This article is devoted to studying the initial-boundary value problem for an ideal polytropic model of non-viscous and compressible gas. We focus our attention on the outflow problem when the flow velocity on the boundary is negative and give a rigorous proof of the asymptotic stability of both the degenerate boundary layer and its superposition with the 3-rarefaction wave under some smallness conditions. New weighted energy estimates are introduced, and the trace of the density and velocity on the boundary are handled by some subtle analysis. The decay properties of the boundary layer and the smooth rarefaction wave also play an important role.
Lili FAN
,
Meichen HOU
. ASYMPTOTIC STABILITY OF A BOUNDARY LAYER AND RAREFACTION WAVE FOR THE OUTFLOW PROBLEM OF THE HEAT-CONDUCTIVE IDEAL GAS WITHOUT VISCOSITY[J]. Acta mathematica scientia, Series B, 2020
, 40(6)
: 1627
-1652
.
DOI: 10.1007/s10473-020-0602-y
[1] Fan L, Liu H, Wang T, Zhao H. Inflow problem for the one-dimensional compressible Navier-Stokes equations under large initial perturbation. J Diff Eqns, 2014, 257:3521-3553
[2] Fan L, Liu H, Yin H. Decay estimates of planar stationary waves for damped wave equations with nonlinear convection in multi-dimensional half space. Acta Mathematica Scientia, 2011, 31B(4):1389-1401
[3] Fan L, Matsumura A. Asymptotic stability of a composite wave of two viscous shock waves for the equation of non-viscous and heat-conductive ideal gas. J Diff Eqns, 2015, 258:1129-1157
[4] Fan L, Gong G Q, Shao S J. Asymptotic stability of viscous contact wave and rarefaction waves for the system of heat-conductive ideal gas without viscosity. Anal Appl, 2019, 258:211-234
[5] He L, Tang S, Wang T. Stability of viscous shock waves for the one-dimensional compressible Navier-Stokes equations with density-dependent viscosity. Acta Mathematica Scientia, 2016, 36B(1):34-48
[6] Hong H, Huang F. Asymptotic behavior of solutions toward the superposition of contact discontinuity and shock wave for compressible Navier-Stokes equations with free boundary. Acta Mathematica Scientia, 2012, 32B(1):389-412
[7] Huang F, Li J, Shi X. Asymptotic behavior of solutions to the full compressible Navier-Stokes equations in the half space. Commun Math Sci, 2010, 8:639-654
[8] Huang F, Matsumura A. Stability of a composite wave of two viscous shock waves for the full compresible Navier-Stokes equation. Comm Math Phys, 2009, 289:841-861
[9] Huang F, Matsumura A, Shi X. Viscous shock wave and boundary layer solution to an inflow problem for compressible viscous gas. Comm Math Phys, 2003, 239:261-285
[10] Huang F, Matsumura A, Shi X. On the stability of contact discontinuity for compressible Navier-Stokes equations with free boundary. Osaka J Math, 2004, 41:193-210
[11] Huang F, Matsumura A, Xin Z. Stability of Contact discontinuties for the 1-D Compressible Navier-Stokes equations. Arch Ration Mech Anal, 2005, 179:55-77
[12] Huang F, Qin X. Stability of boundary layer and rarefactionwave to an outflow problem for compressible Navier-Stokes equations under large perturbation. J Diff Eqns, 2009, 246:4077-4096
[13] Huang F. Yang T, Xin Z. Contact discontinuity with general perturbations for gas motions. Adv Math, 2008, 219:1246-1297
[14] Kawashima S, Matsumura A. Asymptotic stability of travelling wave solutions of systems for onedimensional gas motion. Comm Math Phys, 1985, 101:97-127
[15] Kawashima S, Nakamura T, Nishibata S, Zhu P. Stationary waves to viscous heat-conductive gases in half space:existence, stability and convergence rate. Math Models Methods Appl Sci, 2011, 20:2201-2235
[16] Kawashima S, Nishibata S, Zhu P. Asymptotic stability of the stationary solution to the compressible Navier-Stokes equations in the half space. Commun Math Phys, 2003, 240:483-500
[17] Kawashima S, Zhu P. Asymptotic stability of rarefaction wave for the Navier-Stokes equations for a compressible fluid in the half space. Arch Ration Mech Anal, 2009, 194:105-132
[18] Liu T. Shock wave for compresible Navier-Stokes equations are stable. Comm Math Phys, 1986, 50:565-594
[19] Matsumura A. Inflow and outflow problems in the half space for a one-dimensional isentropic model system of compressible viscous gas. Methods Appl Anal, 2001, 8:645-666; IMS Conference on Differential Equations from Mechanics. Hong Kong, 1999
[20] Matsumura A. Large-time behavior of solutions for a one-dimensional system of non-viscous and heatconductive ideal gas. Private Communication, 2016
[21] Matsumura A, Mei M. Convergence to travelling fronts of solutions of the p-system with viscosity in the presence of a boundary. Arch Ration Mech Anal, 1999, 146:1-22
[22] Matsumura A, Nishihara K. Large time behaviors of solutions to an inflow problem in the half space for a one-dimensional system of compressible viscous gas. Comm Math Phys, 2001, 222:449-474
[23] Nakamura T, Nishibata S. Energy estimate for a linear symmetric hyperbolic-parabolic system in half line. Kinet Relat Models, 2013, 6(4):883-892
[24] Nakamura T, Nishibata S. Existence and asymptotic stability of stationary waves for symmetric hyperbolicparabolic systems in half line. Math Models Methods Appl Sci, 2017, 27:2071-2110
[25] Nishihara K, Yang T, Zhao H. Nonlinear stability of strong rarefaction waves for compressible Navier-Stokes equations. SIAM J Math Anal, 2004, 35:1561-1597
[26] Qin X. Large-time behaviour of solution to the outflow problem of full compressible Navier-Stokes equations. Nonlinearity, 2011, 24:1369-1394
[27] Qin X, Wang Y. Stability of wave patterns to the inflow problem of full compressible Navier-Stokes equations. SIAM J Math Anal, 2009, 41:2057-2087
[28] Qin X, Wang Y. Large-time behavior of solutions to the inflow problem of full compressible Navier-Stokes equations. SIAM J Math Anal, 2011, 43:341-366
[29] Qin X, Wang T, Wang Y. Global stability of wave patterns for compressible Navier-Stokes system with free boundary. Acta Mathematica Scientia, 2016, 36B(4):1192-1214
[30] Smoller J. Shock Wave and Reaction-Diffusion Equations. 2nd edn. New York:Springer-Verlag, 1994
[31] Wan L, Wang T, Zou Q. Stability of stationary solutions to the outflow problem for full compressible Navier-Stokes equations with large initial perturbation. Nonlinearity, 2016, 29:1329-1354
[32] Wang T, Zhao H. One-dimensional compressible heat-conducting gas with temperature-dependent viscosity. Math Models Methods Appl Sci, 2016, 26:2237-2275
