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Time Decay Rate of Solutions Toward the Viscous Shock Waves under Periodic Perturbations for the Scalar Conservation Law with Nonlinear Viscosity

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Abstract

In this paper, it is proved that the weak solution to the Cauchy problem for the scalar viscous conservation law, with nonlinear viscosity, different far field states and periodic perturbations, not only exists globally in time, but also converges towards the viscous shock wave of the corresponding Riemann problem as time goes to infinity. Furthermore, the decay rate is shown. The proof is given by a technical energy method.

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References

  1. Dafermos, C.M. Large time behavior of periodic solutions of hyperbolic systems of conservation laws. J. diff. Eqns., 121: 183–202 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  2. Dong, Z., Huang, F., Su, H. Large time behavior of strong solutions for stochastic Burgers equation, arXiv:2103.06608v1.

  3. Glimm, J., Lax, P.D. Decay of solutions of systems of nonlinear hyperbolic conservation laws. Memoirs of the American Mathematical Society, Providence, R.I., 1970(101)

    Book  MATH  Google Scholar 

  4. Goodman, J., Xin, Z. Viscous limits for piecewise smooth solutions to systems of conservation laws. Arch. Rat. Mech. Anal., 121: 235–265 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  5. Huang, F., Xu, L. Convergence rate toward the viscous shock wave for Burger equation. Preprint?

  6. Il’in, A.M., Oleĭnik, O.A. Asymptotic behavior of the solutions of the Cauchy problem for some quasi-linear equations for large time (Russian.) Mat. Sbornik, 51: 191–216 (1960)

    Google Scholar 

  7. Kawashima, S., Matsumura, A. Asymptotic stability of travleing waves for 1-dimensional gas motion. Comm. Math. Phys., 101(1985): 97–127

    Article  MathSciNet  MATH  Google Scholar 

  8. Ladyzhenskaia, O.A., Solonikov, V.A., Ural’steva, N.N. Linear and quasi-linear equations of parabolic type. A.M.S. Providence, RI, 1968

    Book  Google Scholar 

  9. Lax, P.D. Hyperbolic systems of conservation laws II. Comm. Pure Appl. Math., 10: 537–566 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  10. Matsumura, A., Nishihara, K. Asymptotic stability of traveling waves for scalar viscous conservation laws with non-convex nonlinearity. Comm. Math. Phys., 165: 83–96 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  11. Smoller, J. Shock wave and reaction-diffusion equations. Springer-Verlag, New York, 1983

    Book  MATH  Google Scholar 

  12. Xin, Z., Yuan, Q., Yuan, Y. Asymptotic stability of shock profiles and rarefaction waves under periodic perturbations for 1-d convex scalar viscous conservation laws, Preprint

  13. Yoshida, N. Asymptotic behavior of solutions toward a multiwave pattern to the Cauchy problem for the scalar conservation law with the Ostwald-de Waele-type viscosity. SIAM J. Math. Anal., 49: 2009–2036 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  14. Yoshida, N. Asymptotic behavior of solutions toward the viscous shock waves to the Cauchy problem for the scalar conservation law with nonlinear flux and viscosity, SIAM J. Math. Anal., 50: 891–932 (2018)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Academy of Mathmatics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

    Ye-chi Liu

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  1. Ye-chi Liu
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Liu, Yc. Time Decay Rate of Solutions Toward the Viscous Shock Waves under Periodic Perturbations for the Scalar Conservation Law with Nonlinear Viscosity. Acta Math. Appl. Sin. Engl. Ser. 39, 28–48 (2023). https://doi.org/10.1007/s10255-023-1028-9

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  • DOI: https://doi.org/10.1007/s10255-023-1028-9

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