Colloquium
3:00 p.m., Friday (Jan. 24th)
Math Annex 1100
Tao Luo
Department of Mathematics
University of Michigan, Ann Arbor
Qualitative behaviour of Solutions of
Nonlinear Partial Differential Equations of Balance Laws
Fundamental balance laws of mass, momentum and energy can be
described by nonlinear partial differential equations. Prototypical
examples of these equations are celebrated Euler equations and
NavierStokes equations for fluids, EulerPoisson equations for
fluids with selfgravitation and hydrodynamical model of
semiconductors and their variation by taking into account of
various additional effects, such as relaxation in gas flow which
is not in local thermodynamical equilibrium, and reaction in
combustion, etc. The solutions of these nonlinear equations
always exhibit very singular behavior, such as shock waves,
rarefaction waves, turbulence, phase transition, vacuum states
and detonation waves. Understanding these nonlinear phenomena
poses challenging problems in mathematical theory and applications,
and thus has been one of the driving forces of modern applied
mathematics.
In this talk, I will recall background of the subject, and survey
my works, joint with L. Hsiao, D. Serre, J. Smoller, Z. Xin and
T. Yang etc, on the following topics
1) Shock wave theory for hyperbolic systems with relaxation;
2) Phase transition in nonlinear elasticity with viscosity
and heat conductivity;
3) Free boundary problem of compressible NavierStokes equations;
4) Rotating fluids with selfgravitation.
Refreshments will be served at 2:45 p.m. in the Faculty Lounge,
Math Annex (Room 1115).
