Diffusions with Rough Drifts and Navier-Stokes Equation

Speaker: 

Fraydoun Rezakhanlou

Institution: 

UC Berkeley

Time: 

Wednesday, November 12, 2014 - 3:00pm to 4:00pm

Host: 

Location: 

RH440R

 According to DiPerna-Lions theory, velocity fields with weak
derivatives in L^ p   spaces possess weakly regular flows. When a velocity
field is perturbed by a white noise, the corresponding (stochastic) flow
is far more regular in spatial variables; a diffusion with a drift in a
suitable L p   space possesses weak derivatives with exponential bounds.
As an application we show that a Hamiltonian system that is perturbed by a
white noise produces a symplectic flow for a Hamiltonian function that is
merely in W^{ 1,p}  for p  strictly larger than dimension. I also discuss
the potential application of such regularity bounds to study solutions of
Navier-Stokes equation with the aid of Constantin-Iyer's circulation
formula.

 

Estimates for the homogeneous Landau equation with Coulomb potential

Speaker: 

Maria Gualdani

Institution: 

George Washington University

Time: 

Tuesday, January 20, 2015 - 3:00am to 4:00am

Host: 

Location: 

RH306

We present  conditional existence results for  the Landau equation with
Coulomb potential. Despite lack of a comparison principle for the equation, the
proof of existence relies on barrier arguments and parabolic regularity theory. The
Landau equation arises in kinetic theory of plasma physics. It was derived by Landau
and serves as a formal approximation to the Boltzmann equation when grazing
collisions are predominant. 
We also present long-time existence results for the isotropic version of the Landau
equation with Coulomb potential.

 

 

Unstable manifolds and nonlinear instability of Euler equations

Speaker: 

Chongchun Zeng

Institution: 

Georgia Institute of Technology

Time: 

Tuesday, February 3, 2015 - 3:00pm to 4:00pm

Host: 

Location: 

RH306

Abstract: We consider the nonlinear instability of a steady state
$v_*$ of the Euler equation in an $n$-dim fixed smooth bounded domain. When
considered in $H^s$, $s>1$, at the linear level, the stretching of the
steady fluid trajectories induces unstable essential spectrum which
corresponds to linear instability at small spatial scales and the
corresponding growth rate depends on the choice of the space $H^s$.
More physically interesting linear instability relies on the unstable
eigenvalues which correspond to large spatial scales. In the case when
the linearized Euler equation at $v_*$ has an exponential dichotomy of
center-stable and unstable (from eigenvalues) directions, most of the
previous results obtaining the expected nonlinear instability in $L^2$
(the energy space, large spatial scale) were based on the vorticity
formulation and therefore only work in 2-dim. In this talk, we prove,
in any dimensions, the existence of the unique local unstable manifold
of $v_*$, under certain conditions, and thus its nonlinear
instability. Our approach is based on the observation that the Euler
equation on a fixed domain is an ODE on an infinite dimensional
manifold of volume preserving maps in function spaces. This is a
joint work with Zhiwu Lin.

Continuous maximal regularity on manifolds with singularities and applications to geometric flows

Speaker: 

Yuanzhen Shao

Institution: 

Vanderbilt University

Time: 

Tuesday, September 30, 2014 - 3:00pm

Location: 

RH 306

In this talk, we study continuous maximal regularity theory for a class of degenerate or singular differential operators on manifolds with singularities. Based on this theory, we show local existence and uniqueness of solutions for several nonlinear geometric flows and diffusion equations on non-compact, or even incomplete, manifolds, including the Yamabe flow and parabolic p-Laplacian equations. In addition, we also establish regularity properties of solutions by means of a technique consisting of continuous maximal regularity theory, a parameter-dependent diffeomorphism and the implicit function theorem.

Magnetohydrodynamic fluids with zero magnetic diffusivity

Speaker: 

Xianpeng Hu

Institution: 

Courant Institute

Time: 

Tuesday, May 20, 2014 - 3:00pm to 4:00pm

Host: 

Location: 

RH306

 

Understanding the incompressible/compressible fluid is a fundamental, but
challenging, project not only in numerical analysis, but also in
theoretical analysis, especially when extra effects, such as the elastic
deformation or the magnetic field, interact with the flow. In this talk,
the incompressible fluid and its associated flow map will be reviewed first.
The main object of this talk devotes to a recent work in understanding
incompressible/compressible magnetohydrodynamic fluids with zero magnetic
diffusivity (which is equivalent to infinite conductivity). This is a
joint work with Fanghua Lin.

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