| Abstract: |
| The Korteweg--de Vries--Burgers (KdVB) equation is a fundamental model capturing the interplay of nonlinearity, viscosity (dissipation), and dispersion, with broad physical relevance. It is well known that the KdVB equation admits traveling wave solutions, called viscous-dispersive shocks.
These shock profiles are monotone in the viscosity-dominant regime, while they exhibit infinitely many oscillation when dispersion dominates.
In this talk, we study the stability of such viscous-dispersive shocks, focusing on an $L^2$-contraction property under arbitrarily large perturbations, up to a time-dependent shift. We begin with viscous shocks of the viscous Burgers equation (i.e., the KdVB equation without dispersion), then treat monotone viscous-dispersive shocks and finally address oscillatory shocks. We also present detailed structural properties of the oscillatory profiles.
This is joint work with Geng Chen (University of Kansas), Moon-Jin Kang (KAIST), and Yannan Shen (University of Kansas). |
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