Formation of Singularities for a Conservation Law with Memory.
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WISCONSIN UNIV-MADISON MATHEMATICS RESEARCH CENTER
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Problems arising in continuum mechanics can often be modeled by quasilinear hyperbolic systems in which the characteristic speeds are not constant. Such systems have the property that waves may be amplified and solutions that were initially smooth may develop discontinuities shocks in finite time. Of particular interest are situations in which the destabilizing mechanism arising from nonlinear effects can coexist and compete with dissipative effects. An interesting situation arises when the amplification and dissipative mechanisms are nearly balanced and the outcome of their confrontation cannot be predicted at the outset. Examples are provided by quasilinear second order wave equations with first order frictional damping it has been shown that when the initial data are sufficiently smooth and small in suitable norms, classical solutions exist globally in time. However, if the smooth initial data become sufficiently large in a precise sense, the smooth solution develops a singularity in finite time, no matter how smooth one takes the data. Thus the dissipative mechanism is not sufficiently powerful to prevent the breaking of waves for large enough data.
- Numerical Mathematics