分数阶不可压缩 Navier-Stokes-Coriolis 方程解的整体适定性

Abstract

Abstract:

We consider the Cauchy problem of fractional Navier-Stokes equations with the Coriolis force. Combining the $L^p-L^q$ and $\dot{H}^{\frac{5}{2}-2\alpha}-L^q$ smooth estimates of semiggroup $S$, it is proved that the well-posedness of the fractional Navier-Stokes equations with the Coriolis force and these equations possess a unique global mild solution for arbitrary speed of rotation provided the initial data $u_0$ is small enough in $\dot{H}_{\sigma}^{\frac{5}{2}-2\alpha}(\mathbb{R}^3)$.

Key words: Global well-posedness, Fractional Navier-Stokes equation, Homogeneous Sobolev spaces, Coriolis force

CLC Number:

O174.2

Sun Xiaochun, Wu Yulian, Xu Gaoting. Global Well-Posedness for the Fractional Navier-Stokes Equations with the Coriolis Force[J].Acta mathematica scientia,Series A, 2024, 44(3): 737-745.

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References 24

[1]	Abidi H, Gui G, Zhang P. Well-posedness of 3-D inhomogeneous Navier-Stokes equations with highly oscillatory initial velocity field. J Math Pures Appl, 2013, 100(2): 166-203
[2]	Babin A, Mahalov A, Nicolaenko B. Regularity and integrability of 3D Euler and Navier-Stokes equations for rotating fluids. Asymptot Anal, 1997, 15(2): 103-150
[3]	Bourgain J, Pavlović N. Ill-posedness of the Navier-Stokes equations in a critical space in 3D. J Funct Anal, 2008, 255(9): 2233-2247
[4]	Chemin J F, Desjardins B, Gallagher I, Grenier E. Anisotropy and Dispersion in Rotating Fluids. Amsterdam: North-Holland, 2002
[5]	Ding Y, Sun X. Uniqueness of weak solutions for fractional Navier-Stokes equations. Front Math China, 2015, 10(1): 33-51 doi: 10.1007/s11464-014-0370-x
[6]	Fang D, Han B, Hieber M. Local and global existence results for the Navier-Stokes equations in the rotational framework. Commun Pure Appl Anal, 2015, 14(2): 609-622
[7]	Galdi G P, Silvestre A L. Strong solutions to the Navier-Stokes equations around a rotating obstacle. Arch Ration Mech Anal, 2005, 176(3): 331-350
[8]	Giga Y, Inui K, Mahalov A, et al. Rotating Navier-Stokes equations in $\mathbb{R}^3_+$ with initial data nondecreasing at infinity: The Ekman boundary layer problem. Arch Ration Mech Anal, 2007, 186(2): 177-224
[9]	Giga Y, Inui K, Mahalov A, et al. Uniform global solvability of the rotating Navier-Stokes equations for nondecaying initial data. Indiana Univ Math J, 2008, 57(6): 2775-2791
[10]	Giga Y, Inui K, Matsui S. On the {C}auchy problem for the {N}avier-{S}tokes equations with nondecaying initial data. Hokkaido University Preprint Series in Mathematics, 410: 1-34
[11]	Hieber M, Shibata Y. The Fujita-Kato approach to the Navier-Stokes equations in the rotational framework. Math Z, 2010, 265(2): 481-491
[12]	Iwabuchi T, Takada R. Global solutions for the Navier-Stokes equations in the rotational framework. Arch Ration Math Ann, 2013, 357(2): 727-741
[13]	Iwabuchi T, Takada R. Global well-posedness and ill-posedness for the Navier-Stokes equations with the Coriolis force in function spaces of Besov type. J Funct Anal, 2014, 267(5): 1321-1337
[14]	Kozono H, Ogawa T, Taniuchi Y. Navier-Stokes equations in the Besov space near $L^{\infty}$ and BMO. Kyushu J Math, 2003, 57(2): 303-324
[15]	Majda A. Introduction to PDEs and Waves for the Atmosphere and Ocean. New York: AMS & Courant Institute of Mathematical Sciences at Noew York University, 2003
[16]	Miao C, Yuan B, Zhang B. Well-posedness of the Cauchy problem for the fractional power dissipative equations. Nonlinear Anal, 2008, 68(3): 461-484
[17]	Ru S, Abidin M Z. Global well-posedness of the incompressible fractional Navier-Stokes equations in Fourier-Besov spaces with variable exponents. Comput Math Appl, 2019, 77(4): 1082-1090
[18]	Solonnikov V A. On nonstationary Stokes problem and Navier-Stokes problem in a half-space with initial data nondecreasing at infinity. J Math Sci (NY), 2003, 114(5): 1726-1740
[19]	Sun X, Ding Y. Dispersive effect of the Coriolis force and the local well-posedness for the fractional Navier-Stokes-Coriolis system. J Evol Equ, 2020, 20(2): 335-354
[20]	Sun X, Liu H. Uniqueness of the weak solution to the fractional anisotropic Navier-Stokes equations. Math Methods Appl Sci, 2021, 44(1): 253-264
[21]	Sun X, Liu J. Long time decay of the fractional Navier-Stokes equations in Sobolev-Gevery spaces. J Nonlinear Evol Equ Appl, 2021, 6: 119-135
[22]	Sun X, Liu M, Zhang J. Global well-posedness for the generalized Navier-Stokes-Coriolis equations with highly oscillating initial data. Math Methods Appl Sci, 2023, 46(1): 715-731
[23]	Wu J. The generalized incompressible Navier-Stokes equations in Besov spaces. Dyn Partial Differ Equ, 2004, 1(4): 381-400
[24]	Yu X, Zhai Z. Well-posedness for fractional Navier-Stokes equations in the largest critical spaces $\mathcal{\dot{B}}_{\infty,\infty}^{-(2\beta-1) }(\mathbb{R}^n)$. Math Methods Appl Sci, 2012, 35(6): 676-683

Global Well-Posedness for the Fractional Navier-Stokes Equations with the Coriolis Force

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