A vanishing dynamic capillarity limit equation with discontinuous flux
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, ${\mathfrak f}_{\varepsilon,δ}$ and $u_0$ are smooth functions while $\varepsilon$ and $δ=δ(\varepsilon)$ are fixed constants. Assuming ${\mathfrak f}_{\varepsilon,δ} \to {\mathfrak f} \in L^p( \mathbb{R}^d\times \mathbb{R};\mathbb{R}^d)$ for some $1<p<\infty$, strongly as $\varepsilon\to 0$, we prove that, under an appropriate relationship between $\varepsilon$ and $δ(\varepsilon)$ depending on the regularity of the flux ${\mathfrak f}$, the sequence of solutions $(u_{\varepsilon,δ})$ 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.