Analyticity and Existence of the Keller–Segel–Navier–Stokes Equations in Critical Besov Spaces
-
Minghua Yang
,
Zunwei Fu
und
Suying Liu
Abstract
This paper deals with the Cauchy problem to the Keller–Segel model coupled with the incompressible 3-D Navier–Stokes equations. Based on so-called Gevrey regularity estimates, which are motivated by the works of Foias and Temam [20], we prove that the solutions are analytic for a small interval of time with values in a Gevrey class of functions. As a consequence of Gevrey estimates, we particularly imply higher-order derivatives of solutions in Besov and Lebesgue spaces. Moreover, we prove that the existence of a positive constant
for certain conditions on
1 Introduction and Main Results
Consider the following model proposed in [40]:
where the functions
When
Lorz [32] studied local-in-time weak solutions in a bounded domain in
In particular, if the chemotaxis effects are ignored (
which have been widely studied during the past eighty or more years. Leray [29] proved the global existence of weak solutions of (1.2), but the uniqueness and regularity of weak solutions remains a big open problem. It has been proved that the Cauchy problem (1.2) is globally well-posed for the small initial data in series of function spaces including particularly the following critical spaces:
see Fujita and Kato [21], Kato [25], Kozono and Yamazaki [27] and Koch and Tataru [26]. Xiao [45, 46] proved this property in the space
Recently, the authors of [11, 12, 22, 23, 34, 35, 36, 49] proved the global well-posedness of (1.2) with initial data in some critical spaces and having a large vertical component provided that the horizontal component is small enough. For example, for any
whereas
Zhai and Zhang [49] proved that there exists a global solution of (1.2) satisfying
Biswas [4] introduced
In this article, we shall consider
We recall that system (1.4) enjoys nice scaling properties. If
with initial data
We say that
and
For any operator
Let Λ be the Fourier multiplier whose symbol is given by
for some
Let
and
Meanwhile, let
For simplicity, when
Inspired by the seminal work of Foias and Temam in [20], and also motivated by the papers [5, 1, 2, 4, 6] concerning the Gevrey regularity of solutions, we obtain the Gevrey class regularity for system (1.4) in the context of critical Besov spaces.
Theorem 1.1.
Let
such that system (1.4) admits a unique local solution such that
The solution
Corollary 1.2.
For
If
If
If
If
It is worth mentioning that for any
Motivated by the paper [49] concerning the global well-posedness of 3D-incompressible Navier–Stokes equations with the third component of the initial velocity field being large, applying the local existence mentioned in Theorem 1.1 (in this case
Theorem 1.3.
Let
satisfy
where
then system (1.4) admits a unique global-in-time solution such that
We mention that our results do not have any smallness conditions imposed on the third component
Notation.
Throughout the paper,
2 Preliminaries
Let
More generally, the Fourier transform of any f is in
for all
and
Define the set
Furthermore, for all
and for all
From now on, we write
Here
is a frequency projection to the ball
where
It is well known that if either
Let us now recall the definition of the Chemin–Lerner space
and
where
and the usual modifications are needed when
with the norm
From Minkowski’s inequality, it is easy to deduce that
If
where C is a positive constant independent of f.
Because
The homogeneous paraproduct of v and u is defined by
The homogeneous remainder of v and u is defined by
We have the following Bony decomposition:
Lemma 2.1 ([3, 41]).
Let
be a ball. A positive constant
Lemma 2.2.
Assume that
Then we have
Proof.
When
Lemma 2.3.
Let
and u solves
Denote
In particular, there holds
Lemma 2.4.
Let
Proof.
We refer the reader to [48, 38] for the proof of Lemma 2.4. For the convenience of the reader, we prove Lemma 2.4 here. Let
Using (2.1) and Lemmas 2.1 and 2.2, we have
Thanks to
If
In the case
If
Multiplying both sides of the above inequality with
Lemma 2.5.
If
Proof.
