Nir Gavish, Tel Aviv University

The study of singular solutions of the NLS goes back to the 1960s, with applications in nonlinear optics and, more recently, in BEC. Until recently, the only known singular solutions had a self-similar "peak-type" profile that approaches a delta function near the singularity.

In this talk I will present new families of singular solutions of the NLS that collapse with a self-similar ring profile, and whose blowup rate is different from the one of the "old" singular solutions. I will also show, both theoretically and experimentally, that these new blowup profiles are attractors for large super-Gaussian initial conditions.

This is joint work with Gadi Fibich and Xiao-Ping Wang

Sam Walsh, Brown University

Takafumi Akahori, Ehime University, Japan

March

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We present recent results allowing us to estimate the time of existence of small data semilinear Klein-Gordon equations on Zoll manifolds (e.g. spheres of arbitrary dimension). The proof relies on the normal form method and on the specific distribution of the eigenvalues of the Laplacian perturbed by a potential on Zoll manifolds.

Doyoon Kim, USC

Elliptic and parabolic equations with coefficients measurable in one spatial variable and VMO (vanishing mean oscillation) in the other spatial variables will be presented. For parabolic equations, coefficients are further allowed to be measurable in time. Equations are of non-divergence form and solutions are found in Sobolev spaces (with mixed norms). Equations in the whole space are first considered.

Then using this result, Dirichlet and Neumann boundary value problems in a half-space are solved without having any boundary a priori L_p estimates. This allows us to deal with, for example, parabolic equations in a bounded domain with coefficients measurable in time and VMO in spatial variables.

Hao Wu, PSU

In this talk I will present the Lojasiewicz-Simon approach in the study on convergence of solution to equilibrium for nonlinear evolution equations including Cahn-Hilliard equation, parabolic-hyperbolic phase-field equations, etc..

Umberto Mosco, Worcester Polytechnic Institute

Certain fractals are known to behave like elastic bodies capable of storing intrinsic energy. In our talk we address the question of whether such a fractal is also apt to absorb energy from the surrounding space. As the fractal has a singular embedding in space, this process is not trivial. We show that absorption is possible, while keeping the spectra stable.

Note that this talk was originally scheduled for 11 April.

Eugen Varvaruca, University of Bath, England

We present some recent results on singular solutions of the problem of traveling gravity water waves on flows with vorticity. We show that, for a certain class of vorticity functions, a sequence of regular waves converges to an extreme wave with stagnation points at its crests. We also show that, for any vorticity function, the profile of an extreme wave must have either a corner of 120 degrees or a horizontal tangent at any stagnation point about which it is supposed symmetric. Moreover, the profile necessarily has a corner of 120 degrees if the vorticity is nonnegative near the free surface.

Note special day (Monday) and time (11:00 AM). The room is the usual one, though: Kassar 105.

Gustavo Perla Menzala, LNCC, Brazil

Gerhard Rein, University of Bayreuth

Justin Holmer, UC Berkeley

Gilles Francfort, University of Paris XIII

Hans Knuepfer, Courant Institute

Andrej Zlatos, University of Chicago

We consider a passive scalar advection-diffusion equation in 2D with a periodic incompressible advecting vector field (flow). We provide a sharp characterization of all flow profiles that optimally enhance dissipation in the sense that solutions with compactly supported initial data become arbitrarily small arbitrarily quickly, provided the flow amplitude is large enough. Our characterization is expressed in terms of simple geometric and spectral conditions on the flow. Extensions to higher dimensions and applications to reaction-advection-diffusion equations will also be mentioned.

October

19

Ideal Incompressible Fluids with Continuous Vorticity

We study the regularity of solutions to the two-dimensional Euler equations when initial vorticity is uniformly continuous and in a critical or subcritical Besov space. We show that under these assumptions on the initial data, the solution will lose at most an arbitrarily small amount of Besov regularity over any finite time interval.

Hongjun Gao, Nanjing Normal University, CHINA

Exponential Attractors for a Fully Hyperbolic Phase-Field System

This talk will be held at a special time and place: 11:00 AM in Room 104, 37 Manning Street.

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September

Thread-wire surfaces: Near-wire minimizers and topological finiteness

Alt's thread problem asks for least-area surfaces bounding a fixed "wire" curve and a movable "thread" curve of length L. Near-wire minimizers are important to a finiteness conjecture. For near-wire minimizers on a generic wire we show C^1-regularity where the thread and wire join at cusp-corners. Our result confirms a prediction of the normal vector limit at such points. It is an example of the local geometry dominating other influences in a free boundary problem. Finally, we use a weighted isoperimetric inequality to prove that all minimizers are near-wire minimizers when the thread is near the wire length.

For physical examples of the thread-wire system, feel free to visit http://www.bkstephens.net.

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