Optimal decay for the compressible Navier-Stokes equations without additional smallness assumptions

Journal of Differential Equations - Tập 274 - Trang 543-575 - 2021
Zhouping Xin1, Jiang Xu2
1The Institute of Mathematical Sciences and Department of Mathematics, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong
2Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 211106, PR China

Tài liệu tham khảo

Bahouri, 2011, Fourier Analysis and Nonlinear Partial Differential Equations, vol. 343 Cannone, 1997, A generalization of a theorem by Kato on Navier-Stokes equations, Rev. Mat. Iberoam., 13, 515, 10.4171/RMI/229 Chemin, 1999, Théorèmes d'unicité pour le systèm de Navier-Stokes tridimensionnel, J. Anal. Math., 77, 27, 10.1007/BF02791256 Charve, 2010, A global existence result for the compressible Navier-Stokes equations in the critical Lp framework, Arch. Ration. Mech. Anal., 198, 233, 10.1007/s00205-010-0306-x Chemin, 1995, Flot de champs de vecteurs non lipschitziens et équations de Navier-Stokes, J. Differ. Equ., 121, 314, 10.1006/jdeq.1995.1131 Chen, 2010, Global well-posedness for compressible Navier-Stokes equations with highly oscillating initial velocity, Commun. Pure Appl. Math., 63, 1173, 10.1002/cpa.20325 Danchin, 2000, Global existence in critical spaces for compressible Navier-Stokes equations, Invent. Math., 141, 579, 10.1007/s002220000078 Danchin, 2016, Fourier analysis methods for the compressible Navier-Stokes equations Danchin, 2010, On the well-posedness of the incompressible density-dependent Euler equations in the Lp framework, J. Differ. Equ., 248, 2130, 10.1016/j.jde.2009.09.007 Danchin, 2005, On the uniqueness in critical spaces for compressible Navier-Stokes equations, Nonlinear Differ. Equ. Appl., 12, 111, 10.1007/s00030-004-2032-2 Danchin, 2016, The incompressible limit in Lp type critical spaces, Math. Ann., 366, 1365, 10.1007/s00208-016-1361-x Danchin, 2017, Optimal time-decay estimates for the compressible Navier-Stokes equations in the critical Lp framework, Arch. Ration. Mech. Anal., 224, 53, 10.1007/s00205-016-1067-y Feireisl, 2001, On the existence of globally defined weak solutions to the Navier-Stokes equations, J. Math. Fluid Mech., 3, 358, 10.1007/PL00000976 Fiszdon, 1983, Existence and uniqueness of solutions of the initial boundary value problem for the flow of a baratropic viscous fluid, local in time, Arch. Mech., 35, 497 Fujita, 1964, On the Navier-Stokes initial value problem I, Arch. Ration. Mech. Anal., 16, 269, 10.1007/BF00276188 Giga, 2003, On regularizing-decay rate estimates for solutions to the Navier-Stokes initial value problem, Nonlinear Anal., 1, 549 Guo, 2012, Decay of dissipative equations and negative Sobolev spaces, Commun. Partial Differ. Equ., 37, 2165, 10.1080/03605302.2012.696296 Haspot, 2011, Existence of global strong solutions in critical spaces for barotropic viscous fluids, Arch. Ration. Mech. Anal., 202, 427, 10.1007/s00205-011-0430-2 Hoff, 1995, Global solutions of the Navier-Stokes equations for multidimensional compressible flow with discontinuous initial data, J. Differ. Equ., 120, 215, 10.1006/jdeq.1995.1111 Hoff, 1995, Multidimensional diffusion waves for the Navier-Stokes equations of compressible flow, Indiana Univ. Math. J., 44, 604, 10.1512/iumj.1995.44.2003 Huang, 2012, Global well-posedness of classical solutions with large oscillations and vacuum to the three-dimensional isentropic compressible Navier-Stokes equations, Commun. Pure Appl. Math., 65, 549, 10.1002/cpa.21382 Jiang, 2003, Axisymmetric solutions of the 3D Navier-Stokes equations for compressible isentropic fluids, J. Math. Pures Appl., 82, 949, 10.1016/S0021-7824(03)00015-1 Kagei, 2002, On large time behavior of solutions to the compressible Navier-Stokes equations in the half space in R3, Arch. Ration. Mech. Anal., 165, 89, 10.1007/s00205-002-0221-x Kagei, 2005, Asymptotic behavior