Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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{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.

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The categories of this paper are: mtrl-th
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The categories of this paper are: eess
10
The categories of this paper are: q-bio
26
The categories of this paper are: bayes-an
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The categories of this paper are: acc-phys
31
The categories of this paper are: ao-sci
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The categories of this paper are: q-fin
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The categories of this paper are: hep-th
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The categories of this paper are: math-ph
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The categories of this paper are: physics
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The categories of this paper are: alg-geom
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The categories of this paper are: q-alg
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The categories of this paper are: chao-dyn
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The categories of this paper are: supr-con
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The categories of this paper are: nlin
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The categories of this paper are: comp-gas