Mathematics Colloquia and Seminars

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The classical solutions to many partial differential equations fail to exist in finite time even if the initial data are smooth. Classical examples include hyperbolic conservation laws and Hamilton-Jacobi equations. In such cases viscosity solutions have been introduced which allow the selection of physically relevant weak solutions beyond the singularity time.

We introduce the relaxation approximation to such partial differential equations by replacing such equations with a semi-linear hyperbolic systems with stiff relaxations. While the viscosity regularization arises from the Navier-Stokes approximation to the Euler equations, the relaxation approximation is analogous to the regularization of the Euler equations by the more fundamental Boltzmann equation.

The relaxation approximation introduces a physically natural way to select the entropy or viscosity solution to hyperbolic conservation laws and Hamilton-Jacobi equation. The semilinear nature of the relaxation system paves a new way to derive numerical schemes ( known as the relaxation schemes) that are simple, efficient, and Riemann solver free, with a high resolution, for problems involving shocks, fronts, or other type of discontinuous solutions.