A dynamic capillarity equation with stochastic forcing on manifolds: a singular limit problem

Abstract

We consider a dynamic capillarity equation with stochastic forcing on a compact Riemannian manifold $(M,g)$. \begin{equation*}\tag{P} d \left(u_{\varepsilon,δ}-δΔ u_{\varepsilon,δ}\right) +\operatorname{div} f_{\varepsilon}(x, u_{\varepsilon,δ})\, dt =\varepsilon Δu_{\varepsilon,δ}\, dt Φ(x, u_{\varepsilon,δ})\, dW_t, \end{equation*} where $f_{\epsilon }$ is a sequence of smooth vector fields converging in $L^p(M\times \mathbb{R})$ ($p>2$) as $\epsilon \downarrow 0$ towards a vector field $f\in L^p(M;C^1(\mathbb{R}))$, and $W_t$ is a Wiener process defined on a filtered probability space. First, for fixed values of $\epsilon$ and $\delta$, we establish the existence and uniqueness of weak solutions to the Cauchy problem for (P). Assuming that $f$ is non-degenerate and that $\epsilon$ and $\delta$ tend to zero with $\delta /{\epsilon }^2$ bounded, we show that there exists a subsequence of solutions that strongly converges in $L_{\omega ,t,x}^1$ to a martingale solution of the following stochastic conservation law with discontinuous flux: $$du+\mathrm{div}f(x,u)dt=\Phi (u)d{W}_t.$$ The proofs make use of Galerkin approximations, kinetic formulations as well as $H$-measures and new velocity averaging results for stochastic continuity equations. The analysis relies in an essential way on the use of a.s.~representations of random variables in some particular quasi-Polish spaces. The convergence framework developed here can be applied to other singular limit problems for stochastic conservation laws.