Formation of singularities for a class of nonlinear, hyperbolically degenerate initial-boundary value problems (original) (raw)

1992, Applied Mathematics Letters

The singular limit of a hyperbolic system and the incompressible limit of solutions with shocks and singularities in nonlinear elasticity

Quarterly of Applied Mathematics, 1997

Discontinuous solutions with shocks for a family of almost incompressible hyperelastic materials are studied. An almost incompressible material is one whose deformations are not a priori constrained but whose stress response reacts strongly (of order ε − 1 {\varepsilon ^{ - 1}} ) to deformations that change volume. The material class considered is isotropic and admits motions that are self-similar, exhibit cavitation, and are energy minimizing. For the initial-value problem when considering the entire material, the solutions converge (as ε \varepsilon tends to zero) to an isochoric solution of the limit (incompressible) system with the corresponding arbitrary hydrostatic pressure being the singular limit of the pressures in the almost incompressible materials. The shocks, if they exist, disappear: their speed tends to infinity and their strength tends to zero.

Singularity Formation in Systems of Non-strictly Hyperbolic Equations

1995

We analyze nite time singularity formation for two systems of hyperbolic equations. Our results extend previous proofs of breakdown concerning 2 2 non-strictly hyperbolic systems to nn systems, and to a situation where, additionally, the condition of genuine nonlinearity is violated throughout phase space. The systems we consider include as special cases those examined by Keytz and Kranzer and by Serre. They take the form ut +( (u )u )x =0 ; where is a scalar-valued function of the n-dimensional vector u ,a nd u t +(u )ux =0 ;

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