Stability of traveling waves for Hamilton -Jacobi equations and mesoscopic modeling for diffusion dynamics

The focus of this thesis is the study of deterministic and stochastic models that involve multiple interrelated scales. In the first part we study the stability of planar traveling waves for hyperbolic approximations of Hamilton-Jacobi equations. Such models were first introduced in the context of relaxation approximations for Hamilton-Jacobi equations in [6], while [4] treated the convergence of the relaxation approximation as the regularization parameter tends to zero. These convergence results are limited to smooth solutions, while it is well-known that solutions to Hamilton-Jacobi equations develop singularities in the gradient in finite time, even for smooth initial data. Here we extend the analysis to convergence and stability results in the case where singularities in the gradient are present. Related results on the stability and large time behavior of viscous approximations of hyperbolic conservation laws in one and several dimensions were obtained in [2, 3]. Furthermore, in [7], the stability of planar shocks was shown for relaxation approximations of scalar conservation laws. In these works an essential ingredient of the proof involves a derived integrated form of the conservation laws, reminiscent of a Hamilton-Jacobi equation, which necessitates the use of a shift function. Here the arguments simplify substantially since we already deal with a Hamilton-Jacobi equation and a shift function is not necessary. Furthermore, due the hyperbolic nature of the approximation, we obtain improved decay properties of the solution. In the second part of this thesis, we study mesoscopic models of particle diffusion in several interacting particle systems. These mesoscopic models are stochastic or deterministic integrodifferential equations and are derived through an exact coarse graining, directly from microscopic lattice models, and include detailed microscopic information on particle-particle interactions and particle dynamics. Previous results in [9, 14, 15] are limited to one particle species, however in many applications the diffusion mechanism involves several types of particles. In Chapter 2, we focus on deriving the mesoscopic theories for such complex multi-species dynamics. Starting from microscopic dynamics we derive mesoscopic models for both Metropolis and Arrhenius rules. Also we extend our results to the case where the external driving force is no longer a constant. In Chapter 3, we derive the mesoscopic theory for parabolic Arrhenius dynamics which typically models transport and diffusion in zeolites [26]. Finally, in Chapter 4, we discuss pattern formation in systems with both attractive and repulsive interactions, and determine how the competing microscopic interactions affect the overall morphology