Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Quasilinear rough partial differential equations with transport noise   We investigate the Cauchy problem for a quasilinear equation with transport
rough input of the form $\mathrm{d} u-\partial_i(a^{ij}(u)\partial_j
u)\mathrm{d} t =\mathrm{d} \mathbf{X}_t^i(x)\partial_i u_t,$ $u_0\in L^2$ on
the torus $\mathbb T^d$, where $\mathbf{X}$ is two-step enhancement of a family
of coefficients $(X^i_t(x))_{i=1,\dots d}$, akin to a geometric rough path with
H\"older regularity $\alpha>1/3.$ Using energy estimates, we provide sufficient
conditions that guarantee existence in any dimension, and uniqueness in the
case when $X$ is divergence-free. We then focus on the one-dimensional
scenario, with slightly more regular coefficients. Improving the a priori
estimates of the first results, we prove existence of a class of solutions
whose spatial derivatives satisfy a Ladyzhenskaya-Prodi-Serrin type condition.
Uniqueness is shown in the same class, by obtaining an $L^\infty(L^1)$ estimate
on the difference of two solutions. The latter is obtained by establishing a
link with a certain backward dual equation combined with a (rough) iteration
lemma \`a la Moser.

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The categories of this paper are: ao-sci
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The categories of this paper are: dg-ga
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The categories of this paper are: hep-ph
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The categories of this paper are: physics
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The categories of this paper are: supr-con
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The categories of this paper are: cond-mat
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The categories of this paper are: eess
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The categories of this paper are: acc-phys
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The categories of this paper are: q-fin
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The categories of this paper are: nucl-th
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The categories of this paper are: astro-ph
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The categories of this paper are: funct-an
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The categories of this paper are: cmp-lg
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The categories of this paper are: patt-sol
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The categories of this paper are: q-alg
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The categories of this paper are: atom-ph