Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Sharp Regularity for Weak Solutions to the Porous Medium Equation   Let $u$ be a nonnegative, local, weak solution to the porous medium equation
for $m\ge2$ in a space-time cylinder $\Omega_T$. Fix a point
$(x_o,t_o)\in\Omega_T$: if the average \[
a{\buildrel\mbox{def}\over{=}}\frac1{|B_r(x_o)|}\int_{B_r(x_o)}u(x,t_o)\,dx>0,
\] then the quantity $|\nabla u^{m-1}|$ is locally bounded in a proper
cylinder, whose center lies at time $t_o+a^{1-m}r^2$. This implies that in the
same cylinder the solution $u$ is H\"older continuous with exponent
$\alpha=\frac1{m-1}$, which is known to be optimal. Moreover, $u$ presents a
sort of instantaneous regularisation, which we quantify.

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The categories of this paper are: patt-sol
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The categories of this paper are: nucl-th
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The categories of this paper are: funct-an
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The categories of this paper are: q-fin
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The categories of this paper are: alg-geom
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The categories of this paper are: econ
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The categories of this paper are: adap-org
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The categories of this paper are: chao-dyn
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The categories of this paper are: acc-phys
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The categories of this paper are: ao-sci
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The categories of this paper are: atom-ph
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The categories of this paper are: plasm-ph
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The categories of this paper are: bayes-an
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The categories of this paper are: chem-ph
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The categories of this paper are: gr-qc
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The categories of this paper are: physics