I. Stability of Waves in Viscous Conservation Laws. II. Phase Transitions and Minimal Surfaces

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9107990 Zumbrun Kevin Zumbrun proposes several projects originating from fluid dynamics. These concern structure and stability of interfacial solutions arising in singular limit problems, specifically SHOCK WAVES and PHASE BOUNDARIES. Stability is a central topic in shock wave theory, connected with such issues as physical admissibility of hyperbolic waves, convergence of difference schemes, and the inviscid limit problem. Zumbrun proposes to study a variety of related questions, from stability of multidimensional viscous shock waves in compressible Navier-Stokes equations to physical significance of oscillatory shock layers in a nonlocal, dispersive sedimentation model. The planned methods of analysis include, among others, pointwise, Green's function techniques which have proved to be useful in other situations of delicate stability, spectral analysis using matrix perturbation theory, and Evans function techniques borrowed from the study of reaction diffusion fronts. Likewise, structure of phase boundaries is a central topic in the study of phase transitions. In the limit of zero transition layer thickness, Cahn-Hilliard models for phase transition reduce to idealized minimal surface problems, and the phase boundaries to minimal surfaces. This link is a rich source of intuition, suggesting new problems in both the geometry and pde setting. Zumbrun proposes several of these for study, most notably the regularity of "Neumann" solutions for the minimal surface problem. The planned method of analysis is by a combination of pde and geometric measure theory techniques. The behavior and structure of interfaces is a topic of basic physical interest, as the organizing principle for a variety of effects seen in nature. For example, soap bubbles are well known to form minimum interfacial energy structures identified by the property that they have minimal surface area. These are also STABLE under small perturbations , by the principle that systems move always toward lower-energy states, in this case back toward the minimal energy configuration. At the same time, the large-scale distribution of matter in the visible universe seems to have a similar structure, with vast voids surrounded by thin layers of galaxies. Apparently, we live on an interface--evidently, also, the minimization of quite different energies can lead to similar structure through a common mathematical mechanism. The examples given above are just two instances of the interfaces known as PHASE BOUNDARIES. Other important interfaces are SHOCK WAVES, or moving boundaries separating substances of very different properties (most usually, a "front" separating air of very different temperature or pressure). Like phase boundaries, these are ubiquitous in nature, from sonic booms to "waves" of concentration in a chemical reaction. Again, it is stability or instability that determines whether a particular shock structure will persist or break up. The mathematics governing stability of interfaces is rather subtle and is by no means completely understood. Indeed, the projects proposed by Zumbrun concern basic theoretical questions that must be answered before we can confidently pursue practical applications such as computer modeling of these phenomena in the variety of settings to which they pertain.