Solvability of a Mixed Problem for the Nonlinear Schrödinger Equation

Vladimirov, M. V.

doi:10.1070/sm1987v058n02abeh003118

Cited by 36 publications

(53 citation statements)

References 5 publications

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“…Notice that Theorem 1 was proved in [3] in the particular case M = R 2 /Z 2 . Let us also mention that logarithmic estimates in Brezis-Gallouët [5], Vladimirov [26] and Ogawa-Ozawa [18] allow to prove Theorem 1 on any compact surface M in the particular case α ≤ 2, with the exception of the uniform continuity of the flow map (2.3) if s = 1.…”

Section: Of Course Local Wellposedness Inmentioning

confidence: 99%

The Cauchy Problem for the Nonlinear Schrödinger Equation on a Compact Manifold

Burq¹,

Gérard²,

Tzvetkov³

2003

JNMP

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We discuss the wellposedness theory of the Cauchy problem for the nonlinear Schrö-dinger equation on compact Riemannian manifolds. New dispersive estimates on the linear Schrödinger group are used to get global existence in the energy space on arbitrary surfaces and three-dimensional manifolds, generalizing earlier results by Bourgain on tori. On the other hand, on specific manifolds such as spheres, new instability phenomena are displayed, leading to some kind of illposednesss in higher dimensions.

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Section: Of Course Local Wellposedness Inmentioning

confidence: 99%

The Cauchy Problem for the Nonlinear Schrödinger Equation on a Compact Manifold

Burq¹,

Gérard²,

Tzvetkov³

2003

JNMP

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“…Recent results have shown that the geometry plays a major role : see for example the wellposedness result in H s on a square of R 2 , Bourgain [1], for s > 0, as opposed to an illposedness result in H s on a disc of R 2 , Burq-Gérard-Tzvetkov [8], for s < . For cubic equations a general results on domains of R 2 is due to Brezis-Gallouet [4] and Vladimirov [25]. This result is based on energy methods and logarithmic inequalities.…”

Section: Introductionmentioning

confidence: 99%

Cubic Nonlinear Schrödinger Equation on Three Dimensional Balls with Radial Data

Anton

2008

Communications in Partial Differential Equations

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We prove wellposedness of the Cauchy problem for the cubic nonlinear Schrödinger equation with Dirichlet boundary conditions and radial data on 3D balls. The main argument is based on a bilinear eigenfunction estimate and the use of X s,b spaces. The last part presents a first attempt to study the non radial case. We prove bilinear estimates for the linear Schrödinger flow with particular initial data.

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“…Theorem 2.1 (Brézis-Gallouët [4], Vladimirov [22]). For every u 0 ∈ H 1 (M ), there exists a unique solution u ∈ C(R, H 1 (M )) of (1.1) with…”

Section: A Brief Review Of the Cauchy Problemmentioning

confidence: 99%

High frequency solutions of the nonlinear Schrödinger equation on surfaces

Burq

Gérard²,

Tzvetkov

2009

Quart. Appl. Math.

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Abstract. We address the problem of describing solutions of the nonlinear Schrödin-ger equation on a compact surface in the high frequency regime. In this context, we introduce a nonnegative threshold, depending on the geometry of the surface, which can be seen as a measurement of the nonlinear character of the equation, and we compute this number for the torus and for the sphere, as a consequence of earlier arguments. The last part is devoted to the study, on the sphere, of the critical regime associated to this threshold. We prove that the effective dynamics are described by a new evolution equation, the Resonant Hermite-Schrödinger equation.

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Solvability of a Mixed Problem for the Nonlinear Schrödinger Equation

Cited by 36 publications

References 5 publications

The Cauchy Problem for the Nonlinear Schrödinger Equation on a Compact Manifold

The Cauchy Problem for the Nonlinear Schrödinger Equation on a Compact Manifold

Cubic Nonlinear Schrödinger Equation on Three Dimensional Balls with Radial Data

High frequency solutions of the nonlinear Schrödinger equation on surfaces

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