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Partial Differential Equations and Applications in Geometry and Physics
The mathematical theory of systems of time-dependent nonlinear hyperbolic and mixed type partial differential equations, more specifically conservation laws, in more than one space dimension is in a very unsatisfactory state. The basic issue of global in time existence of solutions is still an open problem. Since the 1950s the existence and uniqueness for scalar equations was solved in the seminal work of Oleinik and Kruzkov. For systems in one space dimension there is an existence theorem of Glimm for data with small total variation since 1965. The small data requirement was only relaxed for some $2\times 2$ systems by DiPerna in the early eighties. Uniqueness is not completely understood, even in the one-dimensional case, despite some recent progress by Bressan, T.-P.\ Liu and T.\ Yang. This field offers a wealth of open problems for future research. Shock waves are discontinuous weak solutions of the equations. This generalization of solutions in the sense of distributions leads to a serious non-uniqueness problem which necessitates the use of additional so-called entropy conditions in order to select the physically meaningful solutions. Any approximation has to be checked whether it leads to these meaningful solutions. Wang Jinghua (Beijing) and Warnecke (Magdeburg) started their collaboration by partially proving the entropy consistency of large time step schemes. Later the convergence of finite difference approximations for relaxation systems and the Ruijgrok-Wu model in kinetic theory were proven. Also results on convergence rates and error estimates were achieved, some jointly with Wang"s former student Liu H. Traveling wave solutions for conservation laws with viscous and dispersive perturbations are smooth approximations of shock waves. The existence and stability of solutions that are perturbations of such traveling waves was proven by Pan Jun, Chinese doctoral student of Warnecke (Magdeburg).
Shock waves, conservation laws
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