Curves of steepest descent are entropy solutions for a class of degenerate convection-diffusion equations

TL;DR

This paper demonstrates that entropy solutions for a class of degenerate convection-diffusion equations can be obtained as limits of a variational scheme, despite the lack of convexity properties.

Contribution

It establishes the convergence of the JKO variational approximation scheme to entropy solutions for degenerate convection-diffusion equations without relying on convexity.

Findings

Entropy solutions are limits of the JKO scheme.

Uniqueness of solutions is proved via doubling of variables.

The approach bypasses the need for convexity in the equation.

Abstract

We consider a nonlinear degenerate convection-diffusion equation with inhomogeneous convection and prove that its entropy solutions in the sense of Kru\v{z}kov are obtained as the - a posteriori unique - limit points of the JKO variational approximation scheme for an associated gradient flow in the $L^2$-Wasserstein space. The equation lacks the necessary convexity properties which would allow to deduce well-posedness of the initial value problem by the abstract theory of metric gradient flows. Instead, we prove the entropy inequality directly by variational methods and conclude uniqueness by doubling of the variables.