Regularization by noise for scalar conservation laws

Joint work with Benjamin Gess

We say that a regularization by noise phenomenon occurs for a possibly ill-posed differential equation if this equation becomes well-posed (in a pathwise sense) under addition of noise. Most of the results in this direction are limited to SDEs and associated linear SPDEs.

In this talk, we show a regularization by noise result for a nonlinear SPDE, namely a stochastic scalar conservation law on ${\text{mr}}^d$ with a space-irregular flux: $${\partial }_{t}v+b\cdot \mathrm{\nabla }[v^2]+\mathrm{\nabla }v\circ \dot{W}=0,$$ where $b=b(x)$ is a given deterministic, possibly irregular vector field, $W$ is a $d$-dimensional Brownian motion ($\circ$ denotes Stratonovich integration) and $v=v(t,x,\omega )$ is the scalar solution. More precisely we prove that, under suitable Sobolev assumptions on $b$ and integrability assumptions on its divergence, the equation admits a unique entropy solution. The result is false without noise.

The proof of uniqueness is based on a careful combination of arguments used in the linear case: first we show the renormalization property for the kinetic formulation of the equation, then we use second order PDE estimates and a duality argument to conclude.

If time permits, we will discuss also some open questions.