We establish the time-asymptotic stability of generic Riemann solutions to the one-dimensional compressible Navier–Stokes–Fourier equations. The Riemann solution under consideration is a generic combination of a shock, a contact discontinuity, and a rarefaction wave. We prove that the perturbed solution of Navier–Stokes–Fourier converges, uniformly in space as time goes to infinity, to an ansatz composed of viscous shock with time-dependent shift, a viscous contact wave and an inviscid rarefaction wave. This is a first resolution of the time-asymptotic stability of three waves of different kinds associated with the generic Riemann solutions. Our approach relies on the method of a-contraction with shifts and relative entropy, specifically applied to both the shock wave and the contact wave. It enables the application of a global energy method for the generic combination of three waves.

Publication:

Arch. Rational Mech. Anal. (2025)

https://doi.org/10.1007/s00205-025-02116-w

Author:

M.-J. Kang

Department of Mathematical Sciences, Korea Advanced Institute of Science and Technology, Daejeon 34141 Korea.

e-mail: moonjinkang@kaist.ac.kr

A. F. Vasseur

Department of Mathematics, The University of Texas at Austin, Austin TX 78712 USA.

e-mail: vasseur@math.utexas.edu

Y. Wang

State Key Laboratory of Mathematical Sciences and Institute of Applied Mathematics, AMSS, Chinese Academy of Sciences, Beijing 100190 People’s Republic of China.

School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing，100049 People’s Republic of China.

e-mail: wangyi@amss.ac.cn