Dispersive estimates in Rn n for the Schrodinger and the wave equations

Simon Moulin

Description

Dispersive estimates in Rn (n ≥ 2) for the Schrodinger and the wave equations Simon Moulin Nantes, Laboratoire de Mathematiques Jean Leray 11 avril 2008 - Orleans Simon Moulin (Nantes, LMJL) Dispersive estimates 11 avril 2008 - Orleans 1 / 43

• hardy-littlewood- sobolev inequality


• problem associated


• dispersive estimates


• sobolev embeddings


• linear problems


• schrodinger equation

Sujets

• Nantes
• Moulin
• Schrödinger equation

Informations

• Publié par: pefav
• Publié le: 01 avril 2008
• Nombre de lectures: 12
• Langue: English

Extrait

Dispersive estimates inR n (n≥2) fortheSchr ¨ odingerandthewaveequations

Simon Moulin

Nantes, Laboratoire de Mathe´  matiques Jean Leray

Simon Moulin (Nantes, LMJL)

11 avril 2008 - Orle´  ans

Dispersive estimates

11 avril 2008 - Orle´  ans

1 / 43

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Introduction

New results Wave equation (n≥3) Schro¨ dinger  equation (n≥4) Dimensionn=2, high frequencies

Ideas of the proofs

Conclusion

Simon Moulin (Nantes, LMJL)

Dispersive estimates

11 avril 2008 - Orle´  ans

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1

2

3

4

Introduction

New results Wave equation (n≥3) Schr ¨ odingerequation(n≥4) Dimensionn=2, high frequencies

Ideas of the proofs

Conclusion

Simon Moulin (Nantes, LMJL)

Dispersive estimates

11avril2008-Orl ´ eans

3 / 43

Goals

Plan of study for Cauchy problem associated to evolution problems.

To study the linear problem. To obtain a priori estimates for the linear problem. To study only some equations and to obtain a priori estimates for these equations. To search what properties could be generalized. To study non-linear problems by perturbative methods.

Simon Moulin (Nantes, LMJL)

Dispersive estimates

11avril2008-Orl ´ eans

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A priori estimates

Energy conservation. Dispersive estimates:L xp (R n )→L xq (R n ). Strichartz estimates.

Strichatz estimates are a priori estimates, for example for one-order PDE:

L 2 x (R n )→L p t (0,t;L q x (R n )).

For two-order PDE 2, one have to have more regularity on initial conditions. Ingredients to obtain these estimates: dispersive estimates, T ∗ Targument, Sobolev embeddings (for example Hardy-Littlewood-Sobolev inequality).

Simon Moulin (Nantes, LMJL)

Dispersive estimates

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