Global existence for defocusing cubic NLS and Gross–Pitaevskii equations in three dimensional exterior domains

Anton, Ramona

doi:10.1016/j.matpur.2007.12.006

Cited by 25 publications

(29 citation statements)

References 26 publications

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“…The estimate (4) was used in [4] to obtain Strichartz estimates with a loss of 1/p derivatives, by combining the gain in regularity in (4) with Sobolev embedding, in order to prove space-time integrability estimates near the obstacle. Improved results were obtained by Anton [1], which show that Strichartz estimates hold with a loss of 1 2p derivatives. The approach in [1] combines the local smoothing estimates (4) with a semi-classical parametrix construction, rather than Sobolev embedding.…”

Section: Introductionmentioning

confidence: 88%

“…See for example Strichartz [21], Ginibre and Velo [8], Keel and Tao [15], and references therein. Scale invariant estimates for s > 0 then follow by Sobolev embedding; such estimates will be called subcritical, as their proof does not use the full rate of dispersion for the equation (1). This paper is primarily concerned with proving scale invariant Strichartz estimates on the domain exterior to a non-trapping obstacle in R n , that is, Ω = R n \ K for some compact set K with smooth boundary.…”

Section: Introductionmentioning

confidence: 99%

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Strichartz estimates and the nonlinear Schrödinger equation on manifolds with boundary

2011

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Abstract. We establish Strichartz estimates for the Schrödinger equation on Riemannian manifolds (Ω, g) with boundary, for both the compact case and the case that Ω is the exterior of a smooth, non-trapping obstacle in Euclidean space. The estimates for exterior domains are scale invariant; the range of Lebesgue exponents (p, q) for which we obtain these estimates is smaller than the range known for Euclidean space, but includes the key L 4 t L ∞ x estimate, which we use to give a simple proof of well-posedness results for the energy critical Schrödinger equation in 3 dimensions. Our estimates on compact manifolds involve a loss of derivatives with respect to the scale invariant index. We use these to establish well-posedness for finite energy data of certain semilinear Schrödinger equations on general compact manifolds with boundary.

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Section: Introductionmentioning

confidence: 88%

Section: Introductionmentioning

confidence: 99%

Strichartz estimates and the nonlinear Schrödinger equation on manifolds with boundary

2011

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“…This is the case of compact surfaces when σ 1, and of compact three dimensional manifolds when σ = 1, see [8]. This is also the case of bounded domains in R 2 for σ 1, see [5], of the ball in R 3 for σ = 1 and radial data [3], and of exterior domains in R 3 , for σ = 1 [4].…”

Section: Final Remarksmentioning

confidence: 99%

Loss of regularity for supercritical nonlinear Schrödinger equations

2008

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Abstract. We consider the nonlinear Schrödinger equation with defocusing, smooth, nonlinearity. Below the critical Sobolev regularity, it is known that the Cauchy problem is ill-posed. We show that this is even worse, namely that there is a loss of regularity, in the spirit of the result due to G. Lebeau in the case of the wave equation. As a consequence, the Cauchy problem for energy-supercritical equations is not well-posed in the sense of Hadamard. We reduce the problem to a supercritical WKB analysis. For super-cubic, smooth nonlinearity, this analysis is new, and relies on the introduction of a modulated energy functionalà la Brenier.

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“…This in turn reduces matters to establishing a parametrix for the equation which may only invert the equation locally, say within the domain of a suitable local diffeomorphism. This approach was then adapted to exterior domain problems by Burq-Gérard-Tzvetkov [5] and Anton [1]. However in each case, the parametrix construction involved did not yield scale invariant estimates.…”

Section: Introductionmentioning

confidence: 99%

On Refined Local Smoothing Estimates for the Schrödinger Equation in Exterior Domains

Blair

2014

Communications in Partial Differential Equations

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Abstract. We consider refinements of the local smoothing estimates for the Schrödinger equation in domains which are exterior to a strictly convex obstacle in R n . By restricting the solution to small, frequency dependent collars of the boundary, it is expected that taking its square integral in space-time should exhibit a larger gain in regularity when compared to the usual gain of half a derivative. By a result of Ivanovici, these refined local smoothing estimates are satisfied by solutions in the exterior of a ball. We show that when such estimates are valid, they can be combined with wave packet parametrix constructions to yield Strichartz estimates. This provides an avenue for obtaining these bounds when Neumann boundary conditions are imposed.

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Global existence for defocusing cubic NLS and Gross–Pitaevskii equations in three dimensional exterior domains

Cited by 25 publications

References 26 publications

Strichartz estimates and the nonlinear Schrödinger equation on manifolds with boundary

Strichartz estimates and the nonlinear Schrödinger equation on manifolds with boundary

Loss of regularity for supercritical nonlinear Schrödinger equations

On Refined Local Smoothing Estimates for the Schrödinger Equation in Exterior Domains

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