421 research outputs found
Concentration Of Laplace Eigenfunctions And Stabilization Of Weakly Damped Wave Equation
- In this article, we prove some universal bounds on the speed of
concentration on small (frequency-dependent) neighborhoods of submanifolds of L
2-norms of quasi modes for Laplace operators on compact manifolds. We deduce
new results on the rate of decay of weakly damped wave equations.
R{\'e}sum{\'e}
Laplace Eigenfunctions And Damped Wave Equation Ii: Product Manifolds
- The purpose of this article is to study possible concentrations of
eigenfunc-tions of Laplace operators (or more generally quasi-modes) on product
manifolds. We show that the approach of the first author and Zworski [10, 11]
applies (modulo rescalling) and deduce new stabilization results for weakly
damped wave equations which extend to product manifolds previous results by
Leautaud-Lerner [12] obtained for products of tori
On well-posedness for the Benjamin-Ono equation
We prove existence of solutions for the Benjamin-Ono equation with data in
, . Thanks to conservation laws, this yields global solutions for
data, which is the natural ``finite energy'' class. Moreover,
inconditional uniqueness is obtained in , which
includes weak solutions, while for , uniqueness holds in a
natural space which includes the obtained solutions.Comment: Important changes. We improved both existence and uniqueness results.
In particular, uniqueness holds in the natural energy
spac
Multilinear Eigenfunction Estimates And Global Existence For The Three Dimensional Nonlinear Schr\"Odinger Equations
We study nonlinear Schr\"odinger equations, posed on a three dimensional
Riemannian manifold . We prove global existence of strong solutions on
and as far as the nonlinearity is defocusing and
sub-quintic and thus we extend the results of Ginibre-Velo and Bourgain who
treated the cases of the Euclidean space and the flat torus \T^3
respectively. The main ingredient in our argument is a new set of multilinear
estimates for spherical harmonics.Comment: Lemma 4.6 in the previous version was false, we made slight
modifications to use only a weaker version of this lemm
Long time dynamics for damped Klein-Gordon equations
For general nonlinear Klein-Gordon equations with dissipation we show that
any finite energy radial solution either blows up in finite time or
asymptotically approaches a stationary solution in . In
particular, any global solution is bounded. The result applies to standard
energy subcritical focusing nonlinearities ,
1\textless{}p\textless{}(d+2)/(d-2) as well as any energy subcritical
nonlinearity obeying a sign condition of the Ambrosetti-Rabinowitz type. The
argument involves both techniques from nonlinear dispersive PDEs and dynamical
systems (invariant manifold theory in Banach spaces and convergence theorems)
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