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

Acronym: TRANSIC.

Scientist in charge: Diego Córdoba.

Experienced Researcher: Omar Lazar

Project Reference: 624450

Start Date: 2014-10-01

End Date: 2016-09-30

Abstract:

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.