
Date Time 
Location  Speaker 
Title – click for abstract 

09/19 3:00pm 
BLOC 628 
James Kelliher UC Riverside 
Nonlinear PDEs Seminar
Title: Bounded Vorticity, unbounded velocity solutions to the 2D Euler equations.
Abstract: The pioneering work on bounded vorticity solutions to the 2D Euler equations was done by Yudovich in the early 1960s, working in a bounded domain. He proved existence and uniqueness of such solutions. Subsequently, existence was shown to hold in much weaker settings (vorticity lying in any Lebesgue space), but uniqueness has only ever been extended to an incrementally larger class of initial data.
Yudovich's theory extends easily to the full plane (indeed it is slightly less technical there) as long as the velocity is assumed to decay sufficiently rapidly at infinity that the BiotSavart law holds. In 1995, Ph. Serfati established the existence and uniqueness of bounded vorticity solutions having no decay at infinity. Such solutions very much violate the BiotSavart law, but Serfati discovered an identity that the solutions hold that can be used as a kind of substitute for that law.
The boundedness of the velocity was very important in Serfati's argument, yet there is room in his identity to accommodate some growth of the velocity at infinity. I will speak on ongoing joint work with Elaine Cozzi in which we exploit Serfati's identity to obtain existence and uniqueness classes allowing growth at infinity as large as possible (without assuming any special symmetry of the initial data). Roughly speaking, we show that existence can be achieved only for very slowly growing velocities, but that uniqueness holds for velocities growing slower than the square root of the distance from the origin. We also consider the issue of continuous dependence on initial data, which is already an interesting problem even in Yudovich's original setting. 

09/26 3:00pm 
BLOC 628 
Giles Auchmuty University of Houston 
Nonlinear PDEs Seminar
Title: Energy Bounds for Planar Divcurl Boundary Value Problems
Abstract: This talk will describe the derivation of sharp L^2 norm estimates for the solutions of divcurl boundary value problems on bounded Lipschitz planar regions Nontrivial prescribed tangential, normal or mixed boundary condition problems are considered of the type that arise in stationary electromagnetic eld modeling. The solutions are found using special decompositions of the vector elds in terms of scalar potentials, stream functions and harmonic elds. Explicit spectral formulae for the solutions are derived that involve various eigenvalues and eigenfunctions of the Laplacian including the Steklov eigenvalues and eigenfunctions. The bounds depend on certain least eigenvalues and the given data. 

10/24 3:00pm 
BLOC 628 
Alex Mahlov Arizona State University 
Nonlinear PDE's seminar
Title:Stochastic ThreeDimensional NavierStokes Equations + Waves: Averaging, Convergence, Regularity and Nonlinear Dynamics
Abstract:
We consider stochastic threedimensional NavierStokes equations + Waves on long time intervals. Regularity results are established by bootstrapping from global regularity of the averaged stochastic resonant equations and convergence theorems. The averaged covariance operator couples stochastic and wave effects. The energy injected in the system by the noise is large, the initial condition has large energy, and the regularization time horizon is long. Regularization is the consequence of precise mechanisms of relevant threedimensional nonlinear interactions. We establish multiscale stochastic averaging, convergence and regularity theorems in a general framework. We also present theoretical and computational results for threedimensional nonlinear dynamics.
