On a nonlinear degenerate parabolic transport-diffusion equation with a discontinuous coefficient

Karlsen, Kenneth H. ; Risebro, Nils H. ; Towers, John D.

Electronic Journal of Differential Equations, Tome 2002 (2002).

Résumé

Summary: We study the Cauchy problem for the nonlinear (possibly strongly) degenerate parabolic transport-diffusion equation $$ \partial_t u + \partial_x \bigl(\gamma(x)f(u)\bigr)=\partial_x^2 A(u), \quad A'(\cdot)\ge 0, $$ where the coefficient $\gamma(x)$ is possibly discontinuous and $f(u)$ is genuinely nonlinear, but not necessarily convex or concave. Existence of a weak solution is proved by passing to the limit as $\varepsilon\downarrow 0$ in a suitable sequence $\{u_{\varepsilon}\}_{\varepsilon}$ of smooth approximations solving the problem above with the transport flux $\gamma(x)f(\cdot)$ replaced by $\gamma_{\varepsilon}(x)f(\cdot)$ and the diffusion function $A(\cdot)$ replaced by $A_{\varepsilon}(\cdot)$, where $\gamma_{\varepsilon}(\cdot)$ is smooth and $A_{\varepsilon}'(\cdot)>0$. The main technical challenge is to deal with the fact that the total variation $|u_{\varepsilon}|_{BV}$ cannot be bounded uniformly in $\varepsilon$, and hence one cannot derive directly strong convergence of $\{u_{\varepsilon}\}_{\varepsilon>0}$. In the purely hyperbolic case ($A'\equiv 0$), where existence has already been established by a number of authors, all existence results to date have used a singular mapping to overcome the lack of a variation bound. Here we derive instead strong convergence via a series of a priori (energy) estimates that allow us to deduce convergence of the diffusion function and use the compensated compactness method to deal with the transport term.

Détail

Classification : 35K65, 35D05, 35R05, 35L80
Keywords: degenerate parabolic equation, nonconvex flux, weak solution, discontinuous coefficient, viscosity method, a priori estimates