Partial Differential Equations Seminar
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Mon, 23/04/2012 17:00 |
John E. Andersson (Warwick) |
Partial Differential Equations Seminar  |
Gibson 1st Floor SR |
| In 1932 Signorini formulated the first variational inequality as a model of an elastic body laying on a rigid surface. In this talk we will revisit this problem from the point of view of regularity theory. We will sketch a proof of optimal regularity and regularity of the contact set. Similar result are known for scalar equations. The proofs for scalar equations are however based on maximum principles and thus not applicable to Signorini's problem which is modelled by a system of equations. | |||
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Mon, 30/04/2012 17:00 |
Jose L. Rodrigo (Warwick) |
Partial Differential Equations Seminar |
Gibson 1st Floor SR |
| based on joint work with Charles Fefferman (Princeton) and Kevin Luli (Yale). | |||
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Mon, 07/05/2012 17:00 |
Dehua Wang (University of Pittsburgh) |
Partial Differential Equations Seminar |
Gibson 1st Floor SR |
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Mon, 14/05/2012 17:00 |
Sergiu Klairneman (Princeton) |
Partial Differential Equations Seminar |
Gibson 1st Floor SR |
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Mon, 21/05/2012 17:00 |
Claude Bardos (Paris VII Denis Diderot) |
Partial Differential Equations Seminar |
Gibson 1st Floor SR |
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Recent results (starting with Scheffer and
Shnirelman and continuing with De Lellis and Szekelhyhidi ) underline
the importance of considering solutions of the incompressible Euler
equations as limits of solutions of more physical examples like
Navier-Stokes or Boltzmann. |
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Mon, 28/05/2012 17:00 |
Richard D. James (University of Minnesota) |
Partial Differential Equations Seminar |
Gibson 1st Floor SR |
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We describe an invariant manifold of the equations of molecular dynamics associated to a given discrete group of isometries. It is a time-dependent manifold, but its dependence on time is explicit. In the case of the translation group, it has dimension 6N, where N is an assignable positive integer. The manifold is independent of the description of the atomic forces within a general framework. Most of continuum mechanics inherits some version of this manifold, as do theories in-between molecular dynamics and continuum mechanics, even though they do not inherit the time reversibility of molecular dynamics on this manifold. The manifold implies a natural statistics of molecular motion, which suggests a simplifying ansatz for the Boltzmann equation which, in turn, leads to new explicit far-from-equilibrium solutions of this equation. In some way the manifold underlies experimental science, i.e., the viscometric flows of fluids and the bending and twisting of beams in solids and the procedures commonly used to measure constitutive relations, this being related to the fact that the form of the manifold can be prescribed independent of the atomic forces. |
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Mon, 04/06/2012 17:00 |
Hoai-Minh Nguyen (University of Minnesota) |
Partial Differential Equations Seminar |
Gibson 1st Floor SR |
| Cloaking recently attracts a lot of attention from the scientific community due to the progress of advanced technology. There are several ways to do cloaking. Two of them are based on transformation optics and negative index materials. Cloaking based on transformation optics was suggested by Pendry and Leonhardt using transformations which blow up a point into the cloaked regions. The same transformations had previously used by Greenleaf et al. to establish the non-uniqueness for Calderon's inverse problem. These transformations are singular and hence create a lot of difficulty in analysis and practical applications. The second method of cloaking is based on the peculiar properties of negative index materials. It was proposed by Lai et al. and inspired from the concept of complementary media due to Pendry and Ramakrishna. In this talk, I will discuss approximate cloaking using these two methods. Concerning the first one, I will consider the situation, first proposed in the work of Kohn et al., where one uses transformations which blow up a small ball (instead of a point) into cloaked regions. Many interesting issues such as finite energy and resonance will be mentioned. Concerning the second method, I provide the (first) rigorous analysis for cloaking using negative index materials by investigating the situation where the loss (damping) parameter goes to 0. I will also explain how the arguments can be used not only to establish the rigor for other interesting related phenomena using negative index materials such as superlenses and illusion optics but also to enlighten the mechanism of these phenomena. | |||
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Mon, 11/06/2012 15:30 |
Emil Wiedemann (Leipzig) |
Partial Differential Equations Seminar |
Gibson Grd floor SR |
| An intriguing, and largely open, question in mathematical fluid dynamics is whether solutions of the Navier-Stokes equations converge in some sense to a solution of the Euler equations in the zero viscosity limit. In fact this convergence could conceivably fail due to oscillations and concentrations occuring in the sequence. In the late 1980s, R. DiPerna and A. Majda extended the classical concept of Young measure to obtain a notion of measure-valued solution of the Euler equations, which records precisely these oscillation and concentration effects. In this talk I will present a result recently obtained in joint work with L. Székelyhidi, which states that any such measure-valued solution is generated by a sequence of distributional solutions of the Euler equations. The result is interesting from two different viewpoints: On the one hand, it emphasizes the huge flexibility of the concept of weak solution for Euler; on the other hand, it provides an example of a characterization theorem for Young measures in the tradition of D. Kinderlehrer and P. Pedregal where the differential constraint on the generating sequence does not satisfy the constant rank condition. | |||
