Reaction-diffusion system of equations in non-stationary medium and arbitrary non-smooth domains.
Regularity and uniqueness in quasilinear parabolic systems
Inspired by a problem in steel metallurgy, we prove the existence, regularity, uniqueness, and continuous data dependence of solutions to a coupled parabolic system in a smooth bounded 3D domain, with nonlinear and nonhomogeneous boundary conditions. The nonlinear coupling takes place in the diffusion coefficient. The proofs are based on anisotropic estimates in tangential and normal directions, and on a refined variant of the Gronwall lemma.
Relaxation of the incompressible porous media equation
It was shown recently by Córdoba, Faraco and Gancedo in [1] that the 2D porous media equation admits weak solutions with compact support in time. The proof, based on the convex integration framework developed for the incompressible Euler equations in [4], uses ideas from the theory of laminates, in particular configurations. In this note we calculate the explicit relaxation of IPM, thus avoiding configurations. We then use this to construct weak solutions to the unstable interface problem (the...
Remark on well-posedness and ill-posedness for the KdV equation.