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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)
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