Viscous Shock Wave to an Inflow Problem for Compressible Viscous Gas with Large Density Oscillations
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摘要
This paper is concerned with the inflow problem for one-dimensional compressible Navier-Stokes equations. For such a problem, Huang, Matsumura, and Shi showed in [4] that there exists viscous shock wave solution to the inflow problem and both the boundary layer solution, the viscous shock wave, and their superposition are time-asymptotically nonlinear stable provided that both the initial perturbation and the boundary velocity are assumed to be sufficiently small. The main purpose of this paper is to show that similar stability results still hold for a class of large initial perturbation which can allow the initial density to have large oscillations. The proofs are given by an elementary energy method and our main idea is to use the smallness of the strength of the viscous shock wave and the boundary velocity to control the possible growth of the solutions induced by the nonlinearity of the compressible Navier-Stokes equations and the inflow boundary condition.The key point in our analysis is to deduce the desired uniform positive lower and upper bounds on the density.
This paper is concerned with the inflow problem for one-dimensional compressible Navier-Stokes equations. For such a problem, Huang, Matsumura, and Shi showed in [4] that there exists viscous shock wave solution to the inflow problem and both the boundary layer solution, the viscous shock wave, and their superposition are time-asymptotically nonlinear stable provided that both the initial perturbation and the boundary velocity are assumed to be sufficiently small. The main purpose of this paper is to show that similar stability results still hold for a class of large initial perturbation which can allow the initial density to have large oscillations. The proofs are given by an elementary energy method and our main idea is to use the smallness of the strength of the viscous shock wave and the boundary velocity to control the possible growth of the solutions induced by the nonlinearity of the compressible Navier-Stokes equations and the inflow boundary condition.The key point in our analysis is to deduce the desired uniform positive lower and upper bounds on the density.
This paper is concerned with the inflow problem for one-dimensional compressible Navier-Stokes equations. For such a problem, Huang, Matsumura, and Shi showed in [4] that there exists viscous shock wave solution to the inflow problem and both the boundary layer solution, the viscous shock wave, and their superposition are time-asymptotically nonlinear stable provided that both the initial perturbation and the boundary velocity are assumed to be sufficiently small. The main purpose of this paper is to show that similar stability results still hold for a class of large initial perturbation which can allow the initial density to have large oscillations. The proofs are given by an elementary energy method and our main idea is to use the smallness of the strength of the viscous shock wave and the boundary velocity to control the possible growth of the solutions induced by the nonlinearity of the compressible Navier-Stokes equations and the inflow boundary condition.The key point in our analysis is to deduce the desired uniform positive lower and upper bounds on the density.
引文
[1]Bianchini,S.,Bressan,A.Vanishing viscosity solutions of nonlinear hyperbolic systems.Ann.Math.161:223-342(2005)
[2]Duan,R.,Liu,H.X.,Zhao,H.J.Nonlinear stability of rarefaction waves for the compressible Navier-Stokes equations with large initial perturbation.Trans.Amer.Math.Soc.,361(1):453-493(2009)
[3]Huang,F.M.,Li,J.,Shi,X.D.Asymptotic behavior of solutions to the full compressible Navier-Stokes equations in the half space.Commun.Math.Sci.,8(3):639-654(2010)
[4]Huang,F.M.,Matsumura,A.,Shi,X.D.Viscous shock wave and boundary layer to an inflow problem for compressible viscous gas.Comm.Math.Phys.,239(1-2):261-285(2003)
[5]Huang,F.M.,Qin,X.H.Stability of boundary layer and rarefaction wave to an outflow problem for compressible Navier-Stokes equations under large perturbation.J.Differential Equations,246(10):4077-4096(2009)
[6]Huang,F.M.,Wang,T.Stability of superposition of viscous contact wave and rarefaction waves for compressible Navier-Stokes system.Indiana Univ.Math.J.,65(6):1833-1875(2016)
[7]Fan,L.L.,Liu,H.X.,Wang,T.,Zhao,H.J.Inflow problem for one-dimensional compressible Navier-Stokes equation with large initial perturbation.J.Differential Equations,257:3521-3553(2014)
