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Series: PDE Seminar

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).

Series: PDE Seminar

Darcy's law was observed in the motion of porous medium flows. This talk aims at the mathematical justification on Darcy's law as long time limit from compressible Euler equations with damping. In particularly, we shall showthat any physical solution with finite total mass shall converges in L^1 distance toward the Barenblatt's solution of the same mass for the Porous Medium Equation. The approach will explore the dissipation of the entropy inequality motivated by the second law of thermodynamics. This is a joint work with Feimin Huang and Zhen Wang.

Series: PDE Seminar

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.

Series: PDE Seminar

The large-time behavior of solutions to Burgers equation with small viscosity isdescribed using invariant manifolds. In particular, a geometric explanation is provided for aphenomenon known as metastability, which in the present context means that solutions spend avery long time near the family of solutions known as diffusive N-waves before finallyconverging to a stable self-similar diffusion wave. More precisely, it is shown that in termsof similarity, or scaling, variables in an algebraically weighted L^2 space, theself-similar diffusion waves correspond to a one-dimensional global center manifold ofstationary solutions. Through each of these fixed points there exists a one-dimensional,global, attractive, invariant manifold corresponding to the diffusive N-waves. Thus,metastability corresponds to a fast transient in which solutions approach this ``metastable"manifold of diffusive N-waves, followed by a slow decay along this manifold, and, finally,convergence to the self-similar diffusion wave. This is joint work with C. Eugene Wayne.

Series: PDE Seminar

I will discuss the intermediate and long time dynamics
of solutions of the nonlinear Schroedinger - Gross Pitaevskii equation,
governing nonlinear dispersive waves in a spatially
non-homogeneous background.
In particular, we present results (with B. Ilan)
on solitons with frequencies near a spectral band edge associated
with periodic potential, and results (with Z. Gang) on large
time energy distribution in systems with multiple bound states.
Finally, we discuss how such results can inform strategies
for control of soliton-like states in optical and quantum systems.

Series: PDE Seminar

In this talk we will present several results concerning the behavior of the Laplace operator with Neumann boundary conditions in a thin domain where its boundary presents a highly oscillatory behavior. Using homogenization and domain perturbation techniques, we obtain the asymptotic limit as the thickness of the domain goes to zero even for the case where the oscillations are not necessarily periodic. We will also indicate how this result can be applied to analyze the asymptotic dynamics of reaction diffusion equations in these domains.

Series: PDE Seminar

One of the challenges in the study of transonic flows is the understanding of
the flow behavior near the sonic state due to the severe degeneracy of the
governing equations. In this talk, I will discuss the well-posedness theory of a
degenerate free boundary problem for a quasilinear second elliptic equation
arising from studying steady subsonic-sonic irrotational compressible flows in a convergent nozzle. The flow speed is sonic at the free boundary where the potential flow equation becomes degenerate. Both existence and uniqueness will be shown and optimal regularity will be obtained. Smooth transonic flows in deLaval nozzles
will also be discussed. This is a joint work with Chunpeng Wang.

Series: PDE Seminar

In a bounded domain with smooth boundary (which can be considered as a
smooth sub-manifold of R3), we consider the Boltzmann equation with
general Maxwell boundary condition---linear combination of specular
reflection and diffusive absorption. We analyze the kinetic (Knudsen
layer) and fluid (viscous layer) coupled boundary layers in both acoustic
and incompressible regimes, in which the boundary layers behave
significantly different. The existence and damping properties of these
kinetic-fluid layers depends on the relative size of accommodation number
and Kundsen number, and the differential geometric property of the
boundary (the second fundamental form.)
As applications, first we justify the incompressible Navier-Stokes-Fourier
limit of the Boltzmann equation with Dirichlet, Navier, and diffusive
boundary conditions respectively, depending on the relative size of
accommodation number and Kundsen number. Using the damping property of the
boundary layer in acoustic regime, we proved the convergence is strong.
The second application is that we derive and justified the higher order
acoustic approximation of the Boltzmann equation.
This is a joint work with Nader Masmoudi.

Series: PDE Seminar

In this talk, we will discuss the global existence and asymptotic behavior of classical solutions for two dimensional inviscid Rotating Shallow Water system with small initial data subject to the zero-relative-vorticity constraint. One of the key steps is a reformulation of the problem into a symmetric quasilinear Klein-Gordon system, for which the global existence of classical solutions is then proved with combination of the vector field approach and the normal forms. We also probe the case of general initial data and reveal a lower bound for the lifespan that is almost inversely proportional to the size of the initial relative vorticity. This is joint work with Bin Cheng.

Series: PDE Seminar

We prove a conjecture of Bryant, Griffiths, and Yang concerning the characteristic variety for the determined isometric embedding system. In particular, we show that the characteristic variety is not smooth for any dimension greater than 3. This is accomplished by introducing a smaller yet equivalent linearized system, in an appropriate way, which facilitates analysis of the characteristic variety.