Fundamental balance laws of mass, momentum and energy can be described by nonlinear partial differential equations. Prototypical examples are the celebrated Euler equations and Navier-Stokes equations for fluids, Euler-Poisson equations for fluids with self-gravitation, and hydrodynamical model of semiconductors, and their variants by taking into account 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 on 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 Navier-Stokes equations;
4. Rotating fluids with self-gravitation.