Applying [47, Lemma 2.6] and choosing
Lemma 2.6.
Let
Proof.
This can be obtained by exploiting the same argument used in [31]. Let
If
If
To estimate the remaining term
Since
By combining all above estimates, multiplying both sides of the above inequality with
Lemma 2.7.
Let
then we have
Proof.
Applying Lemma 2.2 and simple calculations as in proving [36, Lemma 2.2] enable us to conclude the required result, and we omit the details. ∎
Lemma 2.8.
Let
Lemma 2.9.
Let
Proof.
Arguing by interpolation (2.1), the algebra property (2.2) and Lemma 2.2, the proof of Lemma 2.9 consists of an argument similar to the ones in the proofs of Lemmas 2.4–2.6. ∎
Lemma 2.10.
Let
Lemma 2.11.
Let
then we have
3 Proof of Theorem 1.1
To prove the local Gevrey regularity of Theorem 1.1, note that, applying Lemma 2.11 for any given
Borrowing ideas from [14], we can easily prove Theorem 1.1 by a standard appropriate iteration scheme. We omit the details here.
4 Proof of Corollary 1.2: Decay of Besov Norms and Lebesgue Norms
Theorem 1.1 tells us that the solution is locally in the Gevrey regular, that is, the energy bound
As we have showed that the estimate (4.1) holds for the solution
By using the relation between homogeneous Besov spaces and homogeneous Triebel–Lizorkin spaces
By applying (4.3) and (4.1) along the same lines as (4.2), for
By an argument similar to the one in (4.4), there exist positive constants
for
This completes the proof of Corollary 1.2.
5 Proof of Theorem 1.3: Global Existence with Large Vertical Velocity Component
The goal of this section is to present the proof of the global existence of system (1.4) with large vertical velocity component. Let
Lemma 5.1 ([49]).
Let
with
and
where
Lemma 5.2 ([49]).
Let
Lemma 5.3 ([49]).
Let
where
for
The estimate of the pressure ℙ
Taking div to the first equations of system (1.4) yields that
By virtue of
Proposition 5.4.
Let
where
for
Proof.
Applying the operator
where
from which the desired estimate (5.2) follows readily. Finally, according to the definition of Chemin–Lerner norm, we complete the proof. ∎
The estimate of u h u 3
The first equation of (1.4) implies that
and
Proposition 5.5.
Let
and
Proof.
We first prove (5.5). By Applying the operator
Thanks to [17, 37], there exists a positive constant
Similarly proceeding as in [17], from (5.7) and (5.8) we thus infer the inequality
which after time, by integrating over
from which we obtain the desired estimate (5.5).
Finally, as in the proof of (5.9), let us apply
Multiplying (5.10) by
The estimate of n.
Let us consider the second equation of (1.4); we have
Proposition 5.6.
Let
Proof.
Applying the operator
Thanks to [17, Proposition 2.1], there exists a positive constant
It is clear that we have the following homogeneous Bony’s paraproduct decomposition:
Besides, because
Therefore, substituting (5.14)–(5.16) into (5.13), and using an argument for the
Applying the classical estimate on commutators [17] and Lemma 2.1 yields
Now, using the same type of computations as in (5.18), we get
Meanwhile, we obtain
So plugging the above inequalities (5.18)–(5.20) into (5.17) and using Lemma 2.5, we conclude
Then multiplying both sides by
The estimate of c.
Thanks to the second equation of (1.4), we have
Proposition 5.7.
Let
Proof.
Applying the operator
Repeating the process as in Proposition 5.6, from Lemma 2.5 we conclude
Multiplying both sides of the above inequality with
Proof of Theorem 1.3.
