Georgia Institute of Technology College of Sciences

The nonlinear wave equation: u_{tt} - c(u)[c(u)u_x]_x = 0 is a natural generalization of the linear wave equation. In this talk, we will discuss a recent breakthrough addressing the Lipschitz continuous dependence of solutions on initial data for this quasi-linear wave equation. Our earlier results showed that this equation determines a unique flow of conservative solution within the natural energy space H^1(R). However, this flow is not Lipschitz continuous with respect to the H^1 distance, due to the formation of singularity. To prove the desired Lipschitz continuous property, we constructed a new Finsler type metric, where the norm of tangent vectors is defined in terms of an optimal transportation problem. For paths of piecewise smooth solutions, we carefully estimated how the distance grows in time. To complete the construction, we proved that the family of piecewise smooth solutions is dense, following by an application of Thom's transversality theorem. This is a collaboration work with Alberto Bressan.

The study of random Hamilton-Jacobi PDE is motivated by mathematical physics, and in particular, the study of random Burgers equations. We will show that, almost surely, there is a unique stationary solution, which also has better regularity than expected. The solution to any initial value problem converges to the stationary solution exponentially fast. These properties are closely related to the hyperbolicity of global minimizer for the underlying Lagrangian system. Our result generalizes the one-dimensional result of E, Khanin, Mazel and Sinai to arbitrary dimensions. Based on joint works with K. Khanin and R. Iturriaga.

We prove results concerning the equidistribution of some "sparse" subsets of orbits of horocycle flows on $SL(2, R)$ mod lattice. As a consequence of our analysis, we recover the best known rate of growth of Fourier coefficients of cusp forms for arbitrary noncompact lattices of $SL(2, R)$, up to a logarithmic factor. This talk addresses joint work with Livio Flaminio, Giovanni Forni and Pankaj Vishe.

I present a formalism and an computational scheme to quantify the dynamics of grain boundary migration in polycrystalline materials, applicable to three-dimensional microstructure data obtained from non-destructive coarsening experiments. I will describe a geometric technique of interface tracking using well-established optimization algorithms and demonstrate how, when coupled with very basic physical assumptions, one can effectively measure grain boundary energy density and mobility of a given misorientation type in the two-parameter subspace of boundary inclinations. By doing away with any specific model or parameterization for the energetics, I seek to have my analysis applicable to general anisotropies in energy and mobility. I present results in two proof-of-concept test cases, one first described in closed form by J. von Neumann more than half a century ago, and the other which assumes analytic but anisotropic energy and mobility known in advance.