Global existence for energy critical waves in 3-d domains

By Nicolas Burq, Gilles Lebeau, F. Planchon. 2006

134 citations recorded by the source corpus.

We prove that the defocusing quintic wave equation, with Dirichlet boundary conditions, is globally well posed on for any smooth (compact) domain . The main ingredient in the proof is an spectral projector estimate, obtained recently by Smith and Sogge, combined with a precise study of the boundary value problem.

Subfields: Control and Systems Engineering, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/1e100d05f12acf56a4a1e0b631a81b85bb512927

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Control and Systems Engineering

In Mathematical Physics

Citations in the corpus

Decay estimates for variable coefficient wave equations in exterior domains (2008, background): “Certain global-in-time Strichartz estimates are also known in exterior domains, but, except for certain very special cases (see [4], [2], which are closely based on [21]), require that the obstacle be strictly convex.”
Stabilization for the semilinear wave equation with geometric control condition (2012, methodology): “On a bounded domain, the first estimates were proved by Burq, Lebeau and Planchon [Burq et al. 2008] for q ∈ [5,+∞] and extended to a larger range by Blair, Smith and Sogge in [Blair et al.”
On abstract Strichartz estimates and the Strauss conjecture for nontrapping obstacles (2008, background): “Recently, there has been work on proving local Strichartz estimates when X = L r (Ω) for more general exterior domains ([5], [6], [3], [34]), but only partial results for a more restrictive range of exponents than the ones described in Corollary 1.6 have been obtained.”
Strichartz estimates for the wave equation on manifolds with boundary (2008, background): “Precisely, one rescales R1+n by θj , and considers the symbol q(x, ξ ′) = θjpj ( θjx, θ −1ξ ′ ) , j where pj is such that q̂(x1, ζ, ξ ′) is supported in |ζ | cμ1/2, where μ = θjλ is the frequency scale at which uj (θj x) is localized.”
Quintic NLS in the exterior of a strictly convex obstacle (2012, background): “More recently, this result (plus scattering) has been shown to hold outside convex obstacles [63] and (without scattering) in interior domains [12].”
Random data Cauchy theory for supercritical wave equations I: local theory (2007, result): “From the results by Lebeau, Planchon and the first author [7] (see also section 6), one can show that the Cauchy problem is well posed in H D,N(M).”
Dispersion for the wave equation inside strictly convex domains I: the Friedlander model case (2012, unknown): “Appendix A provides details on how to combine these new estimates with arguments from [4] to obtain the large data case.”
Invariant measure for a three dimensional nonlinear wave equation (2007, background): “We hope that our elementary proof may be of independent interest (see [4] for Strichartz inequalities for the wave equation posed on a domain with boundary).”
On existence and uniqueness of solutions for semilinear fractional wave equations (2015, methodology): “In the case of Ω a smooth bounded domain of R3, [2] proved the global well-posedness in the critical case b = 5.”
Injections de Sobolev probabilistes et applications (2011, background): “Il est possible que les résultats de la section 5 puissent aussi être étendus à ce cadre, mais la plage des estimations de Strichartz valides dans ce cadre des problèmes aux limites étant plus restreinte (voir [11, 20, 6]), cela nécessiterait une analyse plus précise.”
On the global well-posedness of energy-critical Schr\"odinger equations in curved spaces (2010, background): “See for example [3, 4, 16, 17, 27, 28, 34, 35, 36, 38, 41, 43, 44, 48, 49, 52] and the book [54] for further discussion and references.”