If we take
Let
where η is a positive constant which will be determined later on. We define
Applying (5.12), we have
Applying Propositions 5.6 and 5.7, we obtain
Noting the definition of the Besov spaces norms and using the Cauchy–Schwarz inequality
for
Thanks to (5.24), (5.26) and Proposition 5.5, we have
We deduce from (5.24), (5.26) and Proposition 5.5 that
Let
Thanks to (5.23), taking
Applying (5.25) and (5.27)–(5.29), we obtain
Thus (5.30) contradicts the definition (5.23). In view of the standard continuity argument, we conclude that
Funding source: National Science Foundation
Award Identifier / Grant number: 11671185
Award Identifier / Grant number: 11771195
Funding source: Natural Science Foundation of Shandong Province
Award Identifier / Grant number: ZR2017MA041
Funding statement: This work was partially supported by NSF of China (grant nos. 11671185, 11771195) and NSF Shandong Province (grant no. ZR2017MA041).
References
[1] H. Bae, Existence and analyticity of Lei–Lin solution to the Navier–Stokes equations, Proc. Amer. Math. Soc. 143 (2015), no. 7, 2887–2892. 10.1090/S0002-9939-2015-12266-6Suche in Google Scholar
[2] H. Bae, A. Biswas and E. Tadmor, Analyticity and decay estimates of the Navier–Stokes equations in critical Besov spaces, Arch. Ration. Mech. Anal. 205 (2012), no. 3, 963–991. 10.1007/s00205-012-0532-5Suche in Google Scholar
[3] H. Bahouri, J.-Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren Math. Wiss. 343, Springer, Heidelberg, 2011. 10.1007/978-3-642-16830-7Suche in Google Scholar
[4] A. Biswas, Gevrey regularity for a class of dissipative equations with applications to decay, J. Differential Equations 253 (2012), no. 10, 2739–2764. 10.1016/j.jde.2012.08.003Suche in Google Scholar
[5] A. Biswas, V. R. Martinez and P. Silva, On Gevrey regularity of the supercritical SQG equation in critical Besov spaces, J. Funct. Anal. 269 (2015), no. 10, 3083–3119. 10.1016/j.jfa.2015.08.010Suche in Google Scholar
[6] A. Biswas and D. Swanson, Gevrey regularity of solutions to the 3-D Navier–Stokes equations with weighted
[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. 10.1016/S1874-5792(05)80006-0Suche in Google Scholar
[8] M. Chae, K. Kang and J. Lee, Existence of smooth solutions to coupled chemotaxis-fluid equations, Discrete Contin. Dyn. Syst. 33 (2013), no. 6, 2271–2297. 10.3934/dcds.2013.33.2271Suche in Google Scholar
[9] M. Chae, K. Kang and J. Lee, Global existence and temporal decay in Keller–Segel models coupled to fluid equations, Comm. Partial Differential Equations 39 (2014), no. 7, 1205–1235. 10.1080/03605302.2013.852224Suche in Google Scholar
[10] M. Chae, K. Kang and J. Lee, Asymptotic behaviors of solutions for an aerotaxis model coupled to fluid equations, J. Korean Math. Soc. 53 (2016), no. 1, 127–146. 10.4134/JKMS.2016.53.1.127Suche in Google Scholar
[11] J.-Y. Chemin, Théorèmes d’unicité pour le système de Navier–Stokes tridimensionnel, J. Anal. Math. 77 (1999), 27–50. 10.1007/BF02791256Suche in Google Scholar
[12] J.-Y. Chemin, M. Paicu and P. Zhang, Global large solutions to 3-D inhomogeneous Navier–Stokes system with one slow variable, J. Differential Equations 256 (2014), no. 1, 223–252. 10.1016/j.jde.2013.09.004Suche in Google Scholar
