TRANSIC

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

Coordinatore	AGENCIA ESTATAL CONSEJO SUPERIOR DE INVESTIGACIONES CIENTIFICAS    more  Organization address address: CALLE SERRANO 117 city: MADRID postcode: 28006 contact info Titolo: Ms. Nome: Ana María Cognome: De La Fuente Email: send email Telefono: 34915681709
Nazionalità Coordinatore	Spain [ES]
Totale costo	166˙336 €
EC contributo	166˙336 €
Programma	FP7-PEOPLE Specific programme "People" implementing the Seventh Framework Programme of the European Community for research, technological development and demonstration activities (2007 to 2013)
Code Call	FP7-PEOPLE-2013-IEF
Funding Scheme	MC-IEF
Anno di inizio	2014
Periodo (anno-mese-giorno)	2014-10-01   -   2016-09-30

 Partecipanti

#	participant	country	role	EC contrib. [€]
1	AGENCIA ESTATAL CONSEJO SUPERIOR DE INVESTIGACIONES CIENTIFICAS  Organization address address: CALLE SERRANO 117 city: MADRID postcode: 28006 contact info Titolo: Ms. Nome: Ana María Cognome: De La Fuente Email: send email Telefono: 34915681709	ES (MADRID)	coordinator	166˙336.20

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 Obiettivo del progetto (Objective)

'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.'