Vinh Nguyen (Cardiff, UK)

With C. IMBERT, we study degenerate parabolic equations in multi-domains whose coefficients are discontinuous along interfaces.  We observe that the approach proposed by IMBERT and Monneau (2014) for Hamilton-Jacobi equations can be further developed to handle generalized junction conditions (such as the generalized Kirchoff ones) and second order terms. We first prove that generalized junction conditions reduce to flux-limited ones.  We then use then vertex test function (Imbert, Monneau — 2014) to prove a comparison principle.  We  then determine the vanishing viscosity limit associated with Hamilton-Jacobi equations posed on multi-domains and networks.  In the two-domain and convex case, the maximal Ishii solution identified by Barles, Briani and Chasseigne (2012) is selected. Finally, we give a short and simple PDE proof for the large deviation result of Boue, Dupuis and Ellis (2000).

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Matteo Novaga (Pisa, Italy)

Electrified liquids are well known to be prone to a variety of interfacial instabilities that result in the onset of apparent interfacial singularities and liquid fragmentation. In the case of electrically conducting liquids, one of the basic models describing the equilibrium interfacial configurations and the onset of instability assumes the liquid to be equipotential and interprets those configurations as local minimizers of the energy consisting of the sum of the surface energy and the electrostatic energy. Surprisingly, this classical geometric variational model is mathematically ill-posed irrespectively of the degree to which the liquid is electrified.

Specifically, an isolated spherical droplet is never a local minimizer, no matter how small is the total charge on the droplet, since the energy can always be lowered by a smooth, arbitrarily small distortion of the droplet’s surface. This is in sharp contrast with the experimental observations that a critical amount of charge is needed in order to destabilize a spherical droplet. We discuss some possible regularization mechanisms for the considered free boundary problem

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Panagiotis E. Souganidis (Chicago, USA)

I will present a recently developed theory for scalar conservation laws with nonlinear multiplicative rough signal dependence. I will describe the difficulties, introduce the notion of pathwise entropy/kinetic solution and its well-posedness. I will also talk about the long time behavior of the solutions as well as some regularization by noise type results.

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Bianca Stroffolini (Napoli, Italy)

The purpose of the Lipschitz truncation is to regularize a given function by a Lipschitz continuous one by changing it only on a small  bad set. It is crucial for the applications that the function is not changed globally, which rules out the possibility of convolutions.

The Lipschitz truncation technique was introduced by Acerbi-Fusco to show lower semicontinuity of certain variational integrals.

Since then this technique has been successfully applied in many different areas: biting lemmas, existence theory  and regularity results of non-linear elliptic PDE . It was also successfully applied in the framework of non-Newtonian fluids of power law type and even in the context of numerical analysis.

I will try to present some Lipschitz truncations Lemmas.
As an application, existence/ regularity of solutions of PDEs will be discussed.

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This round table session is addressed at all the female participants but it is of course open to interested male participants as well (please contact Federica if you are interested in participating).

The aim is to discuss the problem of gender imbalance in mathematical academic positions in particular in UK departments. We will focus on the recruiting problem, in particular from the post-doc level on. This seems to be the academic level where female applicants “disappear”. The contribution to the discussion of the many overseas female participants will be very valuable in understanding if this problem is UK-specific or a general trend across Europe, and in sharing best practise for increasing female representation in maths departments.

Location: Room M/1.25

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