[13] L. Chen, G. Lu and X. Luo, Boundedness of multi-parameter Fourier multiplier operators on Triebel–Lizorkin and Besov–Lipschitz spaces, Nonlinear Anal. 134 (2016), 55–69. 10.1016/j.na.2015.12.016Suche in Google Scholar
[14] H. J. Choe and B. Lkhagvasuren, Global existence result for chemotaxis Navier–Stokes equations in the critical Besov spaces, J. Math. Anal. Appl. 446 (2017), no. 2, 1415–1426. 10.1016/j.jmaa.2016.09.050Suche in Google Scholar
[15] H. J. Choe, B. Lkhagvasuren and M. Yang, Wellposedness of the Keller–Segel Navier–Stokes equations in the critical Besov spaces, Commun. Pure Appl. Anal. 14 (2015), no. 6, 2453–2464. 10.3934/cpaa.2015.14.2453Suche in Google Scholar
[16] W. Dai and G. Lu,
[17] R. Danchin, Local theory in critical spaces for compressible viscous and heat-conductive gases, Comm. Partial Differential Equations 26 (2001), no. 7–8, 1183–1233. 10.1081/PDE-100106132Suche in Google Scholar
[18] R. Danchin, Fourier analysis methods for PDEs, Lecture Notes (2005). Suche in Google Scholar
[19] R. Duan, A. Lorz and P. Markowich, Global solutions to the coupled chemotaxis-fluid equations, Comm. Partial Differential Equations 35 (2010), no. 9, 1635–1673. 10.1080/03605302.2010.497199Suche in Google Scholar
[20] C. Foias and R. Temam, Gevrey class regularity for the solutions of the Navier–Stokes equations, J. Funct. Anal. 87 (1989), no. 2, 359–369. 10.1016/0022-1236(89)90015-3Suche in Google Scholar
[21] H. Fujita and T. Kato, On the Navier–Stokes initial value problem. I, Arch. Ration. Mech. Anal. 16 (1964), 269–315. 10.1007/BF00276188Suche in Google Scholar
[22] G. Gui and P. Zhang, Stability to the global large solutions of 3-D Navier–Stokes equations, Adv. Math. 225 (2010), no. 3, 1248–1284. 10.1007/978-3-642-36028-2_2Suche in Google Scholar
[23] J. Huang, M. Paicu and P. Zhang, Global well-posedness of incompressible inhomogeneous fluid systems with bounded density or non-Lipschitz velocity, Arch. Ration. Mech. Anal. 209 (2013), no. 2, 631–682. 10.1007/s00205-013-0624-xSuche in Google Scholar
[24] J. Jiang, H. Wu and S. Zheng, Global existence and asymptotic behavior of solutions to a chemotaxis-fluid system on general bounded domains, Asymptot. Anal. 92 (2015), no. 3–4, 249–258. 10.3233/ASY-141276Suche in Google Scholar
[25] T. Kato, Strong
[26] H. Koch and D. Tataru, Well-posedness for the Navier–Stokes equations, Adv. Math. 157 (2001), no. 1, 22–35. 10.1006/aima.2000.1937Suche in Google Scholar
[27] H. Kozono and M. Yamazaki, Semilinear heat equations and the Navier–Stokes equation with distributions in new function spaces as initial data, Comm. Partial Differential Equations 19 (1994), 959–1014. 10.1080/03605309408821042Suche in Google Scholar
[28] Z. Lei and F. Lin, Global mild solutions of Navier–Stokes equations, Comm. Pure Appl. Math. 64 (2011), no. 9, 1297–1304. 10.1002/cpa.20361Suche in Google Scholar
[29] J. Leray, Sur le mouvement d’un liquide visqueux emplissant l’espace, Acta Math. 63 (1934), no. 1, 193–248. 10.1007/BF02547354Suche in Google Scholar
[30] J.-G. Liu and A. Lorz, A coupled chemotaxis-fluid model: Global existence, Ann. Inst. H. Poincaré Anal. Non Linéaire 28 (2011), no. 5, 643–652. 10.1016/j.anihpc.2011.04.005Suche in Google Scholar
