Georgia Institute of Technology College of Sciences

Landau damping is a collisionless stability result of considerable importance in plasma physics, as well as in galactic dynamics. Roughly speaking, it says that spatial waves are damped in time (very rapidly) by purely conservative mechanisms, on a time scale much lower than the effect of collisions. We shall present in this talk a recent work (joint with C. Villani) which provides the first positive mathematical result for this effect in the nonlinear regime, and qualitatively explains its robustness over extremely long time scales. Physical introduction and implications will also be discussed.

A classic story of nonlinear science started with the particle-like water wave that Russell famously chased on horseback in 1834. I will recount progress regarding the robustness of solitary waves in nonintegrable model systems such as FPU lattices, and discuss progress toward a proof (with Shu-Ming Sun) of spectral stability of small solitary waves for the 2D Euler equations for water of finite depth without surface tension.

Let $\mathbb{H}$ be a Hilbert space and $h: \mathbb{H} \times \mathbb{H} \rightarrow \mathbb{R}$ be such that $h(x, \cdot)$ is uniformly convex and grows superlinearly at infinity, uniformy in $x$. Suppose $U: \mathbb{H} \rightarrow \mathbb{R}$ is strictly convex and grows superlinearly at infinity. We assume that both $H$ and $U$ are smooth. If $\mathbb{H}$ is of infinite dimension, the initial value problem $\dot x= -\nabla_p h(x, -\nabla U(x)), \; x(0)=\bar x$ is not known to admit a solution. We study a class of parabolic equations on $\mathbb{R}^d$ (and so of infinite dimensional nature), analogous to the previous initial value problem and establish existence of solutions. First, we extend De Giorgi's interpolation method to parabolic equations which are not gradient flows but possess an entropy functional and an underlying Lagrangian. The new fact in the study is that not only the Lagrangian may depend on spatial variables, but it does not induce a metric. These interpolation reveal to be powerful tool for proving convergence of a time discrete algorithm. (This talk is based on a joint work with A. Figalli and T. Yolcu).

Couette flows are shear flows with a linear velocity profile. Known by Orr in 1907, the vertical velocity of the linearized Euler equations at Couette flows is known to decay in time, for L^2 vorticity. It is interesting to know if the perturbed Euler flow near Couette tends to a nearby shear flow. Such problems of nonlinear inviscid damping also appear for other stable flows and are important to understand the appearance of coherent structures in 2D turbulence. With Chongchun Zeng, we constructed non-parallel steady flows arbitrarily near Couette flows in H^s (s<3/2) norm of vorticity. Therefore, the nonlinear inviscid damping is not true in (vorticity) H^s (s<3/2) norm. We also showed that in (vorticity) H^s (s>3/2) neighborhood of Couette flows, the only steady structures (including travelling waves) are stable shear flows. This suggests that the long time dynamics near Couette flows in (vorticity) H^s (s>3/2) space might be simpler. Similar results will also be discussed for the problem of nonlinear Landau damping in 1D electrostatic plasmas.