Conservation Laws with Partial Dissipation in Continuum Mechanics

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This project is to study laws for continuum mechanics, describing the conservationor balance of physical quantities such as mass, momentum, energy, etc. Themathematical description of those laws are complicated nonlinear systems ofpartial differential equations. The systems contain certain terms representingthe dissipative mechanism such as viscosity, heat conduction, frictional damping,relaxation, species diffusion, etc. On one hand, the nonlinearity of the systemstends to generating singularities (shock waves) in the flows. On the other hand,the dissipative mechanism eases such a tendency, but gives rise to richer wavepatterns at the same time. The overall behavior of the flows then depends onthe competition between the two. The difficulty of the problem lies in the factthat the dissipation exists only in certain equations (laws) of a system. Forinstance, physics dictates that the conservation of mass should not have anydissipation. Such a fact then further complicates the competition betweennonlinearity and dissipation. In different wave directions, one or the otherdominates. It is then easy to understand that in general there is no explicitformula for a solution. In fact, even the existence of a solution can be an openquestion. In this project, the awardee will study when a solution canexist all the time. Furthermore, if a solution exists all the time, what isits qualitative behavior? To this aim, the awardee will first reduce the systeminto a set of simplified, decoupled equations. Each of them represents a wavealong a particular direction, and can be solved explicitly. These waves togethergive the time asymptotic wave pattern of the original system. Next, the awardeewill study how well the asymptotic solution approximates the actual solution.Through such a study, the understanding of the underlying physical phenomena canbe obtained.The problems addressed in this research can be illustrated by an example. When a space shuttle returns to the earth, the temperature of the air around it becomes so high that the internal structure of the molecules in the air gets excited, and chemical reactions occur. The air then loses its local thermodynamic equilibrium state. The departure from equilibrium in turn provides the ``driving force" for internal changes. The air relaxes towards its local equilibrium state throughmolecular collisions. Such relaxation processes have significant influence inthe acoustic directions, but not the particle path direction. The awardeewill study how the relaxation processes change the overall behavior ofthe air flow in this situation.