[31] Q. Liu, Gevrey analyticity of solutions to the 3D nematic liquid crystal flows in critical Besov space, Nonlinear Anal. Real World Appl. 31 (2016), 431–451. 10.1016/j.nonrwa.2016.02.008Suche in Google Scholar
[32] A. Lorz, Coupled chemotaxis fluid model, Math. Models Methods Appl. Sci. 20 (2010), no. 6, 987–1004. 10.1142/S0218202510004507Suche in Google Scholar
[33] A. Lorz, A coupled Keller–Segel–Stokes model: Global existence for small initial data and blow-up delay, Commun. Math. Sci. 10 (2012), no. 2, 555–574. 10.4310/CMS.2012.v10.n2.a7Suche in Google Scholar
[34] M. Paicu, Équation anisotrope de Navier–Stokes dans des espaces critiques, Rev. Mat. Iberoam. 21 (2005), no. 1, 179–235. 10.4171/RMI/420Suche in Google Scholar
[35] M. Paicu and P. Zhang, Global solutions to the 3-D incompressible anisotropic Navier–Stokes system in the critical spaces, Comm. Math. Phys. 307 (2011), no. 3, 713–759. 10.1007/s00220-011-1350-6Suche in Google Scholar
[36] M. Paicu and P. Zhang, Global solutions to the 3-D incompressible inhomogeneous Navier–Stokes system, J. Funct. Anal. 262 (2012), no. 8, 3556–3584. 10.1016/j.jfa.2012.01.022Suche in Google Scholar
[37] F. Planchon, Sur un inégalité de type Poincaré, C. R. Acad. Sci. Paris Sér. I Math. 330 (2000), no. 1, 21–23. 10.1016/S0764-4442(00)88138-0Suche in Google Scholar
[38] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Math. Ser. 30, Princeton University Press, Princeton, 1970. 10.1515/9781400883882Suche in Google Scholar
[39] Y. Tao, Boundedness in a chemotaxis model with oxygen consumption by bacteria, J. Math. Anal. Appl. 381 (2011), no. 2, 521–529. 10.1016/j.jmaa.2011.02.041Suche in Google Scholar
[40] I. Tuval, L. Cisneros, C. Dombrowski, W. Wolgemuth, O. Kessler and E. Goldstein, Bacterial swimming and oxygen transport near contact lines, Proc. Natl. Acad. Sci. USA 102 (2005), no. 7, 2277–2282. 10.1073/pnas.0406724102Suche in Google Scholar PubMed PubMed Central
[41] B. Wang, Z. Huo, C. Hao and Z. Guo, Harmonic Analysis Method for Nonlinear Evolution Equations. I, World Scientific Publishing, Hackensack, 2011. 10.1142/8209Suche in Google Scholar
[42] M. Winkler, Global large-data solutions in a chemotaxis-Navier–Stokes system modeling cellular swimming in fluid drops, Comm. Partial Differential Equations 37 (2012), no. 2, 319–351. 10.1080/03605302.2011.591865Suche in Google Scholar
[43] M. Winkler, Stabilization in a two-dimensional chemotaxis-Navier–Stokes system, Arch. Ration. Mech. Anal. 211 (2014), no. 2, 455–487. 10.1007/s00205-013-0678-9Suche in Google Scholar
[44] M. Winkler, Global weak solutions in a three-dimensional chemotaxis–Navier–Stokes system, Ann. Inst. H. Poincaré Anal. Non Linéaire 33 (2016), no. 5, 1329–1352. 10.1016/j.anihpc.2015.05.002Suche in Google Scholar
[45] J. Xiao, Homothetic variant of fractional Sobolev space with application to Navier–Stokes system, Dyn. Partial Differ. Equ. 4 (2007), no. 3, 227–245. 10.4310/DPDE.2007.v4.n3.a2Suche in Google Scholar
[46] J. Xiao, Homothetic variant of fractional Sobolev space with application to Navier–Stokes system revisited, Dyn. Partial Differ. Equ. 11 (2014), no. 2, 167–181. 10.4310/DPDE.2014.v11.n2.a3Suche in Google Scholar
[47] M. Yang, Z. Fu and J. Sun, Existence and Gevrey regularity for a two-species chemotaxis system in homogeneous Besov spaces, Sci. China Math. 60 (2017), no. 10, 1837–1856. 10.1007/s11425-016-0490-ySuche in Google Scholar
