Renormalized entropy solutions of stochastic scalar conservation laws with boundary condition (Q306571)

Let \(D\subseteq\mathbb R^N\) be a bounded open set with a Lipschitz continuous boundary. The equation NEWLINE\[NEWLINE du-\operatorname{div}(f(u))\,dt=h(u)dw NEWLINE\]NEWLINE is considered with initial condition \(u(0)=u_0\) and boundary condition \(u=a\) on \((0,T)\times\partial D\). Here, \(a\) is a measurable function, \(f:\mathbb R\to\mathbb R^N\) is a \(C^2\)-smooth function with derivatives of polynomial growth, \(f(0)=0\) and \(\overline f(a,x)\in L^1((0,T)\times\partial D)\), where NEWLINE\[NEWLINE\overline f(s,x)=\sup\,\{|f(r)\cdot\vec{n}(x)|:r\in[-s^-,s^+]\},NEWLINE\]NEWLINE \(h:\mathbb R\to\mathbb R\) is a Lipschitz continuous function with \(h(0)=0\), \(u_0\in L^1(D)\) and \(w\) is a one-dimensional Wiener process. The authors first introduce certain so called renormalized stochastic entropy solutions and relate them to stochastic entropy solutions. Next, they prove existence and uniqueness of renormalized stochastic entropy solutions, the latter via a comparison principle.