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,\delta} +$\mathrm{d}\mathrm{i}\mathrm{v}$ {\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, ${\mathfrak{f}}_{\epsilon ,\delta }$ and $u_0$ are smooth functions while $\epsilon$ and $\delta =\delta (\epsilon )$ are fixed constants. Assuming ${\mathfrak f}{\varepsilon,\delta} \to {\mathfrak f} \in L^p( $\mathbb{R}$^d\times $\mathbb{R}$;$\mathbb{R}$^d)$forsome$1<p<\infty$,stronglyas$\varepsilon\to 0$, 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 $${\mathrm{\partial }}_{t}u+\mathrm{d}\mathrm{i}\mathrm{v}\mathfrak{f}(\mathbf{x},u)=0.$$ 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.