OxPDE Lunchtime Seminar
|
Tue, 03/11/2009 14:00 |
Gabriel Koch (University of Oxford) |
OxPDE Lunchtime Seminar  |
Gibson 1st Floor SR |
We present an alternative viewpoint on recent studies of regularity of solutions to the Navier-Stokes equations in critical spaces. In particular, we prove that mild solutions which remain bounded in the
space  do not become singular in finite time, a result which was proved in a more general setting by L. Escauriaza, G. Seregin and V. Sverak using a different approach. We use the method of "concentration-compactness" + "rigidity theorem" which was recently developed by C. Kenig and F. Merle to treat critical dispersive equations. To the authors' knowledge, this is the first instance in which this method has been applied to a parabolic equation. This is joint work with Carlos Kenig. |
|||
|
Mon, 09/11/2009 11:00 |
Valeriy Slastikov (University of Bristol) |
OxPDE Lunchtime Seminar |
Gibson 1st Floor SR |
| We address the effect of extreme geometry on a non-convex variational problem motivated by recent investigations of magnetic domain walls trapped by sharp thin necks. We prove the existence of local minimizers representing geometrically constrained walls under suitable symmetry assumptions on the domains and provide an asymptotic characterization of the wall profile. The asymptotic behavior, which depends critically on the scaling of length and width of the neck, turns out to be qualitatively different from the higher-dimensional case and a richer variety of regimes is shown to exist. | |||
|
Fri, 13/11/2009 14:00 |
Marius Paicu (University of Paris XI) |
OxPDE Lunchtime Seminar |
Gibson 1st Floor SR |
| We consider the three dimensional Navier-Stokes equations with a large initial data and we prove the existence of a global smooth solution. The main feature of the initial data is that it varies slowly in the vertical direction and has a norm which blows up as the small parameter goes to zero. In the language of geometrical optics, this type of initial data can be seen as the “ill prepared" case. Using analytical-type estimates and the special structure of the nonlinear term of the equation we obtain the existence of a global smooth solution generated by this large initial data. This talk is based on a work in collaboration with J.-Y. Chemin and I. Gallagher and on a joint work with Z. Zhang. | |||
|
Thu, 19/11/2009 12:30 |
Luc Nguyen (University of Oxford) |
OxPDE Lunchtime Seminar |
Gibson 1st Floor SR |
| According to the Ernst-Geroch reduction, in an axially symmetric stationary electrovac spacetime, the Einstein-Maxwell equations reduce to a harmonic map problem with singular boundary data. I will discuss the “regularity” of the reduced harmonic maps near the boundary and its implication on the regularity of the corresponding spacetimes. | |||
|
Mon, 23/11/2009 13:00 |
Tatyana Shaposhnikova (Linköping University, Sweden) |
OxPDE Lunchtime Seminar |
Gibson 1st Floor SR |
| Given a bounded Lipschitz domain, we consider the Dirichlet problem with boundary data in Besov spaces for divergence form strongly elliptic systems of arbitrary order with bounded complex-valued coefficients. The main result gives a sharp condition on the local mean oscillation of the coefficients of the differential operator and the unit normal to the boundary (automatically satisfied if these functions belong to the space VMO) which guarantee that the solution operator associated with this problem is an isomorphism. | |||
|
Thu, 26/11/2009 11:00 |
Eduard Kirr (University of Illinois at Urbana Champaign, USA) |
OxPDE Lunchtime Seminar |
Gibson 1st Floor SR |
| I will discuss recent results on dispersive estimates for linear PDE's with time dependent coefficients. Then I will discuss how such estimates can be used to study stability of nonlinear solitary waves and resonance phenomena. | |||
