On the Defect of Compactness for the Strichartz Estimates of the Schrödinger Equations

By Sahbi Keraani. 2001

298 citations recorded by the source corpus.

Subfields: Applied Mathematics, Control and Systems Engineering, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/40390816e0c6dddf424cd2b937aff6378f2e0770

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Citations in the corpus

The linear profile decomposition for the Airy equation and the existence of maximizers for the Airy Strichartz inequality (2008, methodology): “For the wave equations, Bahouri-Gérard [1] established a linear profile decomposition for the energy critical wave equation in R3 (their argument can be generalized to higher dimensions).”
On the global well-posedness of energy-critical Schr\"odinger equations in curved spaces (2010, methodology): “In other words, we prove the analogue of Keraani’s theorem [42] in the hyperbolic geometry.”
The cubic nonlinear Schr\"odinger equation in two dimensions with radial data (2007, unknown): “Earlier steps in this direction include [1, 5, 36, 37, 45].”
Nondispersive radial solutions to energy supercritical non-linear wave equations, with applications (2008, methodology): “The profile decomposition was simultaneously discovered by Merle–Vega [26] in the mass critical NLS for d = 2 context and later developed by Keraani [20] for the energy critical NLS.”
Maximizers for the Strichartz Inequalities for the wave equation (2009, methodology): “We first prove the profile decomposition with the diagonal pair q = r = 2(d+1) d−2 , based on the arguments given in [1], [15] and [8].”
Global well-posedness and scattering for the defocusing, $L^{2}$-critical nonlinear Schrödinger equation when $d =3$ (2009, methodology): “See [18], [19], [20] for more information on this method.”
An alternative approach to regularity for the Navier–Stokes equations in critical spaces (2009, methodology): “For example, in [28] the following decomposition (based on [29]) was used for treatment of the NLS case: There exists a sequence {V0,j }∞j=1 ⊆ Ḣ1/2, with associated linear solutions V l j (x, t) = e V0,j , and sequences of scales λj,n ∈ R+ and shifts xj,n ∈ R3 and tj,n ∈ R, such that (after a subseq”
On the blow up phenomenon of the critical nonlinear Schrödinger equation (2006, background): “Gérard [1] and for H 1-critical Schrödinger equations by the author [8] (see also [5,6]).”
Blowup theory for the critical nonlinear Schrödinger equations revisited (2005, unknown): “Gérard [4] (see also [1] and [6]).”
Quintic NLS in the exterior of a strictly convex obstacle (2012, background): “Existence of a minimal blowup solution for the Euclidean energy-critical problem was proved by Kenig–Merle [41] (see also [2, 42] for some ingredients), who were also the first to realize the value of this result for well-posedness arguments.”
Scattering for the non-radial 3D cubic nonlinear Schroedinger equation (2007, methodology): “Remark 2.2. If the assumption that n is uniformly bounded in H1 is weakened to the assumption that n is uniformly bounded in ˙ H1/2, then the above profile decomposition remains valid provided a scaling parameter is also involved, similar to the theorem in [ 11 ].”
The cubic fourth-order Schrödinger equation (2008, methodology): “Now, thanks to (5.4), we may follow the analysis in Bahouri and Gerard [2] and Keraani [21].”
Maximizers for the Strichartz and the Sobolev-Strichartz inequalities for the Schrödinger equation (2009, background): “By following their approaches, we will generalize both Keraani’s and Killip-Visan’s improved Ḣ1x-Sobolev embeddings to those with Ḣ s x norms where 1 r + s d = 1 2 and d ≥ 1 in the appendix of this paper.”
The mass-critical nonlinear Schr\"odinger equation with radial data in dimensions three and higher (2007, methodology): “Building on earlier developments in [1, 4, 24, 25, 30], Kenig and Merle [23] used this method to treat the energycritical focusing problem with radial data in dimensions three, four, and five.”
A refinement of the Strichartz inequality for the wave equation with applications (2012, methodology): “For profile decompositions for the Schrödinger equation, see [2], [8], [35], [45] and [54], and for the Klein–Gordon equation see [38].”
Global well-posedness and scattering for the higher-dimensional energy-critical non-linear Schrodinger equation for radial data (2004, unknown): “4For some results in the same spirit, showing that “bubbles” are the only obstruction to global existence, see [15].”