This project is to study laws for continuum mechanics, describing the conservation or balance of physical quantities such as mass, momentum, energy, etc. The mathematical description of those laws are complicated nonlinear systems of partial differential equations. The systems contain certain terms representing the dissipative mechanism such as viscosity, heat conduction, frictional damping, relaxation, species diffusion, etc. On one hand, the nonlinearity of the systems tends to generating singularities (shock waves) in the flows. On the other hand, the dissipative mechanism eases such a tendency, but gives rise to richer wave patterns at the same time. The overall behavior of the flows then depends on the competition between the two. The difficulty of the problem lies in the fact that the dissipation exists only in certain equations (laws) of a system. For instance, physics dictates that the conservation of mass should not have any dissipation. Such a fact then further complicates the competition between nonlinearity and dissipation. In different wave directions, one or the other dominates. It is then easy to understand that in general there is no explicit formula for a solution. In fact, even the existence of a solution can be an open question. In this project, the awardee will study when a solution can exist all the time. Furthermore, if a solution exists all the time, what is its qualitative behavior? To this aim, the awardee will first reduce the system into a set of simplified, decoupled equations. Each of them represents a wave along a particular direction, and can be solved explicitly. These waves together give the time asymptotic wave pattern of the original system. Next, the awardee will study how well the asymptotic solution approximates the actual solution. Through such a study, the understanding of the underlying physical phenomena can be 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 through molecular collisions. Such relaxation processes have significant influence in the acoustic directions, but not the particle path direction. The awardee will study how the relaxation processes change the overall behavior of the air flow in this situation.
