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Initial boundary value problem for generalized Zakharov equations

Shujun You, Boling Guo, Xiaoqi Ning (2012)

Applications of Mathematics

This paper considers the existence and uniqueness of the solution to the initial boundary value problem for a class of generalized Zakharov equations in ( 2 + 1 ) dimensions, and proves the global existence of the solution to the problem by a priori integral estimates and the Galerkin method.

Injections de Sobolev probabilistes et applications

Nicolas Burq, Gilles Lebeau (2013)

Annales scientifiques de l'École Normale Supérieure

On démontre dans cet article des versions probabilistes des injections de Sobolev sur une variété riemannienne compacte, ( M , g ) . Plus précisément on démontre que pour des mesures de probabilité naturelles sur l’espace L 2 ( M ) , presque toute fonction appartient à tous les espaces L p ( M ) , p < + ∞ . On donne ensuite des applications à l’étude des harmoniques sphériques sur la sphère 𝕊 d  : on démontre (encore pour des mesures de probabilité naturelles) que presque toute base hilbertienne de L 2 ( 𝕊 d ) formée d’harmoniques sphériques...

Invariants, conservation laws and time decay for a nonlinear system of Klein-Gordon equations with Hamiltonian structure

Changxing Miao, Youbin Zhu (2006)

Applicationes Mathematicae

We discuss invariants and conservation laws for a nonlinear system of Klein-Gordon equations with Hamiltonian structure ⎧ u t t - Δ u + m ² u = - F ₁ ( | u | ² , | v | ² ) u , ⎨ ⎩ v t t - Δ v + m ² v = - F ₂ ( | u | ² , | v | ² ) v for which there exists a function F(λ,μ) such that ∂F(λ,μ)/∂λ = F₁(λ,μ), ∂F(λ,μ)/∂μ = F₂(λ,μ). Based on Morawetz-type identity, we prove that solutions to the above system decay to zero in local L²-norm, and local energy also decays to zero if the initial energy satisfies E ( u , v , ℝ ⁿ , 0 ) = 1 / 2 ∫ ℝ ⁿ ( | ∇ u ( 0 ) | ² + | u t ( 0 ) | ² + m ² | u ( 0 ) | ² + | ∇ v ( 0 ) | ² + | v t ( 0 ) | ² + m ² | v ( 0 ) | ² + F ( | u ( 0 ) | ² , | v ( 0 ) | ² ) ) d x < ∞ , and F₁(|u|²,|v|²)|u|² + F₂(|u|²,|v|²)|v|² - F(|u|²,|v|²) ≥ aF(|u|²,|v|²) ≥ 0, a > 0.

Currently displaying 201 – 220 of 475