Asymptotic stability of a boundary layer and rarefaction wave for the outflow problem of the heat-conductive ideal gas without viscosity. (English) Zbl 1499.35498
Summary: 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.
MSC:
| 35Q35 | PDEs in connection with fluid mechanics |
| 35B35 | Stability in context of PDEs |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35B65 | Smoothness and regularity of solutions to PDEs |
| 35M33 | Initial-boundary value problems for mixed-type systems of PDEs |
| 76L05 | Shock waves and blast waves in fluid mechanics |
| 76N15 | Gas dynamics (general theory) |
References:
| [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, 257, 3521-3553 (2014) · Zbl 1301.35106 · doi:10.1016/j.jde.2014.07.001 |
| [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, 31B, 4, 1389-1401 (2011) · Zbl 1249.35197 |
| [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, 258, 1129-1157 (2015) · Zbl 1309.35047 · doi:10.1016/j.jde.2014.10.010 |
| [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, 258, 211-234 (2019) · Zbl 1417.35118 · doi:10.1142/S0219530518500239 |
| [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, 36B, 1, 34-48 (2016) · Zbl 1363.35034 · doi:10.1016/S0252-9602(15)30076-X |
| [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, 32B, 1, 389-412 (2012) · Zbl 1265.35247 · doi:10.1016/S0252-9602(12)60025-3 |
| [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, 8, 639-654 (2010) · Zbl 1213.35101 · doi:10.4310/CMS.2010.v8.n3.a2 |
| [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, 289, 841-861 (2009) · Zbl 1172.35054 · doi:10.1007/s00220-009-0843-z |
| [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, 239, 261-285 (2003) · Zbl 1048.35083 · doi:10.1007/s00220-003-0874-9 |
| [10] | Huang, F.; Matsumura, A.; Shi, X., On the stability of contact discontinuity for compressible Navier-Stokes equations with free boundary, Osaka J Math, 41, 193-210 (2004) · Zbl 1062.35066 |
| [11] | Huang, F.; Matsumura, A.; Xin, Z., Stability of Contact discontinuties for the 1-D Compressible Navier-Stokes equations, Arch Ration Mech Anal, 179, 55-77 (2005) · Zbl 1079.76032 · doi:10.1007/s00205-005-0380-7 |
| [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, 246, 4077-4096 (2009) · Zbl 1167.35032 · doi:10.1016/j.jde.2009.01.017 |
| [13] | Huang, F.; Yang, T.; Xin, Z., Contact discontinuity with general perturbations for gas motions, Adv Math, 219, 1246-1297 (2008) · Zbl 1155.35068 · doi:10.1016/j.aim.2008.06.014 |
| [14] | Kawashima, S.; Matsumura, A., Asymptotic stability of travelling wave solutions of systems for one-dimensional gas motion, Comm Math Phys, 101, 97-127 (1985) · Zbl 0624.76095 · doi:10.1007/BF01212358 |
| [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, 20, 2201-2235 (2011) · Zbl 1213.35104 · doi:10.1142/S0218202510004908 |
| [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, 240, 483-500 (2003) · Zbl 1038.35057 · doi:10.1007/s00220-003-0909-2 |
| [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, 194, 105-132 (2009) · Zbl 1273.76353 · doi:10.1007/s00205-008-0191-8 |
| [18] | Liu, T., Shock wave for compresible Navier-Stokes equations are stable, Comm Math Phys, 50, 565-594 (1986) · Zbl 0617.76069 |
| [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, 8, 645-666 (2001) · Zbl 1161.76555 |
| [20] | Matsumura A. Large-time behavior of solutions for a one-dimensional system of non-viscous and heat-conductive 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, 146, 1-22 (1999) · Zbl 0957.76072 · doi:10.1007/s002050050134 |
| [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, 222, 449-474 (2001) · Zbl 1018.76038 · doi:10.1007/s002200100517 |
| [23] | Nakamura, T.; Nishibata, S., Energy estimate for a linear symmetric hyperbolic-parabolic system in half line, Kinet Relat Models, 6, 4, 883-892 (2013) · Zbl 1284.35063 · doi:10.3934/krm.2013.6.883 |
| [24] | Nakamura, T.; Nishibata, S., Existence and asymptotic stability of stationary waves for symmetric hyperbolic-parabolic systems in half line, Math Models Methods Appl Sci, 27, 2071-2110 (2017) · Zbl 1387.35054 · doi:10.1142/S0218202517500397 |
| [25] | Nishihara, K.; Yang, T.; Zhao, H., Nonlinear stability of strong rarefaction waves for compressible Navier-Stokes equations, SIAM J Math Anal, 35, 1561-1597 (2004) · Zbl 1065.35057 · doi:10.1137/S003614100342735X |
| [26] | Qin, X., Large-time behaviour of solution to the outflow problem of full compressible Navier-Stokes equations, Nonlinearity, 24, 1369-1394 (2011) · Zbl 1223.35074 · doi:10.1088/0951-7715/24/5/001 |
| [27] | Qin, X.; Wang, Y., Stability of wave patterns to the inflow problem of full compressible Navier-Stokes equations, SIAM J Math Anal, 41, 2057-2087 (2009) · Zbl 1200.35219 · doi:10.1137/09075425X |
| [28] | Qin, X.; Wang, Y., Large-time behavior of solutions to the inflow problem of full compressible Navier-Stokes equations, SIAM J Math Anal, 43, 341-366 (2011) · Zbl 1241.35022 · doi:10.1137/100793463 |
| [29] | Qin, X.; Wang, T.; Wang, Y., Global stability of wave patterns for compressible Navier-Stokes system with free boundary, Acta Mathematica Scientia, 36B, 4, 1192-1214 (2016) · Zbl 1363.35037 · doi:10.1016/S0252-9602(16)30062-5 |
| [30] | Smoller, J., Shock Wave and Reaction-Diffusion Equations (1994), New York: Springer-Verlag, New York · Zbl 0807.35002 · doi:10.1007/978-1-4612-0873-0 |
| [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, 29, 1329-1354 (2016) · Zbl 1338.76109 · doi:10.1088/0951-7715/29/4/1329 |
| [32] | Wang, T.; Zhao, H., One-dimensional compressible heat-conducting gas with temperature-dependent viscosity, Math Models Methods Appl Sci, 26, 2237-2275 (2016) · Zbl 1353.35241 · doi:10.1142/S0218202516500524 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
