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

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Viscous conservation laws arise in a wide variety of physical applications, including fluid dynamics, magnetohydrodynamics,and materials science. Of particular importance are solutionsof such equations that are stable and hence typically correspond with observable phenomena. Unfortunately, establishingthe stability of these solutions has proven to be a quitedifficult 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 ofarbitrary 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 ofresearch in three directions. First, new techniques recently developed by Howard and Zumbrun appear suitablefor extension to (i) systems of viscous conservation laws admitting degenerate viscous shock waves, and (ii)systems of viscous conservation laws with high orderviscosity. 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 techniquesrecently developed in the context of perturbation theoryfor 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 andmomentum often leads to partial differential equationsthat 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: thosewhose principal structure is robust to minor environmentalfluctuations. We propose to continue and extend a promising line of research that has been extraordinarily successful in establishing a clear criterion for suchstability. A direct consequence of the approach is a detailed understanding of certain fundamental partialdifferential equations.