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Series: PDE Seminar

We discuss how to solve a Hamilton-Jacobi-Bellman equation ``at resonance." Our characterization is in terms of invariant measures and is analogous to the Fredholm alternative in the linear case.

Series: PDE Seminar

Consider a nonlinear Schrodinger
equation in $R^3$
whose linear part has three or more eigenvalues
satisfying some resonance
conditions. Solutions which are initially small in
$H^1 \cap
L^1(R^3)$ and inside a neighborhood of the first excited state
family are shown to converge to either a first excited state or a
ground state at time infinity. An essential part of our analysis
is on the linear and nonlinear estimates near nonlinear excited
states, around which the linearized operators have eigenvalues
with nonzero real parts and their corresponding eigenfunctions
are
not uniformly localized in space. This is a joint work with Kenji Nakanishi
and Tuoc Van Phan.The preprint of the talk is available at
http://arxiv.org/abs/1008.3581

Series: PDE Seminar

Series: PDE Seminar

In the simplest form, our result gives a characterization of bounded,divergence-free vector fields on the plane such that the Cauchyproblem for the associated continuity equation has a unique boundedsolution (in the sense of distribution).Unlike previous results in this directions (Di Perna-Lions, Ambrosio,etc.), the proof does not rely on regularization, but rather on adimension-reduction argument which allows us to prove uniqueness usingwell-known one-dimensional results (it is indeed a variant of theclassical method of characteristics).Note that our characterization is not given in terms of functionspaces, but using a qualitative property which is completelynon-linear in character, namely a suitable weak formulation of theSard property.This is a joint work with Giovanni Alberti (University of Pisa) andStefano Bianchini (SISSA, Trieste).

Series: PDE Seminar

Fokker-Planck equation is a linear parabolic equation which describes
the time evolution of of probability distribution of a stochastic
process defined on a Euclidean space. Moreover, it is the gradient flow
of free energy functional. We will present a Fokker-Planck equation
which is a system of ordinary differential equations and describes the
time evolution of probability distribution of a stochastic process on a
graph with a finite number of vertices. It is shown that there is a
strong connection but also substantial differences between the ordinary
differential equations and the usual Fokker-Planck equation on Euclidean
spaces. Furthermore, the ordinary differential equation is in fact a
gradient flow of free energy on a Riemannian manifold whose metric is
closely related to certain Wasserstein metrics. Some examples will also be discussed.

Series: PDE Seminar

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.

Series: PDE Seminar

We discuss a non-linear eigenvalue problem where the eigenvalue has a natural control-theoretic interpretation as an optimal "long-time averaged cost." We also show how such problems arise in financial market models with small transaction costs.

Series: PDE Seminar

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.

Series: PDE Seminar

We consider the three dimensional Navier-Stokes equations with a large initial data and we prove the existence of a global smooth solution. The main feature of the initial data is that it varies slowly in the vertical direction and has a norm which blows up as the small parameter goes to zero. Using the language of geometrical optics, this type of initial data can be seen as the ``ill prepared" case. Using analytical-type estimates and the special structure of the nonlinear term of the equation we obtain the existence of a global smooth solution generated by this large initial data. This talk is based on a work in collaboration with J.-Y. Chemin and I. Gallagher and on a joint work with Z. Zhang.

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).