Spectral analysis and stability for wave patterns and multidimensional waves

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This project concerns stability of solutions to systems of nonlinear conservation laws. Though the methods under development will be broadly applicable to general conservation laws, the project focuses on three particular models: (1) the reacting Navier-Stokes model of combustion theory; (2) the Cahn-Hilliard equation of phase separation; and (3) a recent model of thin-film dynamics. For (1) (combustion), the project goal is to identify distinguished solutions corresponding to combustive behavior (such as an explosion), and to categorize each as stable or unstable. Each such solution corresponds with a particular speed of combustion, and so the rate of combustion can be found by determining which of these solutions is stable. For (2) (phase separation), the project goal is to determine rates of phase separation. In practice, the properties (flexibility, hardness, etc.) of a particular compound will depend on the amount of phase separation that has occurred in its development, and so the rate of phase separation is particularly important to industrial manufacturing. For (3) (thin films), the project goal is to determine when a "fingering" instability will occur. While the practical applications considered are on the scale of silicon chips, the basic idea can be understood on the level of painting a wall. If a thin layer of paint is brushed across the top of a wall, we expect it to drip down in long vertical lines, or fingers. The goal of this project is to identify conditions, in certain applications, under which such fingers do not form. This project focuses on certain partial differential equations (PDE) that describe conserved quantities such as mass, biomass, energy, or charge. Such equations are often quite complicated, and generally speaking cannot be solved explicitly for general initial conditions. Moreover, numerical evaluation of such equations can be extremely time-consuming, and results can be highly dependent on parameter values, which may not be accurately known. In light of these difficulties, PDE of this type are often studied through consideration of certain distinguished solutions that represent specialized modes of behavior. Once such a distinguished solution has been identified, a natural and important question regards its stability: roughly speaking, do general conditions exist under which the solution occurs/persists in nature? The primary goal of this project is to categorize distinguished solutions of certain standard PDE models as stable or unstable, and to use this information to understand the dynamics modeled by these equations. The work will be of use in modeling a variety of important physical processes, including combustion dynamics and thin-film flow.