A vanishing dynamic capillarity limit equation with discontinuous flux

TL;DR

This paper establishes the existence and uniqueness of solutions to a complex PDE with discontinuous flux, and shows convergence to a conservation law as parameters vanish, using fixed point and kinetic methods.

Contribution

It introduces a novel analysis of a vanishing capillarity limit equation with discontinuous flux, proving convergence to a conservation law under specific parameter relationships.

Findings

Solutions exist and are unique for the PDE with discontinuous flux.

Solutions converge strongly to the conservation law solution as parameters tend to zero.

The analysis employs fixed point and kinetic formulation techniques.

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,\delta} +\mathrm{div} {\mathfrak f}_{\varepsilon,\delta}({\bf x}, u_{\varepsilon,\delta})=\varepsilon \Delta u_{\varepsilon,\delta}+\delta(\varepsilon) \partial_t \Delta u_{\varepsilon,\delta}, \ \ {\bf x} \in M, \ \ t\geq 0 u|_{t=0}=u_0({\bf x}). \end{cases} \end{equation*} Here, ${\mathfrak f}_{\varepsilon,\delta}$ and $u_0$ are smooth functions while $\varepsilon$ and $\delta=\delta(\varepsilon)$ are fixed constants. Assuming ${\mathfrak f}_{\varepsilon,\delta} \to {\mathfrak f} \in L^p( \mathbb{R}^d\times \mathbb{R};\mathbb{R}^d)$ for some $1<p<\infty$, strongly as $\varepsilon\to 0$, we prove that, under an appropriate relationship between $\varepsilon$ and $\delta(\varepsilon)$ depending on the regularity of the flux ${\mathfrak f}$, the sequence of solutions $(u_{\varepsilon,\delta})$ strongly converges in $L^1_{loc}(\mathbb{R}^+\times \mathbb{R}^d)$ towards a solution to the conservation law $$ \partial_t u +\mathrm{div} {\mathfrak f}({\bf x}, u)=0. $$ 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.