Mathematical Sciences: Problems in Conservation Laws

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9404384 Zumbrun This project concerns the study of nonlinear systems of partial differential equations. It consists of two main programs, in parabolic and hyperbolic conservation laws, respectively. The first project involves the study of stability of waves in viscous conservation laws. This project is intimately connected with the central problems of hyperbolic admissibility and the inviscid limit. Current theory, based on energy methods, remains ad hoc and incomplete. A new, pointwise stability analysis is proposed for the treatment of several open problems, including: Lebesgue integrability behavior, rarefactions, multiple wave patterns, undercompressive and "fake Lax" shocks, weak deflagration waves, multi-dimensional fronts in MHD, and nonuniform convergence of shock capturing schemes. The second project involves refined wave tracing methods for hyperbolic conservation laws. Wave tracing gives a great deal of information about approximate solutions obtained by the Glimm random choice scheme. It is proposed that, by refined accounting techniques, more of this information can be extracted in the limiting process. Previously, decay and convergence to N-waves have been established for nonconvex systems. More recently, existence and decay have been shown for periodic solutions of nxn, nonresonant systems, generalizing the work of Glimm and Lax for 2x2 systems. It is planned to study periodic solutions of nonconvex and of resonant systems, and, ultimately, continuous dependence on initial data. This project deals with equations of applied mathematics. In particular, the equations of continuum mechanics will be study. Emphasis will be placed on the study of stability and convergence of viscous shock and rarefaction waves. The analysis can be applied to real world problems encountered, for example, in gas dynamics. ***