Endpoint Sobolev Theory for the Muskat Equation

By Thomas Alazard, Quoc-Hung Nguyen. Communications in Mathematical Physics, 2020

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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 $$L^2$$ L 2 functions with three-half derivative in $$L^2$$ 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.

Subfields: Applied Mathematics, Mathematical Physics

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