A vanishing dynamic capillarity limit equation with discontinuous flux

Abstract

We prove existence and uniqueness of a solution to the Cauchy problem corresponding to the equation \begin{equation*} \begin{cases} \partial_t u_{\varepsilon,δ} +\mathrm{div} {\mathfrak f}_{\varepsilon,δ}({\bf x}, u_{\varepsilon,δ})=\varepsilon Δu_{\varepsilon,δ}+δ(\varepsilon) \partial_t Δu_{\varepsilon,δ}, \ \ {\bf x} \in M, \ \ t\geq 0 u|_{t=0}=u_0({\bf x}). \end{cases} \end{equation*} Here, ${\mathfrak{f}}_{\epsilon ,\delta }$ and $u_0$ are smooth functions while $\epsilon$ and $\delta =\delta (\epsilon )$ are fixed constants. Assuming ${\mathfrak{f}}_{\epsilon ,\delta }\to \mathfrak{f}\in L^p({\mathbb{R}}^d\times \mathbb{R};{\mathbb{R}}^d)$ for some $1<p<\infty$, strongly as $\epsilon \to 0$, we prove that, under an appropriate relationship between $\epsilon$ and $\delta (\epsilon )$ depending on the regularity of the flux $\mathfrak{f}$, the sequence of solutions $(u_{\epsilon ,\delta })$ strongly converges in $L_{loc}^1({\mathbb{R}}^{+}\times {\mathbb{R}}^d)$ towards a solution to the conservation law $${\partial }_{t}u+\mathrm{div}\mathfrak{f}(\mathbf{x},u)=0.$$ The main tools employed in the proof are the Leray-Schauder fixed point theorem for the first part and reduction to the kinetic formulation combined with recent results in the velocity averaging theory for the second.

23 pages, minor corrections, final version