[8]Kagei,Y.,Kawashima,S.Stability of planar stationary solutions to the compressible Navier-Stokes equation on the half space.Comm.Math.Phys.,266(2):401-430(2006)
[9]Kanel’,Ya.A model system of equations for the one-dimensional motion of a gas.Differencial’nye Uravnenija 4:721-734(1968)(in English Transl.),Diff.Eqns.,4:374-380(1968)(Russian)
[10]Kawashima,S.,Nakamura,T.,Nishibata,S.,Zhu,P.C.Stationary waves to viscous heat-conductive gases in half-space:existence,stability and convergence rate.Math.Models Methods Appl.Sci.,20(12):2201-2235(2010)
[11]Kawashima,S.,Nishibata,S.,Zhu,P.C.Asymptotic stability of the stationary solution to the compressible Navier-Stokes equations in the half space.Comm.Math.Phys.,240(3):483-500(2003)
[12]Kawashima.S.,Zhu,P.C.Asymptotic stability of nonlinear wave for the compressible Navier-Stokes equations in the half space.J.Differential Equations 244(12):3151-3179(2008)
[13]Kruzhkov,S.N.First-order quasilinear equations in several independent variables.Mat.Sbornik,81:228-255(1970)
[14]Liu,T.P.,Zeng,Y.N.Shock waves in conservation laws with physical viscosity.Memoirs of the American Mathematical Society,234:No.1105(2015)
[15]Matsumura,A.Inflow and outflow problems in the half space for a one-dimensional isentropic model system of compressible viscous gas.IMS Conference on Differential Equations from Mechanics(Hong Kong,1999).Methods Appl.Anal.,8(4):645-666(2001)
[16]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):1-22(1999)
[17]Matsumura,A.,Nishihara,K.On the stability of travelling wave solutions of a one-dimensional model system for compressible viscous gas.Japan J.Appl.Math.,2(1):17-25(1985)
[18]Matsumura,A.,Nishihara,K.Global stability of the rarefaction wave of a one-dimensional model system for compressible viscous gas.Comm.Math.Phys.,144(2):325-335(1992)
[19]Matsumura,A.,Nishihara,K.Global asymptotics toward the rarefaction wave for solutions of viscous p-system with boundary effect.Quart.Appl.Math.,58(1):69-83(2000)
[20]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(3):449-474(2001)
[21]Nakamura,T.,Nishibata,S.Stationary wave associated with an inflow problem in the half line for viscous heat-conductive gas.J.Hyperbolic Differ.Equ.,8(4):651-670(2011)
[22]Nakamura,T.,Nishibata,S.Convergence rate toward planar stationary waves for compressible viscous fluid in multidimensional half space.SIAM J.Math.Anal.,41:(5):1757-1791(2009)
[23]Nakamura,T.,Nishibata,S.,Yuge,T.Convergence rate of solutions toward stationary solutions to the compressible Navier-Stokes equation in a half line.J.Differential Equations,241(1):94-111(2007)
[24]Qin,X.H.Large-time behaviour of solutions to the outflow problem of full compressible Navier-Stokes equations.Nonlinearity,24(5):1369-1394(2011)
[25]Qin,X.H.,Wang,Y.Stability of wave patterns to the inflow problem of full compressible Navier-Stokes equations.SIAM J.Math.Anal.,41(5):2057-2087(2009)
[26]Qin,X.H.,Wang,Y.Large-time behavior of solutions to the inflow problem of full compressible NavierStokes equations.SIAM J.Math.Anal.,43(1):341-366(2011)
[27]Shi,X.D.On the stability of rarefaction wave solutions for viscous p-system with boundary effect.Acta Math.Appl.Sin.Engl.Ser.,19(2):341-352(2003)
[28]Wan,L.,Wang,T.,Zhao,H.J.Asymptotic stability of wave patterns to compressible viscous and heatconducting gases in the half space.J.Differential Equations,261(11):5949-5991(2016)
[29]Wan,L.,Wang,T.,Zou,Q.Y.Stability of stationary solutions to the outflow problem for full compressible Navier-Stokes equations with large initial perturbation.Nonlinearity,29(4):1329-1354(2016)
[30]Wang,T.,Zhao,H.J.,Zou,Q.Y.One-dimensional compressive Navier-Stocks equations with large density oscillation.Kinetic and Related Models,3(6):547-571(2013)
[31]Zhu,P.C.Nonlinear Waves for the Compressible Navier-Stokes Equations in the Half Space.The report for JSPS postdoctoral research at Kyushu University,Fukuoka,Japan,August 2001
[2]Duan,R.,Liu,H.X.,Zhao,H.J.Nonlinear stability of rarefaction waves for the compressible Navier-Stokes equations with large initial perturbation.Trans.Amer.Math.Soc.,361(1):453-493(2009)
[3]Huang,F.M.,Li,J.,Shi,X.D.Asymptotic behavior of solutions to the full compressible Navier-Stokes equations in the half space.Commun.Math.Sci.,8(3):639-654(2010)
[4]Huang,F.M.,Matsumura,A.,Shi,X.D.Viscous shock wave and boundary layer to an inflow problem for compressible viscous gas.Comm.Math.Phys.,239(1-2):261-285(2003)
