Zero dissipation limit to rarefaction waves for the 1-D compressible Navier-Stokes equations

Abstract

The zero dissipation limit for the one-dimensional Navier-Stokes equations of compressible, isentropic gases in the case that the corresponding Euler equations have rarefaction wave solutions is investigated in this paper. In a paper (Comm. Pure Appl. Math., 46, 1993, 621–665) by Z. P. Xin, the author constructed a sequence of solutions to one-dimensional Navier-Stokes isentropic equations converging to the rarefaction wave as the viscosity tends to zero. Furthermore, he obtained that the convergence rate is $\varepsilon ^{\tfrac{1}{4}} \left| {\ln \varepsilon } \right|$. In this paper, Xin’s convergence rate is improved to $\varepsilon ^{\tfrac{1}{3}} \left| {\ln \varepsilon } \right|^2$ by different scaling arguments. The new scaling has various applications in related problems.

Keywords

Compressible Navier-Stokes equations / Rarefaction wave / Compressible Euler equations

References

[1] David H., Liu T. P.. The inviscid limit for the Navier-Stokes equations of compressible, isentropic flow with shock data. Indiana Univ. Math. J., 1989, 38(4): 861-915

[2] Huang, F. M., Li, M. J. and Wang, Y., Zero dissipation limit to rarefaction wave with vacuum for 1-D compressible Navier-Stokes equations, preprint.

[3] Jiang S., Ni G. X., Wen J. S.. Vanishing viscosity limit to rarefaction waves for the Navier-Stokes equations of one-dimensional compressible heat-conducting fluids (electronic). SIAM J. Math. Anal., 2006, 38(2): 368-384

[4] Lax, P. D., Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, Slam Reg. Conf. Ser. Appl. Math., Philadelphia, 11, 1973.

[5] Liu T. P., Xin Z. P.. Nonlinear stability of rarefaction waves for compressible Navier-Stokes equation. Comm. Math. Phys., 1988, 118: 451-465

[6] Matsumura A., Nishihara K.. Asymptotics toward the rarefaction waves of the solutions of a onedimensional model system for compressible viscous gas. Japan J. Appl. Math., 1986, 3(1): 1-13

[7] Smoller J.. Shock Waves and Rarefaction-Diffusion Equations, 1983, New York, Berlin: Springer-Verlag

[8] Xin Z. P.. Zero dissipation limit to rarefaction waves for the one-dimensional Navier-Stokes equations of compressible isentropic gases. Comm. Pure Appl. Math., 1993, 46: 621-665

[9] Xin Z. P., Zeng H.. Convergence to rarefaction waves for the nonlinear Boltzmann equation and compressible Navier-Stokes equations. J. Diff. Eq., 2010, 249: 827-871