[HG_AP] Large-Time Behavior towards Composite Waves of Degenerate Shock and Rarefaction Wave for Modified KdV-Burgers Equation
ABSTRACT
In this talk, we study the modified Korteweg–de Vries–Burgers (mKdVB) equation $u_t+(u^3)_x=\mu u_{xx}-\kappa u_{xxx}$. Due to the non-convexity of the flux, 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 strengths of the shock and the rarefaction need not be small. They are restricted only by an explicit ratio condition and by the relative strength of the dispersion, which governs the monotonicity of the shock. The stability estimate is uniform with respect to the dispersion strength $\kappa$. As a key ingredient, we show that the degenerate viscous-dispersive shock profile in the monotone regime approaches its two end states at the same rates as the purely viscous profile. When $\kappa=0$, the time-asymptotic stability of the viscous composite wave was established by Huang, Wang and Zhang. Our result extends it to the viscous-dispersive setting and relaxes the restriction on the rarefaction strength. The proof relies on the method of $a$-contraction with shifts developed by Kang and Vasseur. This is joint work with Namhyun Eun (KIAS) and Jeongho Kim (Kyung Hee University).
