Scattering for H^1/2 bounded solutions to the cubic, defocusing NLS in 3 dimensions
By Carlos E.Kenig, F. Merle. 2007
160 citations recorded by the source corpus.
We show that if a solution of the defocusing cubic NLS in 3d remains bounded in the homogeneous Sobolev norm of order 1/2 in its maximal interval of existence, then the interval is infinite and the solution scatters. No radial assumption is made.
Subfields: Control and Systems Engineering, Mathematical Physics, Statistical and Nonlinear Physics
View Paper: https://www.semanticscholar.org/paper/b49d01169377a691e32981104c2e381ccca7ff3b
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In Control and Systems Engineering
In Mathematical Physics
In Statistical and Nonlinear Physics
Citations in the corpus
- An alternative approach to regularity for the Navier–Stokes equations in critical spaces (2009, methodology): “ space XT for any fixed T < T∗(u 0), where XT denotes a space of functions defined on the space-time region R3 ×(0,T). Define |||u||| := sup t∈[0,T∗(u0)) ku(t)k ˙ H 1 2 . The type of result proved in [28] (variants of which were proved in [26, 27] and which we will prove here as well) is that |||u||”
- Global well-posedness and scattering for the mass critical nonlinear Schr{\"o}dinger equation with mass below the mass of the ground state (2011, unknown): “42] showed that for conjecture 1.3 to fail, there must exist a minimal mass blowup solution with a number of additional properties. We show that such a solution cannot occur, proving theorem 1.6. See [23], [25], [24] for more information on this method. Definition 1.4 A set is precompact in L2(Rd) if”
- Global well-posedness and scattering for the defocusing, $L^{2}$-critical, nonlinear Schrödinger equation when $d=2$ (2010, methodology): “for this reason that research on H˙ sc - critical problems has generally concentrated on the L2 or mass - critical problems (p= 4 d) and the H˙ 1 or energy - critical problems (p= 4 d−2). However see [46], [64], and [65] for global well - posedness and scattering when a solution to (1.6) has an assu”
- Scattering for the radial 3D cubic wave equation (2014, methodology): “follow the scheme from [22], which is a refinement of the methods from [20, 21].”
- Global well-posedness and scattering for the defocusing, L2-critical, nonlinear Schrödinger equation when d = 1 (2010, background): “See [53, 70, 69] for global well-posedness and scattering results when a solution to (1.4) has an assumed bound on the ˙ H s c norm, 0 s c 1.”
- On the Probabilistic Cauchy Theory of the Cubic Nonlinear Schrödinger Equation on Rd, d≥3 (2014, methodology): “ the global well-posedness result by Ryckman-Vi¸san [53] and Vi¸san [60]. For other dimensions d≥ 3 with d6= 4, it is not known whether Hypothesis (B) holds. Let us compa re (1.14) and the results in [35] and [39]. Assuming that w∈ L∞ t H˙ scrit x (I ∗× Rd), where I is a maximal interval of existenc”
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- Nondispersive radial solutions to energy supercritical non-linear wave equations, with applications (2008, methodology): “In fact, the concentrationcompactness/rigidity theorem method that we introduced in [18] to study global well-posedness and scattering for the focusing, energy critical non-linear Schrödinger equation in the radial case and which we also applied in non-radial settings to the energy critical, focusi”