Renormalized entropy solutions for degenerate parabolic-hyperbolic equations with time-space dependent coefficients

Abstract

We study the well-posedness of renormalized entropy solutions to the Cauchy problem for general degenerateparabolic-hyperbolic equations of the form\begin{eqnarray*}\partial_{t}u+ \sum_{i=1}^{d}\partial_{x_{i}f_{i}(u,t,x)}=\sum_{i,j=1}^{d}\partial_{x_j}(a_{ij}(u,t,x)\partial_{x_i}u)+\gamma(t,x)\end{eqnarray*}with initial data $u(0,x)=u_{0}(x)$, where the convection flux function $f$,the diffusion function $a$, and the source term $\gamma$ dependexplicitly on the independent variables $t$ and $x$.We prove the uniqueness by using Kružkov's device of doubling variablesand the existence by using vanishing viscosity method.