TRANSIC · Global existence vs Blow-up in some nonlinear PDEs arising in fluid mechanics

The present project aims at studying qualitative properties of some nonlinear Partial Differential Equations arising in fluid mechanics. It is divided into 3 parts.Part 1 and Part 2 address the study of some classes of 1D hydrodynamic models, namely, the inviscd Surface Quasi-Gesotrophic equation (SQG) and the generalized Constantin-Lax-Majda (gCLM) equation. Both models are closely related to the 3D Euler equation written in terms of the vorticity and are therefore mathematically interesting. More specifically, Part 1 is devoted to the study of particular solutions of the inviscid (SQG) equation which blow up in finite time. Those particular solutions turn out to satisfy a 1D non local equation which are a particular case of (gCLM) equation. Therefore, we focus on that 1D equation and we prove finite time blow-up by using methods coming from harmonic analysis and the so-called ""nonlocal maximal principle"" or the ""modulus of continuity method"" introduced by Kiselev, Nazarov and Volberg.In contrast to Part 1, Part 2 is devoted to the proof of a global existence theorem for another particular case of (gCLM) equation. Unlike Part 1 where the ""modulus of continuity method"" will be used only in one step of the proof, Part 2 is completly based on the use of the ""modulus of continuity method"".Finally, Part 3 deals with the Muskat problem which describes the interface between two fluids of different density but same viscosity. This part is centered around a global existence result due to Constantin, Cordoba, Gancedo, Strain and is based on the use of a new formulation of the Muskat problem recently obtained by Lazar.""