×

Weak solutions and development of singularities of the SU(2)\(\sigma\)- model. (English) Zbl 0686.35081

The classical (i.e. non-quantum) solutions to the nonlinear \(\sigma\)- models are considered. The following facts are proved:
(i) the existence of smooth solutions for arbitrary smooth initial data in \(1+1\) dimensional nonlinear \(\sigma\)-model with fields taking values in complete Riemann manifold.
(ii) the existence of global weak solution for all initial data of finite energy for the O(4) \(\sigma\)-model in \(3+1\) dimension.
(iii) the existence of smooth initial data to the \(3+1\) dimensional O(4) model which develop singularities in finite time.
To prove these results the existence of conserved quantities and standard tools from functional analysis are used.
Reviewer: P.Maslanka

MSC:

35L70 Second-order nonlinear hyperbolic equations
35B40 Asymptotic behavior of solutions to PDEs
35D05 Existence of generalized solutions of PDE (MSC2000)
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Elles, Bull. London Math. Soc. 10 pp 1– (1978)
[2] Ginibre, Ann. Physics 142 pp 393–
[3] Chao-Hoo, Comm. Pure Appl. Math. 33 pp 727–
[4] Long-time existence of solutions of nonlinear wave equations, Ph.D. Thesis, NYU, 1985,.
[5] Pohlmeyer, Comm. Math. Phys. 46 pp 207– (1976)
[6] Rubin, Comm. Pure Appl. Math. 10 pp 65– (1957)
[7] Schoen, Inventiones Mathematicae 78 pp 89– (1984)
[8] Global existence of harmonic maps in Minkowski space, preprint.
[9] Strauss, An. Acad. Brasil Cienc. 42 pp 643– (1970)
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.