Computer-assisted proofs in PDE: a survey

J Gómez-Serrano - SeMA Journal, 2019‏ - Springer
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Paralinearization of the Muskat equation and application to the Cauchy problem

T Alazard, O Lazar - Archive for Rational Mechanics and Analysis, 2020‏ - Springer
We paralinearize the Muskat equation to extract an explicit parabolic evolution equation
having a compact form. This result is applied to give a simple proof of the local well …

Endpoint Sobolev theory for the Muskat equation

T Alazard, QH Nguyen - Communications in Mathematical Physics, 2023‏ - Springer
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 …

Lyapunov functions, identities and the Cauchy problem for the Hele–Shaw equation

T Alazard, N Meunier, D Smets - Communications in Mathematical …, 2020‏ - Springer
This article is devoted to the study of the Hele–Shaw equation. We introduce an approach
inspired by the water-wave theory. Starting from a reduction to the boundary, introducing the …

Well-posedness and stability results for some periodic Muskat problems

BV Matioc - Journal of Mathematical Fluid Mechanics, 2020‏ - Springer
We study the two-dimensional Muskat problem in a horizontally periodic setting and for
fluids with arbitrary densities and viscosities. We show that in the presence of surface …