of solutions of the compressible Navier-Stokes equations on the half space, Arch. Ration. Mech. Anal., 177, 231, 10.1007/s00205-005-0365-6 Kobayashi, 2002, Some estimates of solutions for the equations of motion of compressible viscous fluid in an exterior domain in R3, J. Differ. Equ., 184, 587, 10.1006/jdeq.2002.4158 Kobayashi, 1999, Decay estimates of solutions for the equations of motion of compressible viscous and heat-conductive gases in an exterior domain of R3, Commun. Math. Phys., 200, 621, 10.1007/s002200050543 Kozono, 1994, Semilinear heat equations and the Navier-Stokes equations with distributions in new function spaces as initial data, Commun. Partial Differ. Equ., 19, 959, 10.1080/03605309408821042 Li, 2019, Global well-posedness and large time asymptotic behavior of classical solutions to the compressible Navier-Stokes equations with vacuum, Ann. PDE, 5, 10.1007/s40818-019-0064-5 Lions, 1998, Mathematical Topics in Fluid Mechanics. Vol. 2. Compressible Models, vol. 10 Liu, 1998, The pointwise estimates of diffusion waves for the Navier-Stokes equations in odd multi-dimensions, Commun. Math. Phys., 196, 145, 10.1007/s002200050418 Matsumura, 1979, The initial value problem for the equation of motion of compressible viscous and heat-conductive fluids, Proc. Jpn. Acad., Ser. A, 55, 337, 10.3792/pjaa.55.337 Matsumura, 1980, The initial value problem for the equations of motion of viscous and heat-conductive gases, J. Math. Kyoto Univ., 20, 67, 10.1215/kjm/1250522322 Mitrinović, 1994 Nash, 1962, Le problème de Cauchy pour les équations différentielles d'un fluide général, Bull. Soc. Math. Fr., 90, 487, 10.24033/bsmf.1586 Okita, 2014, Optimal decay rate for strong solutions in critical spaces to the compressible Navier-Stokes equations, J. Differ. Equ., 257, 3850, 10.1016/j.jde.2014.07.011 Paicu, 2012, Global solutions to the 3-D incompressible inhomogeneous Navier-Stokes system, J. Funct. Anal., 262, 3556, 10.1016/j.jfa.2012.01.022 Ponce, 1985, Global existence of small solution to a class of nonlinear evolution equations, Nonlinear Anal. TMA, 9, 339, 10.1016/0362-546X(85)90001-X Serrin, 1959, On the uniqueness of compressible fluid motion, Arch. Ration. Mech. Anal., 3, 271, 10.1007/BF00284180 Sohinger, 2014, The Boltzmann equation, Besov spaces, and optimal time decay rates in Rxn, Adv. Math., 261, 274, 10.1016/j.aim.2014.04.012 Solonnikov, 1973, Estimates of solutions to a nonstationary Navier-Stokes system, Zap. Nauchn. Semin. LOMI, 38, 153 Strain, 2006, Almost exponential decay near Maxwellian, Commun. Partial Differ. Equ., 31, 417, 10.1080/03605300500361545 Vaigant, 1995, On the existence of global solutions of two-dimensional Navier-Stokes of a compressible viscous fluids, Sib. Math. J., 36, 1108, 10.1007/BF02106835 Valli, 1982, An existence theorem for compressible viscous fluids, Ann. Mat. Pura Appl., 130, 197, 10.1007/BF01761495 Xin, 1998, Blowup of smooth solutions to the compressible Navier-Stokes equation with compact density, Commun. Pure Appl. Math., 51, 229, 10.1002/(SICI)1097-0312(199803)51:3<229::AID-CPA1>3.0.CO;2-C Xin, 2013, On blowup of clasical solutions to the compressible Navier-Stokes equations, Commun. Math. Phys., 321, 529, 10.1007/s00220-012-1610-0 Xu, 2019, A low-frequency assumption for optimal time-decay estimates to the compressible Navier-Stokes equations, Commun. Math. Phys., 371, 525, 10.1007/s00220-019-03415-6 Xu, 2015, The optimal decay estimates on the framework of Besov spaces for generally dissipative systems, Arch. Ration. Mech. Anal., 218, 275, 10.1007/s00205-015-0860-3 Zeng, 1994, L1 Asymptotic behavior of compressible isentropic viscous 1-D flow, Commun. Pure Appl. Math., 47, 1053, 10.1002/cpa.3160470804 Zhai, 2017, Global well-posedness for the 3D incompressible inhomogeneous Navier-Stokes equations and MHD equations, J. Differ. Equ., 262, 1359, 10.1016/j.jde.2016.10.016