbol can be broken down into two terms:
The first term is rewritten as
For , by scaling, we only need to show
which is obvious since the symbol is compactly supported and has no singularity at the origin (cf. ). Note that the kernel of can be controlled similarly if . So, without loss of generality, we may assume in the sequel. Write
In view of the previous argument, only the kernel of the last term, denoted by , needs a control. By a simple calculation, we get that
and so, if the multi-index α satisfies , then
Thus, an integration by parts derives that the kernel of enjoys
In order to prove (2.2), an improvement must be made when . Now let . Then
It is easy to see that
Repeatedly using integration by parts, we obtain
reaching the desired estimate (2.2) upon choosing .
(iii) For , estimate (2.3) is obvious. So, it remains to treat . In view of the argument in (ii), it is enough to handle . Since
an integration by parts gives (as estimated in (ii))
and the desired result (2.3) follows. ∎
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 . This is the main reason why our well-posed results fail to cover the case and (the triangle in Figure 1). Note that can be rewritten as , where is well-behaved as an error term, and behaves like
So, in view of the identity (for a dimensional constant )
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.
If , and
If or , and
If and , then ( 2.4 ) still holds.
(i) Suppose and . An application of the definition of , Plancehrel’s formula and Hölder’s inequality gives
(ii) Suppose or and . Using the inner-product in with respect to the spatial variable , we obtain
If is the kernel of , then an application of Lemma 2.1 and Hölder’s inequality derives
This, along with another application of Hölder’s inequality, implies
(iii) Suppose and . In view of the argument used in (ii), we obtain
Denote by and 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
It is clear that and with . So, an integration by parts shows
Thus, (2.5) follows by a scaling argument thanks to ∎
2.2 Proof of Theorem 1.1
where 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 . Thanks to the definition, we have
thus it remains to verify that (2.7) is bounded from to . Of course, it suffices to show both the -bound
and the -bound
Step 1: -bound. Letting be the characteristic function of and I the identity map, we divide into three parts:
For , we use the boundedness of the Riesz transform and Lemma 2.3 (i) to derive
Notice that is bounded on , provided , with its operator norm . Thus, using the Cauchy–Schwarz inequality, we have
In view of the definition of , we conclude
For , 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 , by the decay property of the kernel of we get that if and , then
Then, by Hölder’s inequality, we get
Since , we have
If , then
and hence, for , we have
If , then , and hence
Using the same calculation as in with , we obtain
Meanwhile, utilizing Hölder’s inequality, we derive
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 and , then
Step 2. The desired well-posedness may be viewed as an extension of Kato’s -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 , 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:
whence defining the solution space of all vector-valued functions with the norm
On the one hand, for the initial data in (1.1), we have
On the other hand, for the corresponding bilinear part, a direct computation as in  shows that if , then
This, along with the standard fixed-point argument, as in , completes the proof.
2.4 Proof of Corollary 1.3
thanks to . On the other hand, noting the following two facts:
is translation-invariant and satisfies the scaling (1.3),
Thus, the desired identification follows.
3 Ill-posedness in
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 . Referring to [3, 8], for a large integer , we choose the following initial data:
where and the vectors are parallel to . For and a large integer N dependent on l, let
The following lemma is our main new tool, which asserts that the initial data constructed above is well behaved in our spaces .
Suppose . If is given in (3.1), then
In view of the definition of , we have
So, it remains to show that
for any , provided that , which is sufficient since .
Furthermore, if , then the above estimate for is still valid, and hence it remains to establish a similar estimate for . As a matter of fact, since
we utilize and , to obtain
The proof is completed. ∎
Next, as in , we write
It turns out that gives no trouble as an error term. So, the main contribution comes from the bilinear term . A straightforward calculation derives
Then can be further decomposed according to
This in turn gives
It turns out that only matters, while other terms can be controlled easily under the -norm. More precisely, we have the following two lemmas.
Lemma 3.2 (-estimates from ).
Let and . Then
for all when T is sufficiently small and l is sufficiently large. Actually, one can choose
Let be defined as in (3.3). Then
Furthermore, the solution given by (3.4) is relatively large even in the resolution space:
Next we estimate in . In a similar calculation done as above, we have
Recall that is as in Lemma 3.2 and . Then
Since , we have , 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 with solution such that
However, using (3.6), we have
for and l sufficiently large. Thus, we finish the proof by letting .
4 Application to
In this section, we demonstrate Corollary 1.5.
4.1 Characterization of CS functions
where ϕ is a radial function on such that
we discover an equivalent norm for :
provided that ψ satisfies the above conditions on ϕ, where .
