Large-Time Behavior towards Composite Waves of Degenerate Shock and Rarefaction Wave for Modified KdV-Burgers Equation

Abstract

In this paper, we study the modified Korteweg-de Vries-Burgers (mKdVB) equation \[u_t + (u^3)_x = \mu u_{xx} - \kappa u_{xxx}\] with $\mu>0$ and $\kappa>0$. Since the flux is non-convex, the (inviscid) Riemann problem may be solved by a composite wave of a degenerate Oleinik shock and a rarefaction wave. We prove the global existence of solutions and the time-asymptotic stability of the corresponding viscous-dispersive composite wave, up to a time-dependent shift, under small $H^1$ perturbations. The shock strength $\delta_S$ and the rarefaction strength $\delta_R$ need not be small and are restricted only by the explicit ratio $\delta_R\le\frac{5}{18}\delta_S$ and the quantity $\kappa\delta_S^2/\mu^2$ governing the monotonicity of the shock. The stability estimate is uniform with respect to the dispersion strength $\kappa$. As a key ingredient of the stability analysis, we establish structural properties of the degenerate viscous-dispersive shock profile in the monotone regime, including two-sided pointwise bounds on its derivative and the rates at which it converges to its two end states. These decay rates coincide with those of the corresponding purely viscous profile, with constants independent of the dispersion coefficient. In the absence of dispersion, namely, when $\kappa=0$, the time-asymptotic stability of the corresponding viscous composite wave was established by Huang, Wang and Zhang [26]. The present work extends their result to the viscous-dispersive setting and relaxes the restriction on the rarefaction strength. The proof relies on the method of $a$-contraction with shifts (for viscous conservation laws) developed by Kang and Vasseur [32,33,35].