OxPDE Lunchtime Seminar
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Thu, 19/01/2012 12:30 |
Paolo Antonelli (DAMPT, University of Cambridge) |
OxPDE Lunchtime Seminar  |
Gibson 1st Floor SR |
| In this seminar I will expose some results obtained jointly with P. Marcati, concerning the global existence of weak solutions for the Quantum Hydrodynamics System in the space of energy. We don not require any additional regularity and/or smallness assumptions on the initial data. Our approach replaces the WKB formalism with a polar decomposition theory which is not limited by the presence of vacuum regions. In this way we set up a self consistent theory, based only on particle density and current density, which does not need to define velocity fields in the nodal regions. The mathematical techniques we use in this paper are based on uniform (with respect to the approximating parameter) Strichartz estimates and the local smoothing property. I will then discuss some possible future extensions of the theory. | |||
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Thu, 26/01/2012 12:30 |
Veronique Fischer (University of Padova and guest at King's College London) |
OxPDE Lunchtime Seminar |
Gibson 1st Floor SR |
| Pseudo-differential operators (PDO's) are primarily defined in the familiar setting of the Euclidean space. For four decades, they have been standard tools in the study of PDE's and it is natural to attempt defining PDO's in other settings. In this talk, after discussing the concept of PDO's on the Euclidean space and on the torus, I will present some recent results and outline future work regarding PDO's on Lie groups as well as some of the applications to PDE's | |||
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Thu, 02/02/2012 12:30 |
Laura Caravenna (OxPDE, University of Oxford) |
OxPDE Lunchtime Seminar |
Gibson 1st Floor SR |
| In the talk I will mention two regularity results: the SBV regularity for strictly hyperbolic, genuinely nonlinear 1D systems of conservation laws and the characterization of intrinsic Lipschitz codimension 1 graphs in the Heisenberg groups. In both the contexts suitable scalar, 1D balance laws arise with very low regularity. I will in particular highlight the role of characteristics. This seminar will be based on joint works with G. Alberti, S. Bianchini, F. Bigolin and F. Serra Cassano, and the main previous literature. | |||
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Thu, 09/02/2012 12:30 |
Yves Capdeboscq (OxPDE, University of Oxford) |
OxPDE Lunchtime Seminar |
Gibson 1st Floor SR |
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Consider the solution of a scalar Helmholtz equation where the potential (or index) takes two positive values, one inside a disk or a ball (when d=2 or 3) of radius epsilon and another one outside. For this classical problem, it is possible to derive sharp explicit estimates of the size of the scattered field caused by this inhomogeneity, for any frequencies and any contrast. We will see that uniform estimates with respect to frequency and contrast do not tend to zero with epsilon, because of a quasi-resonance phenomenon. However, broadband estimates can be derived: uniform bounds for the scattered field for any contrast, and any frequencies outside of a set which tends to zero with epsilon. |
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Thu, 16/02/2012 12:30 |
Reto Müller (Imperial College, London) |
OxPDE Lunchtime Seminar |
Gibson 1st Floor SR |
| In this talk, we first study the Mean Curvature Flow, an evolution equation for submanifolds of some Euclidean space. We review a famous monotonicity formula of Huisken and its application to classifying so-called Type I singularities. Then, we discuss the Ricci Flow, which might be seen as the intrinsic analog of the Mean Curvature Flow for abstract Riemannian manifolds. We explain how Huisken's classification of Type I singularities can be adopted to this intrinsic setting, using monotone quantities found by Perelman. | |||
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Thu, 01/03/2012 12:30 |
François Murat (Université Paris VI) |
OxPDE Lunchtime Seminar |
Gibson 1st Floor SR |
In this lecture I will report on joint work with
J. Casado-Díaz, T. Chacáon Rebollo, V. Girault and M.~Gómez Marmol
which was published in Numerische Mathematik,
vol. 105, (2007), pp. 337-510.
We consider, in dimension  , the standard
 finite elements approximation of the
second order linear elliptic equation in divergence form with
coefficients in
 which generalizes Laplace's
equation. We assume that the family of
triangulations is regular and that it satisfies
an hypothesis close to the classical
hypothesis which
implies the discrete maximum principle.
When the right-hand side belongs to  , we prove
that the unique solution of the discrete
problem converges in
 (for every  with
 <  ) to the unique renormalized
solution of the problem. We obtain a weaker result when the
right-hand side is a bounded Radon
measure. In the case where the dimension is
 or  and where the coefficients are smooth, we give an
error estimate in
when the right-hand side belongs to
 for some  >  . |
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Thu, 08/03/2012 12:30 |
Yoshihito Oshita (Okayama University, Japan) |
OxPDE Lunchtime Seminar |
Gibson 1st Floor SR |
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We study the mean-field models describing the evolution of distributions
of particle radii obtained by taking the small volume fraction limit of
the free boundary problem describing the micro phase separation of
diblock copolymer melts, where micro phase separation consists of an
ensemble of small balls of one component. In the dilute case, we
identify all the steady states and show the convergence of solutions. |
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