We estimate the domain of analyticity and Gevrey-class regularity of solutions to the Euler equations on the half-space, and on a three-dimensional bounded domain. We obtain new lower bounds for the rate of decay of the real-analyticity radius of the solution, which depend algebraically on the Sobolev norm. In the case of the bounded domain, using Lagrangian coordinates, we prove the persistence of the non-analytic Gevrey-class regularity.

The 2D Boussinesq system is potentially relevant to the study of atmospheric and oceanographic turbulence, as well as other astrophysical situations where rotation and stratification play a dominant role. In fluid mechanics, the 2D Boussinesq system is commonly used in the field of buoyancy-driven flow. It describes the motion of incompressible inhomogeneous viscous fluid subject to convective heat transfer under the influence of gravitational force. It is well-known that the 2D Boussinesq equations are closely related to 3D Euler or Navier-Stokes equations for incompressible flow, and it shares a similar vortex stretching effect as that in the 3D incompressible flow. In fact, in vortex formulation, the 2D inviscid Boussinesq equations are formally identical to the 3D incompressible Euler equations for axisymmetric swirling flow. Therefore, the qualitative behaviors of the solutions to the two systems are expected to be identical. Better understanding of the 2D Boussinesq system will undoubtedly shed light on the understanding of 3D flows. In this talk, I will discuss some recent results concerning global existence, uniqueness and asymptotic behavior of classical solutions to initial boundary value problems for 2D Boussinesq equations with partial viscosity terms on bounded domains for large initial data.

his talk will be about the interplay between polynomials
whose zero sets interact with the n-torus in C^n (in several natural
ways) and operator related complex function theory (in several natural
settings). Interpolation problems for analytic functions, sums of
(Hermitian) squares formulas, and reproducing kernels all play
important roles. The talk will be relatively non-technical.

This talk will be about a general framework of image processing such as Image denoising, deblurring, segmentation, etc. Variational and also PDE approaches will be considered. Various function spaces will also be mentioned and we will see some advantages and disadvantages of the functions spaces. At the end,
noise - texture characterization or separation techniques will be discussed. Notice that there are no mathematical definitions of noise and texture yet. We will think about this too.