Mathematics > Analysis of PDEs
[Submitted on 7 May 2018 (v1), last revised 4 Nov 2020 (this version, v2)]
Title:A vanishing dynamic capillarity limit equation with discontinuous flux
View PDFAbstract: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.
Submission history
From: Michael Kunzinger [view email][v1] Mon, 7 May 2018 19:57:23 UTC (24 KB)
[v2] Wed, 4 Nov 2020 14:03:31 UTC (25 KB)
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