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Viscous Shock Wave to an Inflow Problem for Compressible Viscous Gas with Large Density Oscillations

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Abstract

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.

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References

  1. Bianchini, S., Bressan, A. Vanishing viscosity solutions of nonlinear hyperbolic systems. Ann. Math. 161: 223–342 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  13. Kruzhkov, S.N. First–order quasilinear equations in several independent variables. Mat. Sbornik, 81: 228–255 (1970)

    MATH  Google Scholar 

  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)

    Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  26. Qin, X.H., Wang, Y. Large–time behavior of solutions to the inflow problem of full compressible Navier–Stokes equations. SIAM J. Math. Anal., 43(1): 341–366 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Google Scholar 

  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

    Google Scholar 

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Authors and Affiliations

  1. School of Mathematics and Statistics, Beijing Institute of Technology, Beijing, 100081, China

    Dong-fen Bian

  2. School of Mathematics and Computer Science, Wuhan Polytechnic University, Wuhan, 430023, China

    Li-li Fan

  3. Institute of Applied Mathematics, Academy of Mathematics and Systems Science, The Chinese Academy of Sciences, Beijing, 100190, China

    Lin He

  4. School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, China

    Hui-jiang Zhao

  5. Computational Science Hubei Key Laboratory, Wuhan University, Wuhan, 430072, China

    Hui-jiang Zhao

Authors
  1. Dong-fen Bian
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  2. Li-li Fan
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  3. Lin He
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  4. Hui-jiang Zhao
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Correspondence to Lin He.

Additional information

This paper is dedicated to Professor Philippe G. Ciarlet on the occasion of his 80th birthday.

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Bian, Df., Fan, Ll., He, L. et al. Viscous Shock Wave to an Inflow Problem for Compressible Viscous Gas with Large Density Oscillations. Acta Math. Appl. Sin. Engl. Ser. 35, 129–157 (2019). https://doi.org/10.1007/s10255-019-0801-2

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  • DOI: https://doi.org/10.1007/s10255-019-0801-2

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