Nonlinear Stability of Strong Planar Rarefaction Waves for the Relaxation Approximation of Conservation Laws in Several Space Dimensions

Abstract

In this paper, we show that a strong planar rarefaction wave is nonlinear stable, namely it is an attractor for the relaxation approximation of the scalar conservation laws in several space dimensions. Compared with former results obtained by T. P. Liu (1987, Comm. Math. Phys. 108, 153 175) and T. Luo (1997, J. Differential Equations 133, 255 279), our main novelty lies in the fact that the planar rarefaction waves do not need to be small, and in the one-dimensional case, the initial disturbance can also be chosen arbitrarily large. Academic Press

Convergence to Strong Nonlinear Diffusion Waves for Solutions of p-System with Damping

“…These results and Sobolev's inequality immediately deduce (1.23). Finally, we point out that similar ideas have been used to study the nonlinear stability of the strong planar rarefaction waves to the solutions of the relaxation approximation to scalar conservation laws in several space dimensions [25] and we believe that they can also be used to study the convergence of the solutions of the following hyperbolic system with relaxation…”

Decay rates of strong planar rarefaction waves to scalar conservation laws with degenerate viscosity in several space dimensions

“…For I 18 , by using the Sobolev inequality, the Gagliardo-Nirenberg inequality and the Cauchy-Schwarz inequality, we have (4.43)…”

“…Ito showed the convergence rate toward the weak planar rarefaction wave (see [6]). Recently, Nishikawa and Nishihara in [11] obtained the decay rate toward the strong planar rarefaction wave, and Zhao in [18] also obtained nonlinear stability of the strong planar rarefaction wave for a relaxation model in several space dimensions.…”

Stability of contact discontinuity for Jin–Xin relaxation system

“…Liu [18] first considered a general 2 × 2 relaxation system in one spatial dimension, and gave the stability criteria for the shock waves, rarefaction waves and also diffusion waves. Since then, many authors have studied the stability of shock waves and rarefaction waves to the relaxation system in one or several space dimensions; see [3][4][5][6][7][8]16,17,[21][22][23][24][25]27,[31][32][33][34], etc. However, as far as we know, there is no result corresponding to contact discontinuities for the relaxation system (1.1).…”