[48] M. Yang and J. Sun, Gevrey regularity and existence of Navier–Stokes–Nernst–Planck–Poisson system in critical Besov spaces, Commun. Pure Appl. Anal. 16 (2017), no. 5, 1617–1639. 10.3934/cpaa.2017078Suche in Google Scholar
[49] C. Zhai and T. Zhang, Global well-posedness to the 3-D incompressible inhomogeneous Navier–Stokes equations with a class of large velocity, J. Math. Phys. 56 (2015), no. 9, Article ID 091512. 10.1063/1.4931467Suche in Google Scholar
[50] Q. Zhang, Local well-posedness for the chemotaxis-Navier–Stokes equations in Besov spaces, Nonlinear Anal. Real World Appl. 17 (2014), 89–100. 10.1016/j.nonrwa.2013.10.008Suche in Google Scholar
[51] Q. Zhang and X. Zheng, Global well-posedness for the two-dimensional incompressible chemotaxis-Navier–Stokes equations, SIAM J. Math. Anal. 46 (2014), no. 4, 3078–3105. 10.1137/130936920Suche in Google Scholar
© 2018 Walter de Gruyter GmbH, Berlin/Boston
This work is licensed under the Creative Commons Attribution 4.0 International License.
Artikel in diesem Heft
- Frontmatter
- Sharpened Adams Inequality and Ground State Solutions to the Bi-Laplacian Equation in ℝ4
- Prescribing Gaussian and Geodesic Curvature on Disks
- Existence and Multiplicity Results for Systems of First-Order Differential Equations via the Method of Solution-Regions
- Singularly Perturbed Fractional Schrödinger Equation Involving a General Critical Nonlinearity
- Finite-Dimensionality and Determining Modes of the Global Attractor for 2D Boussinesq Equations with Fractional Laplacian
- Analyticity and Existence of the Keller–Segel–Navier–Stokes Equations in Critical Besov Spaces
- Polynomial Solutions of Equivariant Polynomial Abel Differential Equations
- Existence of Extremal Functions for Higher-Order Caffarelli–Kohn–Nirenberg Inequalities
- A Diffusion Equation with a Variable Reaction Order
- Concentration-Compactness Principle of Singular Trudinger--Moser Inequalities in ℝn and n-Laplace Equations
- Singularly Perturbed Fractional Schrödinger Equations with Critical Growth
- On a Kirchhoff Equation in Bounded Domains
Artikel in diesem Heft
- Frontmatter
- Sharpened Adams Inequality and Ground State Solutions to the Bi-Laplacian Equation in ℝ4
- Prescribing Gaussian and Geodesic Curvature on Disks
- Existence and Multiplicity Results for Systems of First-Order Differential Equations via the Method of Solution-Regions
- Singularly Perturbed Fractional Schrödinger Equation Involving a General Critical Nonlinearity
- Finite-Dimensionality and Determining Modes of the Global Attractor for 2D Boussinesq Equations with Fractional Laplacian
- Analyticity and Existence of the Keller–Segel–Navier–Stokes Equations in Critical Besov Spaces
- Polynomial Solutions of Equivariant Polynomial Abel Differential Equations
- Existence of Extremal Functions for Higher-Order Caffarelli–Kohn–Nirenberg Inequalities
- A Diffusion Equation with a Variable Reaction Order
- Concentration-Compactness Principle of Singular Trudinger--Moser Inequalities in ℝn and n-Laplace Equations
- Singularly Perturbed Fractional Schrödinger Equations with Critical Growth
- On a Kirchhoff Equation in Bounded Domains
