Stability of viscous shock profile for convective porous-media flow with degenerate viscosity
Yechi Liu
TL;DR
This work analyzes the large-time behavior of viscous shock waves for the convective porous-media equation with degenerate viscosity, $u_t+f(u)_x=(u^m)_{xx}$, in the regime $1<m<2$. It introduces perturbation variables and an integrated perturbation $\Phi$ to develop an energy-based stability framework that accommodates degeneracy via sign-based estimates and a domain decomposition strategy, leading to nonlinear stability of the viscous shock profile $U(\xi)$ under small initial perturbations. The authors derive an $L^2$-decay rate $\|u(t,\cdot)-U(\cdot-\gamma t)\|_{L^2}\le C_\delta(1+t)^{-\frac{1}{4(11m+7)}+\delta}$ and, through interpolation, an $L^\infty$-decay rate $\|u(t,\cdot)-U(\cdot-\gamma t)\|_{L^\infty}\le C_\delta(1+t)^{-\frac{1}{6(11m+7)}+\delta}$. These results extend classical decay analyses to degenerate porous-media flows and provide a robust method for handling viscosity degeneracy in stability proofs. The work thus advances understanding of shock stability and long-time behavior in nonlinear degenerate parabolic systems relevant to porous-media dynamics.
Abstract
In this paper, we are concerned with the large time behavior of viscous shock wave for the convective porous-media equation with degenerate viscosity. We get the regularity of the solution for general initial data and prove the shock wave is nonlinearly stable providing the initial perturbation is small. Moreover, the $L^\infty$ decay rate is obtained, which generalized the famous result \cite{osh82}. Note that the traditional energy method and continuity argument can not be directly used in this paper since the degeneration of viscosity. One need to fully utilize the sign of perturbation and it derivatives, decompose the integral domain to ensure that in each domain the sign is invariant. Then the stability and the decay rate are obtained by energy method and an area inequality.