Well-posedness of 3D incompressible generalized Navier-Stokes system in Fourier-Triebel-Lizorkin spaces

doi:10.7523/j.ucas.2021.0065

Abstract

Abstract: In this paper, we consider the Cauchy problem for the 3D incompressible generalized Navier-Stokes system and study the well-posedness in critical Fourier-Triebel-Lizorkin spaces $\widehat {\dot F}_{p,q}^{4 - \alpha - \frac{3}{p}}$($\mathbb{R}^3$). Making use of Fourier localization method and Banach fixed point theorem, we proved that if $p > \frac{3}{{5 - \alpha }}$, q ≥ 1, the system is locally well-posed for large initial data as well as global well-posed for small initial data. Also we established same result for $p = \frac{3}{{5 - \alpha }}$，q∈[$\frac{3}{{5 - \alpha }}$，$\frac{6}{{5 - \alpha }}$].

Key words: Navier-Stokes system, Fourier-Triebel-Lizorkin spaces, global well-posedness, local well-posedness

CLC Number:

O175.2

MIN Dezai, WU Gang, YAO Zhuoya. Well-posedness of 3D incompressible generalized Navier-Stokes system in Fourier-Triebel-Lizorkin spaces[J]. Journal of University of Chinese Academy of Sciences, 2023, 40(2): 145-154.

References

[1] Jean L. Sur le mouvement d'un liquide visqueux emplissant l'espace[J]. Acta Mathematica, 1934, 63(1):193-248.
[2] Fujita H, Kato T. On the Navier-Stokes initial value problem, I[J]. Archive for Rational Mechanics Analysis, 1963, 16:269-315.
[3] Kato T. Strong L^p-solutions of the Navier-Stokes equation in R^m, with applications to weak solutions[J]. Mathematische Zeitschrift, 1984, 187(4):471-480.
[4] Planchon F. Global strong solutions in Sobolev or Lebesgue spaces to the incompressible Navier-Stokes equations in R³[J]. Annales De l'Institut Henri Poincare' C, Analyse Non Line'aire, 1996, 13(3):319-336.
[5] Cannone M. A generalization of a theorem by Kato on Navier-Stokes equations[J]. Revista Matemática Iberoamericana, 1997, 13(3):515-542.
[6] Chemin J Y. Théorèmes d' unicité pour le système de navier-stokes tridimensionnel[J]. Journal D'analyse Mathématique, 1999, 77(1):27-50.
[7] Koch H, Tataru D. Well-posedness for the Navier-Stokes equations[J]. Advances in Mathematics, 2001, 157(1):22-35.
[8] Bourgain J,Pavlovic'N. Ill-posedness of the Navier-Stokes equations in a critical space in 3D[J]. Journal of Functional Analysis, 2008, 255(9):2233-2247.
[9] Yoneda T. Ill-posedness of the 3D-Navier-Stokes equations in a generalized Besov space near BMO^-1[J]. Journal of Functional Analysis, 2010, 258(10):3376-3387.
[10] Lions J L. Quelques méthodes de résolution des problèmes Aux limites non linéaries (French)[M]. Paris:Dunod Gauthier-Villars, 1969.
[11] Katz N H, Pavlovic'N.A cheap Caffarelli-Kohn-Nirenberg inequality for the Navier-Stokes equation with hyper-dissipation[J]. Geometric & Functional Analysis GAFA, 2002, 12(2):355-379.
[12] Wu J H. The generalized incompressible Navier-Stokes equations in Besov spaces[J]. Dynamics of Partial Differential Equations, 2004, 1(4):381-400.
[13] Wu J H. Lower bounds for an integral involving fractional laplacians and the generalized Navier-Stokes equations in Besov spaces[J]. Communications in Mathematical Physics, 2006, 263(3):803-831.
[14] Wang W H, Wu G. Global mild solution of the generalized Navier-Stokes equations with the Coriolis force[J]. Applied Mathematics Letters, 2018, 76:181-186.
[15] Yao X H, Deng C. Well-posedness and ill-posedness for the 3D generalized Navier-Stokes equations in $\dot F_{\frac{3}{\alpha } - 1}^{ - \alpha ,r}$[J]. Discrete and Continuous Dynamical Systems, 2014, 34(2):437-459.
[16] Cannone M, Karch G. Smooth or singular solutions to the Navier-Stokes system?[J]. Journal of Differential Equations, 2004, 197(2):247-274.
[17] Lei Z, Lin F H. Global mild solutions of Navier-Stokes equations[J]. Communications on Pure and Applied Mathematics, 2011, 64(9):1297-1304.
[18] Bahouri H, Chemin J Y, Danchin R. Fourier analysis and nonlinear partial differential equations[M]. Berlin, Heidelberg:Springer Berlin Heidelberg, 2011.
[19] Nie Y, Zheng X X. Ill-posedness of the 3D incompressible hyperdissipative Navier-Stokes system in critical Fourier-Herz spaces[J]. Nonlinearity, 2018, 31(7):3115-3150.
[20] Nan Z J, Zheng X X. Existence and uniqueness of solutions for Navier-Stokes equations with hyper-dissipation in a large space[J]. Journal of Differential Equations, 2016, 261(6):3670-3703.
[21] Cannone M. Harmonic analysis tools for solving the incompressible Navier-Stokes equations[J]. Handbook of Mathematical Fluid Dynamics, 2005, 3:161-244.