Stephen J. Watson : A Priori Bounds in Thermo-Viscoelasticity with Phase Transitions

The object of our study is the set of equations of thermo-elasticity with viscosity and heat conduction. These equations include, as a special case, the compressible Navier-Stokes equation familiar from gas dynamics, but in addition allow for solid-like materials. We seek to understand the temporal asymptotic fate of large initial data under a variety of boundary conditions. The realm of phase changes, such as occur in Van-der-Waals gases and martensitic transformations, are of especial interest. Now, obtaining point-wise a priori bounds on the density which are time independent is a major analytical obstacle to resolving this question. We present two new results on this issue. First, for specified-stress boundary conditions we give a positive result applicable to a general class of materials. Second, for Dirichlet boundary conditions we derive the estimates for a special class of gaseous materials; p'th power gases. We conclude with a discussion on the relation between asymptotic states and minimization principles of associated free energies. Numerical simulations will highlight some surprising features of the dynamics. In particular, the limiting states are not necessarily strong minimizers, in the sense of the calculus of variations, of the free energy.

• Category: Applied Math and Analysis
• Duration: 01:01:24
• Date:  April 17, 2000 at 4:00 PM
• Views: 26
• Tags:   seminar, Applied Math Seminar