249 citations · 684 across the 32 of their papers we have counts for
6 papers · 2 filters
Endpoint Sobolev theory for the Muskat equation
Thomas Alazard, Quoc-Hung Nguyen
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 Sob…
On the Cauchy problem for the Muskat equation with non-Lipschitz initial data
Thomas Alazard, Quoc-Hung Nguyen
This article is devoted to the study of the Cauchy problem for the Muskat equation. We consider initial data belonging to the critical Sobolev space of functions with three-half de…
On the Cauchy problem for the Muskat equation. II: Critical initial data
Thomas Alazard, Quoc-Hung Nguyen
We prove that the Cauchy problem for the Muskat equation is well-posed locally in time for any initial data in the critical space of Lipschitz functions with three-half derivative…
Cauchy theory for the water waves system in an analytic framework
Thomas Alazard, Nicolas Burq, Claude Zuily
In this paper we consider the Cauchy problem for gravity water waves, in a domain with a flat bottom and in arbitrary space dimension. We prove that if the data are of size $\varep…
Functional inequalities and strong Lyapunov functionals for free surface flows in fluid dynamics
Thomas Alazard, Didier Bresch
This paper is motivated by the study of Lyapunov functionals for four equations describing free surface flows in fluid dynamics: the Hele-Shaw and Mullins-Sekerka equations togethe…
Convexity and the Hele-Shaw equation
Thomas Alazard
Walter Craig's seminal works on the water-waves problem established the importance of several exact identities: Zakharov's hamiltonian formulation, shape derivative formula for the…