Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Endpoint Sobolev theory for the Muskat equation   This paper is devoted to the study of solutions with critical regularity for
the two-dimensional Muskat equation. We prove that the Cauchy problem is
well-posed on the endpoint Sobolev space of $L^2$ functions with three-half
derivative in $L^2$. This result is optimal with respect to the scaling of the
equation. One well-known difficulty is that one cannot define a flow map such
that the lifespan is bounded from below on bounded subsets of this critical
Sobolev space. To overcome this, we estimate the solutions for a norm which
depends on the initial data themselves, using the weighted fractional
Laplacians introduced in our previous works. Our proof is the first in which a
null-type structure is identified for the Muskat equation, allowing to
compensate for the degeneracy of the parabolic behavior for large slopes.

25
The categories of this paper are: patt-sol
12
The categories of this paper are: nucl-th
18
The categories of this paper are: econ
22
The categories of this paper are: chao-dyn
23
The categories of this paper are: solv-int
26
The categories of this paper are: bayes-an
33
The categories of this paper are: plasm-ph
27
The categories of this paper are: comp-gas
16
The categories of this paper are: stat
20
The categories of this paper are: acc-phys
14
The categories of this paper are: math-ph
19
The categories of this paper are: eess
0
The categories of this paper are: hep-ph
37
The categories of this paper are: supr-con
32
The categories of this paper are: atom-ph
31
The categories of this paper are: ao-sci