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We prove existence and uniqueness of a solution to the Cauchy problemcorresponding to the equation \begin{equation*} \begin{cases} \partial_tu_{\varepsilon,\delta} +\mathrm{div} {\mathfrak f}_{\varepsilon,\delta}({\bfx}, u_{\varepsilon,\delta})=\varepsilon \Deltau_{\varepsilon,\delta}+\delta(\varepsilon) \partial_t \Deltau_{\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 anappropriate relationship between $\varepsilon$ and $\delta(\varepsilon)$depending on the regularity of the flux ${\mathfrak f}$, the sequence ofsolutions $(u_{\varepsilon,\delta})$ strongly converges in$L^1_{loc}(\mathbb{R}^+\times \mathbb{R}^d)$ towards a solution to theconservation 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 theoremfor the first part and reduction to the kinetic formulation combined withrecent results in the velocity averaging theory for the second.

A vanishing dynamic capillarity limit equation with discontinuous flux