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

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57 pages
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


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Dispersive estimates inRn(n2) 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 (n3) Schro¨ dinger equation (n4) Dimensionn=2, high frequencies
Ideas of the proofs
Conclusion
Simon Moulin (Nantes, LMJL)
Dispersive estimates
11 avril 2008 - Orle´ ans
2 / 43
1
2
3
4
Introduction
New results Wave equation (n3) Schr¨odingerequation(n4) 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
4 / 43
A priori estimates
Energy conservation. Dispersive estimates:Lxp(Rn)Lxq(Rn). Strichartz estimates.
Strichatz estimates are a priori estimates, for example for one-order PDE:
L2x(Rn)Lpt(0,t;Lqx(Rn)).
For two-order PDE 2, one have to have more regularity on initial conditions. Ingredients to obtain these estimates: dispersive estimates, TTargument, Sobolev embeddings (for example Hardy-Littlewood-Sobolev inequality).
Simon Moulin (Nantes, LMJL)
Dispersive estimates
11avril2008-Orl´eans
5 / 43