Georgia Institute of Technology College of Sciences

I will begin by presenting our recent results on the spherically symmetric Coulomb waves. Specifically, we study the evolution operator of H= -\Delta+q/|x| where q>0. Utilizing a distorted Fourier transform adapted to H, we explicitly compute the evolution kernel. A detailed analysis of this kernel reveals that e^itH satisfies an L^1 \to L^{\infty} dispersive estimate with the natural decay rate t^{-3/2}. This work was conducted in collaboration with Adam Black, Bruno Vergara, and Jiahua Zhou. Following this, I will discuss our ongoing research on the nonlinear Schrödinger equation, where we apply the distorted Fourier transform developed for the Coulomb Hamiltonian. This work is being carried out in collaboration with Mengyi Xie.

The compressible Euler equations readily form shocks, but in 1D the inclusion of viscosity prevents such singularities. In this talk, we will quantitatively examine the interaction between generic shock formation and viscous effects as the viscosity tends to zero. We thereby obtain sharp rates for the vanishing-viscosity limit in Hölder norms, and uncover universal viscous structure near shock formation. The results hold for large classes of viscous hyperbolic conservation laws, including compressible Navier–Stokes with physical rather than artificial viscosity. This is joint work with John Anderson and Sanchit Chaturvedi.

We study nonlinear energy transfer and the existence of stationary measures in a class of degenerately forced SDEs on R^d with a quadratic, conservative nonlinearity B(x, x) constrained to possess various properties common to finite-dimensional fluid models and a linear damping term −Ax that acts only on a proper subset of phase space in the sense that dim(kerA) ≫ 1. Existence of a stationary measure is straightforward if kerA = {0}, but when the kernel of A is nontrivial a stationary measure can exist only if the nonlinearity transfers enough energy from the undamped modes to the damped modes. We develop a set of sufficient dynamical conditions on B that guarantees the existence of a stationary measure and prove that they hold “generically” within our constraint class of nonlinearities provided that dim(kerA) < 2d/3 and the stochastic forcing acts directly on at least two degrees of freedom. We also show that the restriction dim(kerA) < 2d/3 can be removed if one allows the nonlinearity to change by a small amount at discrete times. In particular, for a Markov chain obtained by evolving our SDE on approximately unit random time intervals and slightly perturbing the nonlinearity within our constraint class at each timestep, we prove that there exists a stationary measure whenever just a single mode is damped.