- You are here:
- Home
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
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 characterizatio
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.
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
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
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
In this talk,we study weighted L^p-norm inequalities for general spectralmultipliersfor self-adjoint positive definite operators on L^2(X), where X is a space of homogeneous type. We show that the sharp weighted Hormander-type spectral multiplier theorems follow from the appropriate estimatesof the L^2 norm of the kernel of spectral multipliers and the Gaussian boundsfor the corresponding heat kernels.
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
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