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 ) (M,g) : d ( u ε , δ − δ Δ u ε , δ ) + d i v f ε ( x , u ε , δ ) d t = ε Δ u ε , δ d t + Φ ( x , u ε , δ ) d W t , \begin{equation*} d \left (u_{\varepsilon ,\delta } -\delta \Delta u_{\varepsilon ,\delta }\right ) +div\mathfrak {f}_{\varepsilon }(\mathbf {x}, u_{\varepsilon ,\delta })\, dt =\varepsilon \Delta u_{\varepsilon ,\delta }\, dt + \Phi (\mathbf {x}, u_{\varepsilon ,\delta })\, dW_t, \end{equation*} where f ε \mathfrak {f}_{\varepsilon } is a sequence of smooth vector fields converging in L p ( M × R ) L^p(M\times \mathbb {R}) ( p > 2 p>2 ) as ε ↓ 0 \varepsilon \downarrow 0 towards a vector field f ∈ L p ( M ; C 1 ( R ) ) \mathfrak {f}\in L^p(M;C^1(\mathbb {R})) , and W t W_t is a Wiener process defined on a filtered probability space. First, for fixed values of ε \varepsilon and δ \delta , we establish the existence and uniqueness of weak solutions to the Cauchy problem for the above-stated equation. Assuming that f \mathfrak {f} is non-degenerate and that ε \varepsilon and δ \delta tend to zero with δ / ε 2 \delta /\varepsilon ^2 bounded, we show that there exists a subsequence of solutions that strongly converges in L ω , t , x 1 L^1_{\omega ,t,\mathbf {x}} to a martingale solution of the following stochastic conservation law with discontinuous flux: d u + d i v f ( x , u ) d t = Φ ( u ) d W t . \begin{equation*} d u +div\mathfrak {f}(\mathbf {x}, u)\,dt =\Phi (u)\, dW_t. \end{equation*} The proofs make use of Galerkin approximations, kinetic formulations as well as H H -measures and new velocity averaging results for stochastic continuity equations. The analysis relies 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.