Entropic Burgers equation via a minimizing movement scheme based on the Wasserstein metric

As noted by the second author in the context of unstable two-phase porous medium flow, entropy solutions of Burgers’ equation can be recovered from a minimizing movement scheme involving the Wasserstein metric in the limit of vanishing time step size [4]. In this paper, we give a simpler proof by verifying that the anti-derivative is a viscosity solution of the associated Hamilton Jacobi equation. AMS Subject classification: 76S05 (35Q35 49Q20 76T99) Introduction The aim of this paper is twofold. On one side we give a simpler proof of a result found by the second author ([4]). This amounts in proving that the minimizing movements scheme for the Energy E(θ) = ∫ xθdx on a two-phase Wasserstein space produces the entropy solution of the Burgers’ equation. The difference with respect to the approach of [4] consists in the fact that we pass to the anti-derivative of the limit function and prove that it is the viscosity solution of the associated Hamilton Jacobi equation. This provides some technical simplifications. On the other hand we discuss, mostly at an informal level, the gradient flow structure that this result suggests. Indeed, on an heuristic level the fact that the Burgers’ equation is the equation of the gradient flow of E(θ) = ∫ xθdx on a two-phase Wasserstein space is obvious (see Section 2.3). It is less obvious why the minimizing movement scheme converges to the entropy solution. University of Nice Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany