×

zbMATH — the first resource for mathematics

Large time behavior of isentropic compressible Navier-Stokes system in \(\mathbb R^3\). (English) Zbl 1214.35047
Summary: We consider the long-time behavior and optimal decay rates of global strong solution to three-dimensional isentropic compressible Navier-Stokes (CNS) system in the present paper. When the regular initial data also belong to some Sobolev space \(H^l(\mathbb R^3)\cap\dot B_{1,\infty}^{-s}(\mathbb R^3)\) with \(l\geq 4\) and \(s\in [0,1]\), we show that the global solution to the CNS system converges to the equilibrium state at a faster decay rate in time. In particular, the density and momentum converge to the equilibrium state at the rates \((1+t)^{-3/4-s/2}\) in the \(L^2\)-norm or \((1+t)^{-3/2-s/2}\) in the \(L^\infty\)-norm, respectively, which are shown to be optimal for the CNS system.

MSC:
35Q30 Navier-Stokes equations
76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
35B40 Asymptotic behavior of solutions to PDEs
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Matsumura, The initial value problem for the equation of motion of compressible viscous and heat-conductive fluids, Proceedings of the Japan Academy, Series-A 55 pp 337– (1979) · Zbl 0447.76053 · doi:10.3792/pjaa.55.337
[2] Matsumura, The initial value problem for the equation of motion of viscous and heat-conductive gases, Journal of Mathematics of Kyoto University 20 pp 67– (1980) · Zbl 0429.76040
[3] Ponce, Global existence of small solution to a class of nonlinear evolution equations, Nonlinear Analysis 9 pp 339– (1985) · Zbl 0576.35023 · doi:10.1016/0362-546X(85)90001-X
[4] Zeng, L1 asymptotic behavior of compressible isentropic viscous 1-D flow, Communications on Pure and Applied Mathematics 47 pp 1053– (1994) · Zbl 0807.35110 · doi:10.1002/cpa.3160470804
[5] Hoff, Multi-dimensional diffusion waves for the Navier-Stokes equations of compressible flow, Indiana University Mathematics Journal 44 pp 603– (1995) · Zbl 0842.35076 · doi:10.1512/iumj.1995.44.2003
[6] Hoff, Pointwise decay estimates for multidimensional Navier-Stokes diffusion waves, Zeitschrift für Angewandte Mathematik und Physik 48 pp 597– (1997) · Zbl 0882.76074 · doi:10.1007/s000330050049
[7] Liu, The pointwise estimates of diffusion waves for the Navier-Stokes equations in odd multi-dimensions, Communications in Mathematical Physics 196 pp 145– (1998) · Zbl 0912.35122 · doi:10.1007/s002200050418
[8] Li, The Green’s function of the Navier-Stokes equations for gas dynamics in \(\mathbb{R}\)3, Communications in Mathematical Physics 257 pp 579– (2005) · Zbl 1075.76053 · doi:10.1007/s00220-005-1351-4
[9] Kobayashi, Some estimates of solutions for the equations of motion of compressible viscous fluid in an exterior domain in \(\mathbb{R}\)3, Journal of Differential Equations 184 pp 587– (2002) · Zbl 1069.35051 · doi:10.1006/jdeq.2002.4158
[10] Kobayashi, Decay estimates of solutions for the equations of motion of compressible viscous and heat-conductive gases in an exterior domain in \(\mathbb{R}\)3, Communications in Mathematical Physics 200 pp 621– (1999) · Zbl 0921.35092 · doi:10.1007/s002200050543
[11] Kagei, On large time behavior of solutions to the compressible Navier-Stokes equations in the half space in \(\mathbb{R}\)3, Archive for Rational Mechanics and Analysis 165 pp 89– (2002) · Zbl 1016.35055 · doi:10.1007/s00205-002-0221-x
[12] Kagei, Asymptotic behavior of solutions to the compressible Navier-Stokes equations on the half space, Archive for Rational Mechanics and Analysis 177 pp 231– (2005) · Zbl 1098.76062 · doi:10.1007/s00205-005-0365-6
[13] Kagei, Large time behavior of solutions to the compressible Navier-Stokes equation in an infinite layer, Hiroshima Mathematical Journal 38 (1) pp 95– (2008) · Zbl 1151.35072
[14] Li, Optimal decay rate of the compressible Navier-Stokes-Poisson system in \(\mathbb{R}\)3, Archive for Rational Mechanics and Analysis 196 (2) pp 681– (2010) · Zbl 1205.35201 · doi:10.1007/s00205-009-0255-4
[15] Matsumura, Initial-boundary value problems for the equations of motion of compressible viscous and heat-conductive fluids, Communications in Mathematical Physics 89 (4) pp 445– (1983) · Zbl 0543.76099 · doi:10.1007/BF01214738
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.