Pointwise and Semigroup Methods in Viscous Conservation Laws and Completely Integrable Systems

Texas A&M Research Foundation, College Station TX

Investigators

Abstract

Viscous conservation laws arise in a wide variety of physical applications, including fluid dynamics, magnetohydrodynamics, and materials science. Of particular importance are solutions of such equations that are stable and hence typically correspond with observable phenomena. Unfortunately, establishing the stability of these solutions has proven to be a quite difficult problem. The pointwise Green's function approach, however, initiated by Liu and developed by Liu and his collaborators, has proven quite robust: in applications to viscous shock waves arising in single conservation laws of arbitrary order, viscous shock waves arising in systems with second order diffusion, planar viscous shock waves, degenerate viscous shock waves, and rarefaction waves. We propose to continue and extend this promising line of research in three directions. First, new techniques recently developed by Howard and Zumbrun appear suitable for extension to (i) systems of viscous conservation laws admitting degenerate viscous shock waves, and (ii) systems of viscous conservation laws with high order viscosity. Second, we propose to develop further techniques that will extend the pointwise Green's function approach to the case of viscous rarefaction waves. Finally, we would like to incorporate new techniques recently developed in the context of perturbation theory for completely integrable systems into the study of the necessarily oscillatory dynamics that arise in viscous conservation laws of order higher than two. The conservation of such fundamental properties as energy and momentum often leads to partial differential equations that model some underlying physical process. For example, the Navier-Stokes equations of fluid dynamics and the Maxwell equations of electromagnetism follow this paradigm. Of primary concern are stable phenomena: those whose principal structure is robust to minor environmental fluctuations. We propose to continue and extend a promising line of research that has been extraordinarily successful in establishing a clear criterion for such stability. A direct consequence of the approach is a detailed understanding of certain fundamental partial differential equations.