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


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)
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