×

On nonlinear Schrödinger equations. (English) Zbl 0970.35134

This work is devoted to the study of the solutions of the semilinear Schrödinger equation \[ iz_t-\triangle z+|z|^4z=0\text{ in }\mathbb{R}\times R^3 \] under initial condition \(z(0,\mathbf x)=z_0(\mathbf x)\in C^{\infty }\cap L^2\), \(\mathbf x\in \mathbb{R}^3\). It is shown that assuming radial symmetry, the solution remains regular for all time. This is a different proof of a recent result by J. Bourgain. The idea of the proof consists in the use of a new a priori energy estimate proposed by the author. An important observation with respect to the considered problem is that the total energy \(E(t)\) is conserved in time, i.e. \(E(t)=E(0)\). The energy density is positive definite, which justifies the term repulsive for the nonlinear potential. The energy estimate however, does not preclude the possibility that the solution explodes in finite time. The author shows that this cannot happen.
The equation under consideration has been proposed as multiparticle approximation in quantum mechanics, when there exist a lot of quantum particles acting in unison provided a finite number of particle-particle interactions are taken into account.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35K55 Nonlinear parabolic equations
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Bourgain J. Well posedness of defocusing critical nonlinear Schrödinger equation in the radial case. to appear in J. Amer. Math. Soc · Zbl 0958.35126
[2] Ginibre J, Ann. Inst. H. Poincare, Anal. Non Lin 1 pp 309– (1984)
[3] Ginibre J, Ann. Inst. H. Poincare, Anal. Non Lin. 2 pp 309– (1985)
[4] DOI: 10.1063/1.523491 · Zbl 0372.35009 · doi:10.1063/1.523491
[5] DOI: 10.1215/S0012-7094-77-04430-1 · Zbl 0372.35001 · doi:10.1215/S0012-7094-77-04430-1
[6] Cazenave T. An introduction to nonlinear Schrödinger equations. Textos de methods Mathematicos, 22, Universidade Federal do Rio de Janeiro · Zbl 0584.35022
[7] Kato T, Ann. Inst. Henri Poincare, Physique Theorique 46 pp 113– (1987)
[8] DOI: 10.1016/0022-1236(78)90073-3 · Zbl 0395.35070 · doi:10.1016/0022-1236(78)90073-3
[9] Strauss W, Lecture Notes in Physics 73 (1987)
[10] DOI: 10.1007/BF01208265 · Zbl 0527.35023 · doi:10.1007/BF01208265
[11] Bourgain J. Refinements of Strichartz’ inequality and applications to 2D-NLS with critcal nonlinearity. Preprint · Zbl 0917.35126
[12] Bourgain J. Scattering in the energy space for 3-D NLS. Preprint · Zbl 0972.35141
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.