Optimal time decay of the compressible micropolar fluids

Abstract This paper primarily studies the large-time behavior of solutions to the Cauchy problem on the compressible micropolar fluid system which is a generalization of the classical Navier–Stokes system. The asymptotic stability of the steady state with the strictly positive constant density, the vanishing velocity, and micro-rotational velocity is established under small perturbation in regular Sobolev space. Moreover, it turns out that both the density and the velocity tend time-asymptotically to the corresponding equilibrium state with rate ( 1 + t ) − 3 / 4 in L 2 and the micro-rotational velocity also tends to the equilibrium state with the faster rate ( 1 + t ) − 5 / 4 in L 2 norm. The proof is based on the spectrum analysis and time-weighted energy estimate.

[1]  Nermina Mujaković,et al.  Non-homogeneous boundary value problem for one-dimensional compressible viscous micropolar fluid model: a global existence theorem , 2009 .

[2]  Michael Taylor,et al.  Partial Differential Equations I: Basic Theory , 1996 .

[3]  Mingtao,et al.  Global Strong Solutions for the Viscous, Micropolar, Compressible Flow , 2011 .

[4]  Nermina Mujaković,et al.  One-dimensional flow of a compressible viscous micropolar fluid: a local existence theorem , 1998 .

[5]  Mingtao Chen.  Global Well-Posedness of the 2D Incompressible Micropolar Fluid Flows with Partial Viscosity and Angular Viscosity , 2013 .

[6]  Nermina Mujaković,et al.  Nonhomogeneous Boundary Value Problem for One-Dimensional Compressible Viscous Micropolar Fluid Model: Regularity of the Solution , 2008 .

[7]  Changjiang Zhu,et al.  Darcy's law and diffusion for a two-fluid Euler-Maxwell system with dissipation , 2014, 1403.2528.

[8]  Changjiang Zhu,et al.  OPTIMAL DECAY RATES TO CONSERVATION LAWS WITH DIFFUSION-TYPE TERMS OF REGULARITY-GAIN AND REGULARITY-LOSS , 2011, 1104.1271.

[9]  C. Miao,et al.  Global well-posedness for the micropolar fluid system in critical Besov spaces , 2010, 1008.0219.

[10]  Ivan Dražić,et al.  3-D flow of a compressible viscous micropolar fluid with spherical symmetry: a global existence theorem , 2012 .

[11]  Nermina Mujaković,et al.  Non-homogeneous boundary value problem for one-dimensional compressible viscous micropolar fluid model: a local existence theorem , 2007 .

[12]  Giovanni P. Galdi,et al.  A note on the existence and uniqueness of solutions of the micropolar fluid equations , 1977 .

[13]  A. Eringen,et al.  THEORY OF MICROPOLAR FLUIDS , 1966 .

[14]  Shuichi Kawashima,et al.  Large-time behaviour of solutions to hyperbolic–parabolic systems of conservation laws and applications , 1987, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[15]  Takaaki Nishida,et al.  The Initial Value Problem for the Equations of Motion of compressible Viscous and Heat-conductive Fluids. , 1979 .

[16]  Renjun Duan.  Green's function and large time behavior of the Navier-Stokes-Maxwell system , 2011 .

[17]  N. Mujakovic.  One-dimensional flow of a compressible viscous micropolar fluid: The Cauchy problem , 2005 .

[18]  Jianwen Zhang,et al.  Global weak solutions of 3D compressible micropolar fluids with discontinuous initial data and vacuum , 2015 .

[19]  N. Mujakovic.  GLOBAL IN TIME ESTIMATES FOR ONE-DIMENSIONAL COMPRESSIBLE VISCOUS MICROPOLAR FLUID MODEL , 2005 .

[20]  Takaaki Nishida,et al.  Initial boundary value problems for the equations of motion of compressible viscous and heat-conductive fluids , 1983 .

[21]  Jianwen Zhang,et al.  Blowup criterion for the three-dimensional equations of compressible viscous micropolar fluids with vacuum , 2013 .

[22]  Shuichi Kawashima,et al.  Systems of a Hyperbolic-Parabolic Composite Type, with Applications to the Equations of Magnetohydrodynamics , 1984 .

[23]  Ivan Dražić,et al.  3-D flow of a compressible viscous micropolar fluid with spherical symmetry: Large time behavior of the solution , 2015 .

[24]  Renjun Duan.  Global Smooth Flows for the Compressible Euler-Maxwell System: Relaxation Case , 2010, 1006.3606.

[25]  Ivan Dražić,et al.  3-D flow of a compressible viscous micropolar fluid with spherical symmetry: uniqueness of a generalized solution , 2014 .

[26]  Takaaki Nishida,et al.  The initial value problem for the equations of motion of viscous and heat-conductive gases , 1980 .