Endpoint Sobolev Theory for the Muskat Equation

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