Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case

By Carlos E. Kenig, F. Merle. 2006

978 citations recorded by the source corpus.

We prove, for the energy critical, focusing NLS, that for data whose energy is smaller than that of the standing wave, and whose homogeneous Sobolev norm H^1 is smaller than that of the standing wave and which is radial, we have global well-posedness and scattering in dimensions 3, 4 and 5. This is sharp since if the data is in the inhomogeneous Sobolev space H^1, of energy smaller than the standing wave but of larger homogeneous H^1 norm, we have blow-up in finite time. The result follows from a general method that we introduce into this type of critical problem. By concentration-compactness we produce a critical element, which modulo the symmetries of the equation is compact, has minimal energy among those which fail to have the conclusion of our theorem. In addition, we show that the dilation parameter in the symmetry, for this solution, can be taken strictly positive.We then establish a rigidity theorem that shows that no such compact, modulo symmetries, object can exist. It is only at this step that we use the radial hypothesis.The same analysis, in a simplified form, applies also to the defocusing case, giving a new proof of results of Bourgain and Tao.

Subfields: Applied Mathematics, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/1a655b298ead35c938e860855b800e94ac7f55d1

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In Applied Mathematics

1. B. Dodson
2. Carlos E. Kenig
3. F. Merle
4. Aynur Bulut
5. Pierre Raphael
6. Terence Tao
7. Raphael Cote
8. A. Lawrie

In Mathematical Physics

1. F. Merle
2. Carlos E. Kenig
3. B. Dodson
4. Terence Tao
5. Aynur Bulut
6. Pierre Raphael
7. Rowan Killip
8. Monica Visan

In Statistical and Nonlinear Physics

1. F. Merle
2. Carlos E. Kenig
3. Pierre Raphael
4. B. Dodson
5. Yvan Martel
6. Monica Visan
7. Rowan Killip
8. Terence Tao

Citations in the corpus

• Scattering for the non-radial 3D cubic nonlinear Schroedinger equation (2007, methodology): “The argument of [7] in the radial case followed a strategy introduced by Kenig-Merle [ 9 ] for proving global well-posedness and scattering for the focusing energy-critical NLS.”
• Soliton resolution for equivariant wave maps to the sphere (2013, methodology): “We now do an induction on the energy in the spirit of the Kenig Merle concentration compactness argument [15, 16].”
• MODIFIED SCATTERING FOR THE CUBIC SCHRÖDINGER EQUATION ON PRODUCT SPACES AND APPLICATIONS (2013, background): “jujp 1u; p>1 (1.2) dates back at least to [21]. The rst natural question is the issue of global existence of solutions, and many works have investigated this problem in dierent geometric settings [4, 5, 6, 14, 19, 23, 24, 25, 30, 37, 50, 52, 57, 58, 59, 60, 61, 62, 69, 71, 72, 77, 84, 89]. The co”
• Blow up for the critical gKdV equation II: minimal mass dynamics (2012, methodology): “[14] for similar arguments in the case of nonlinear Schrödinger equation), and means that scattering is a stable regime without any assumption of size on the solution.”