A Conjecture Regarding Optimal Strichartz Estimates for the Wave Equation

2017 ◽  
pp. 293-299 ◽  
Author(s):  
Neal Bez ◽  
Chris Jeavons ◽  
Tohru Ozawa ◽  
Hiroki Saito
Keyword(s):  
Wave Equation ◽  
Strichartz Estimates

scholarly journals Strichartz estimates for the wave equation on a 2D model convex domain

2021 ◽  
Vol 300 ◽  
pp. 830-880
Author(s):  
Oana Ivanovici ◽  
Gilles Lebeau ◽  
Fabrice Planchon
Keyword(s):  
Wave Equation ◽  
Convex Domain ◽  
Strichartz Estimates ◽  
2D Model

Strichartz Estimates for Wave Equations with Charge Transfer Hamiltonians

2021 ◽  
Vol 273 (1339) ◽  
Author(s):  
Gong Chen
Keyword(s):  
Wave Equation ◽  
Charge Transfer ◽  
Wave Equations ◽  
Linear Models ◽  
Energy Estimate ◽  
Strichartz Estimates ◽  
Content Type ◽  
Scattering States ◽  
Scattering State ◽  
A Charge

We prove Strichartz estimates (both regular and reversed) for a scattering state to the wave equation with a charge transfer Hamiltonian in R 3 \mathbb {R}^{3} : \[ ∂ t t u − Δ u + ∑ j = 1 m V j ( x − v → j t ) u = 0. \partial _{tt}u-\Delta u+\sum _{j=1}^{m}V_{j}\left (x-\vec {v}_{j}t\right )u=0. \] The energy estimate and the local energy decay of a scattering state are also established. In order to study nonlinear multisoltion systems, we will present the inhomogeneous generalizations of Strichartz estimates and local decay estimates. As an application of our results, we show that scattering states indeed scatter to solutions to the free wave equation. These estimates for this linear models are also of crucial importance for problems related to interactions of potentials and solitons, for example, in [Comm. Math. Phys. 364 (2018), no. 1, pp. 45–82].

A center-stable manifold for the energy-critical wave equation in ℝ3 in the symmetric setting

2014 ◽  
Vol 11 (03) ◽  
pp. 437-476 ◽  
Author(s):  
Marius Beceanu
Keyword(s):  
Wave Equation ◽  
Elliptic Equation ◽  
Initial Data ◽  
Stable Manifold ◽  
Weighted Sobolev Space ◽  
Stationary Solutions ◽  
Strichartz Estimates ◽  
Critical Nonlinearity ◽  
Local Center ◽  
New Class

Consider the focusing semilinear wave equation in ℝ3 with energy-critical nonlinearity [Formula: see text] This equation admits stationary solutions of the form [Formula: see text] called solitons, which solve the elliptic equation [Formula: see text] Restricting ourselves to the space of symmetric solutions ψ for which ψ(x) = ψ(-x), we find a local center-stable manifold, in a neighborhood of ϕ(x, 1), for this wave equation in the weighted Sobolev space [Formula: see text] Solutions with initial data on the manifold exist globally in time for t ≥ 0, depend continuously on initial data, preserve energy, and can be written as the sum of a rescaled soliton and a dispersive radiation term. The proof is based on a new class of reverse Strichartz estimates, recently introduced by Beceanu and Goldberg and adapted here to the case of Hamiltonians with a resonance.

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