In this talk we present a joint work with Lei Zhen of Fudan University.

Let v=v(x, t) be a solution to the 3 d axis symmetric NS.
Let (x_0, t_0) be a point such that the flow speed |v(x_0,t_0)| is comparable to the maximum speed for time t

We develop the concept of an infinite-energy statistical solution to the Navier-Stokes and Euler equations in the whole plane. We use a velocity formulation with enough generality to encompass initial velocities having bounded vorticity, which includes the important special case of vortex patch initial data. Our approach is to use well-studied properties of statistical solutions in a ball of radius R to construct, in the limit as R goes to infinity, an infinite-energy solution to the Navier-Stokes equations. We then construct an infinite-energy statistical solution to the Euler equations by making a vanishing viscosity argument.

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.

Three different singularly perturbed eigenvalue problems in perforated
domains, or in domains with perforated boundaries, with direct
biological applications, are studied asymptotically. In the context
of cellular signal transduction, a common scenario is that a diffusing
surface-bound molecule must arrive at a localized signalling region,
or trap, on the cell membrane before a signalling cascade can be
initiated. In order to determine the time-scale for this process,
asymptotic results are given for the mean first passage time (MFPT) of
a diffusing particle confined to the surface of a sphere that has
absorbing traps of small radii. In addition, asymptotic results are
given for the related narrow escape problem of calculating the MFPT
for a diffusing particle inside a sphere that has small traps on an
otherwise reflecting boundary. The MFPT for this narrow escape problem
is shown to be minimized for particular trap configurations that
minimize a certain discrete variational problem (DVP). This DVP is
closely related to the classic Fekete point problem of determining the
minimum energy configuration for repelling Coulomb charges on the unit
sphere. Finally, in the context of spatial ecology, a long-standing
problem is to determine the persistence threshold for extinction of a
species in a heterogeneous spatial landscape consisting of either
favorable or unfavorable local habitats. For a 2-D spatial landscape
consisting of such localized patches, the persistence threshold is
calculated asymptotically and the effects of both habitat
fragmentation and habitat location on the persistence threshold is
examined. From a mathematical viewpoint, the persistence threshold
represents the principal eigenvalue of an indefinite weight singularly
perturbed eigenvalue problem, resulting from a linearization of the
diffusive logistic model.

The analysis of these three PDE eigenvalue problems is based on the
development of a common singular perturbation methodology to treat
localized patches or traps in combination with some detailed
analytical properties of the Neumann Green's function for the
Laplacian. With this asymptotic framework, the problem of optimizing
the principal eigenvalue for the each of these three problems is
reduced to the simpler task of determining optimal configurations for
certain discrete variational problems.

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Speaker: Michael Ward (UBC). Joint work with Dan Coombs (UBC), Alexei
Chekhov (U. Sask), Alan Lindsay (UBC), Anthony Peirce (UBC), Samara
Pillay (JP Morgan), Ronny Straube (Max Planck, Magdeburg).