[5]Huang,F.M.,Qin,X.H.Stability of boundary layer and rarefaction wave to an outflow problem for compressible Navier-Stokes equations under large perturbation.J.Differential Equations,246(10):4077-4096(2009)
[6]Huang,F.M.,Wang,T.Stability of superposition of viscous contact wave and rarefaction waves for compressible Navier-Stokes system.Indiana Univ.Math.J.,65(6):1833-1875(2016)
[7]Fan,L.L.,Liu,H.X.,Wang,T.,Zhao,H.J.Inflow problem for one-dimensional compressible Navier-Stokes equation with large initial perturbation.J.Differential Equations,257:3521-3553(2014)
[8]Kagei,Y.,Kawashima,S.Stability of planar stationary solutions to the compressible Navier-Stokes equation on the half space.Comm.Math.Phys.,266(2):401-430(2006)
[9]Kanel’,Ya.A model system of equations for the one-dimensional motion of a gas.Differencial’nye Uravnenija 4:721-734(1968)(in English Transl.),Diff.Eqns.,4:374-380(1968)(Russian)
[10]Kawashima,S.,Nakamura,T.,Nishibata,S.,Zhu,P.C.Stationary waves to viscous heat-conductive gases in half-space:existence,stability and convergence rate.Math.Models Methods Appl.Sci.,20(12):2201-2235(2010)
[11]Kawashima,S.,Nishibata,S.,Zhu,P.C.Asymptotic stability of the stationary solution to the compressible Navier-Stokes equations in the half space.Comm.Math.Phys.,240(3):483-500(2003)
[12]Kawashima.S.,Zhu,P.C.Asymptotic stability of nonlinear wave for the compressible Navier-Stokes equations in the half space.J.Differential Equations 244(12):3151-3179(2008)
[13]Kruzhkov,S.N.First-order quasilinear equations in several independent variables.Mat.Sbornik,81:228-255(1970)
[14]Liu,T.P.,Zeng,Y.N.Shock waves in conservation laws with physical viscosity.Memoirs of the American Mathematical Society,234:No.1105(2015)
[15]Matsumura,A.Inflow and outflow problems in the half space for a one-dimensional isentropic model system of compressible viscous gas.IMS Conference on Differential Equations from Mechanics(Hong Kong,1999).Methods Appl.Anal.,8(4):645-666(2001)
[16]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):1-22(1999)
[17]Matsumura,A.,Nishihara,K.On the stability of travelling wave solutions of a one-dimensional model system for compressible viscous gas.Japan J.Appl.Math.,2(1):17-25(1985)
[18]Matsumura,A.,Nishihara,K.Global stability of the rarefaction wave of a one-dimensional model system for compressible viscous gas.Comm.Math.Phys.,144(2):325-335(1992)
[19]Matsumura,A.,Nishihara,K.Global asymptotics toward the rarefaction wave for solutions of viscous p-system with boundary effect.Quart.Appl.Math.,58(1):69-83(2000)
[20]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(3):449-474(2001)
[21]Nakamura,T.,Nishibata,S.Stationary wave associated with an inflow problem in the half line for viscous heat-conductive gas.J.Hyperbolic Differ.Equ.,8(4):651-670(2011)
[22]Nakamura,T.,Nishibata,S.Convergence rate toward planar stationary waves for compressible viscous fluid in multidimensional half space.SIAM J.Math.Anal.,41:(5):1757-1791(2009)
[23]Nakamura,T.,Nishibata,S.,Yuge,T.Convergence rate of solutions toward stationary solutions to the compressible Navier-Stokes equation in a half line.J.Differential Equations,241(1):94-111(2007)
[24]Qin,X.H.Large-time behaviour of solutions to the outflow problem of full compressible Navier-Stokes equations.Nonlinearity,24(5):1369-1394(2011)
[25]Qin,X.H.,Wang,Y.Stability of wave patterns to the inflow problem of full compressible Navier-Stokes equations.SIAM J.Math.Anal.,41(5):2057-2087(2009)
[26]Qin,X.H.,Wang,Y.Large-time behavior of solutions to the inflow problem of full compressible NavierStokes equations.SIAM J.Math.Anal.,43(1):341-366(2011)
[27]Shi,X.D.On the stability of rarefaction wave solutions for viscous p-system with boundary effect.Acta Math.Appl.Sin.Engl.Ser.,19(2):341-352(2003)
[28]Wan,L.,Wang,T.,Zhao,H.J.Asymptotic stability of wave patterns to compressible viscous and heatconducting gases in the half space.J.Differential Equations,261(11):5949-5991(2016)
[29]Wan,L.,Wang,T.,Zou,Q.Y.Stability of stationary solutions to the outflow problem for full compressible Navier-Stokes equations with large initial perturbation.Nonlinearity,29(4):1329-1354(2016)
[30]Wang,T.,Zhao,H.J.,Zou,Q.Y.One-dimensional compressive Navier-Stocks equations with large density oscillation.Kinetic and Related Models,3(6):547-571(2013)
[31]Zhu,P.C.Nonlinear Waves for the Compressible Navier-Stokes Equations in the Half Space.The report for JSPS postdoctoral research at Kyushu University,Fukuoka,Japan,August 2001