Now, set ϕ be the inverse Fourier transform of and . In view of the above analysis, we have a semi-group characterization for each Campanato–Sobolev (CS) function :
where stands for the spatial gradient.
4.2 Proof of Corollary 1.5
The preceding characterization leads to introducing the space of all functions on with the norm
It is not hard to check the following implication:
Suppose and . If is the j-th Riesz transform, then
Assume . Since
we split into two pieces via the point-mass function δ:
On the one hand, using the fact that the predual of is the homogeneous Besov space , we estimate
On the other hand, noticing that
and that is a convolution operator with its kernel 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 , then , by definition. An application of the estimate for the Riesz transform gives
and consequently . This in turn produces
An application of the triangle inequality implies .
On the other hand, if , then choosing
one has due to the above-proved Riesz transform estimate. This further derives
So, there exist for such that , and then
as desired. ∎
S. Abe and S. Thurner, Anomalous diffusion in view of Einsteins 1905 theory of Brownian motion, Phys. A 365 (2005), 403–407. Google Scholar
P. Auscher and D. Frey, A new proof for Koch and Tataru’s result on the well-posedness of Navier–Stokes equations in , preprint (2013), https://arxiv.org/abs/1310.3783.
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
S. Campanato, Proprietá di una famiglia dispazi funzionali, Ann. Sc. Norm. Super. Pisa, Sci. Fis. Mat. III. Ser. 18 (1964), 137–160. Google Scholar
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
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
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
M. Dai, J. Qing and M. E. Schonbek, Norm inflation for incompressible magneto-hydrodynamic system in , Adv. Differential Equations 16 (2011), 725–746. Google Scholar
C. Deng and X. Yao, Ill-posedness of the incompressible Navier–Stokes equations in , preprint (2013), https://arxiv.org/abs/1302.7084v1.
C. Deng and X. Yao, Well-posedness and ill-posedness for the generalized Navier–Stokes equations in , Discrete Contin. Dyn. Syst. 34 (2014), 437–459. Google Scholar
G. Duvaut and J.-L. Lions, Les inéquations en mécanique et en physique, Trav. Rech. Math. 21, Dunod, Paris, 1972. Google Scholar
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
H. Fujita and T. Kato, On the nonstationary Navier–Stokes system, Rend. Semin. Mat. Univ. Padova 32 (1962), 243–260. Google Scholar
J. Garnett, P. W. Jones, T. M. Le and L. A. Vese, Modeling oscillatory components with the homogeneous spaces and , Pure Appl. Math. Q. 7 (2011), 275–318. Google Scholar
Y. Giga and T. Miyakawa, Solutions in of the Navier–Stokes initial value problem, Arch. Ration. Mech. Anal. 89 (1985), 267–281. Google Scholar
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
T. Kato, Strong -solutions of the Navier–Stokes equation in , with applications to weak solutions, Math. Z. 187 (1984), 471–480. Google Scholar
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
P. G. Lemariè-Rieusset, The Navier–Stokes equations in the critical Morrey–Campanato space, Rev. Mat. Iberoam. 23 (2002), 897–930. Google Scholar
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
P.-L. Lions and N. Masmoudi, Uniqueness of mild solutions of the Navier–Stokes system in , Comm. Partial Differential Equations 26 (2001), 2211–2226. Google Scholar
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
G. Stampacchia, -spaces and interpolation, Comm. Pure Appl. Math. 17 (1964), 293–306. Google Scholar
S. Tourville, Existence and uniqueness of solutions for a modified Navier–Stokes equation in , Comm. Partial Differential Equations 23 (1998), 97–121. Google Scholar
L. Vlahos, H. Isliker, Y. Kominis and K. Hizonidis, Normal and anomalous diffusion: A tutorial, preprint (2008), https://arxiv.org/abs/0805.0419.
T. Yoneda, Ill-posedness of the 3D Navier–Stokes equations in a generalized Besov space near , J. Funct. Anal. 258 (2010), 3376–3387. Google Scholar
X. Yu and Z. Zhai, Well-posedness for fractional Navier–Stokes equations in the largest critical spaces , Math. Methods Appl. Sci. 35 (2012), 676–683. Google Scholar
Z. Zhai, Well-posedness for fractional Navier–Stokes equations in critical spaces close to , Dyn. Partial Differ. Equ. 7 (2010), 25–44. Google Scholar
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
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.
© 2019 Walter de Gruyter GmbH, Berlin/Boston. This work is licensed under the Creative Commons Attribution 4.0 Public License. BY 4.0