Gibson 1st Floor SR

Modern mathematical approaches to plasticity result in non-convex variational problems for which the standard methods of the calculus of variations are not applicable. In this contribution we consider geometrically nonlinear crystal elasto-plasticity in two dimensions with one active slip system. In order to derive information about macroscopic material behavior the relaxation of the corresponding incremental problems is studied. We focus on the question if realistic systems with an elastic energy leading to large penalization of small elastic strains can be well-approximated by models based on the assumption of rigid elasticity. The interesting finding is that there are qualitatively different answers depending on whether hardening is included or not. In presence of hardening we obtain a positive result, which is mathematically backed up by Γ-convergence, while the material shows very soft macroscopic behavior in case of no hardening. The latter is due to the vanishing relaxation for a large class of applied loads.

This is joint work with Sergio Conti and Georg Dolzmann.

The space BD of functions of bounded deformation consists of all L1-functions whose distributional symmetrized derivative (defined by duality with the symmetrized gradient ($\nabla u + \nabla u^T)/2$) is representable as a finite Radon measure. Such functions play an important role in a variety of variational models involving (linear) elasto-plasticity. In this talk, I will present the first general lower semicontinuity theorem for symmetric-quasiconvex integral functionals with linear growth on the whole space BD. In particular we allow for non-vanishing Cantor-parts in the symmetrized derivative, which correspond to fractal phenomena. The proof is accomplished via Jensen-type inequalities for generalized Young measures and a construction of good blow-ups, based on local rigidity arguments for some differential inclusions involving symmetrized gradients. This strategy allows us to establish the lower semicontinuity result even without a BD-analogue of Alberti's Rank-One Theorem in BV, which is not available at present. A similar strategy also makes it possible to give a proof of the classical lower semicontinuity theorem in BV without invoking Alberti's Theorem.

In this talk we will first consider the Allen-Cahn action functional that controls the probability of rare events in an Allen-Cahn type equation with additive noise. Further we discuss a perturbation of the Allen-Cahn equation by a stochastic flow. Here we will present a tightness result in the sharp interface limit and discuss the relation to a version of stochastically perturbed mean curvature flow. (This is joint work with Luca Mugnai, Leipzig, and Hendrik Weber, Warwick.)

The classical isoperimetric inequality states that, given a set $E$ in $R^n$ having the same measure of the unit ball $B$, the perimeter $P(E)$ of $E$ is greater than the perimeter $P(B)$ of $B$. Moreover, when the isoperimetric deficit $D(E)=P(E)-P(B)$ equals 0, than $E$ coincides (up to a translation) with $B$. The sharp quantitative form of the isoperimetric inequality states that $D(E)$ can be bound from below by $A(E)^2$, where the Fraenkel asymmetry $A(E)$ of $E$ is defined as the minimum of the volume of the symmetric difference between $E$ and any translation of $B$. This result, conjectured by Hall in 1990, has been proven in its full generality by Fusco-Maggi-Pratelli (Ann. of Math. 2008) via symmetrization arguments and more recently by Figalli-Maggi-Pratelli (Invent. Math. 2010) through optimal transportation techniques. In this talk I will present a new proof of the sharp quantitative version of the isoperimetric inequality that I have recently obtained in collaboration with G.P.Leonardi (University of Modena e Reggio). The proof relies on a variational method in which a penalization technique is combined with the regularity theory for quasiminimizers of the perimeter. As a further application of this method I will present a positive answer to another conjecture posed by Hall in 1992 concerning the best constant for the quantitative isoperimetric inequality in $R^2$ in the small asymmetry regime.

I will show that one can (at least in theory) guarantee the "validity" of a numerical approximation of a solution of the 3D Navier-Stokes equations using an explicit a posteriori test, despite the fact that the existence of a unique solution is not known for arbitrary initial data.

The argument relies on the fact that if a regular solution exists for some given initial condition, a regular solution also exists for nearby initial data ("robustness of regularity"); I will outline the proof of robustness of regularity for initial data in $H^{1/2}$.

I will also show how this can be used to prove that one can verify numerically (at least in theory) the following statement, for any fixed R > 0: every initial condition $u_0\in H^1$ with $\|u\|_{H^1}\le R$ gives rise to a solution of the unforced equation that remains regular for all $t\ge 0$.

This is based on joint work with Sergei Chernysehnko (Imperial), Peter Constantin (Chicago), Masoumeh Dashti (Warwick), Pedro Marín-Rubio (Seville), Witold Sadowski (Warsaw/Warwick), and Edriss Titi (UC Irivine/Weizmann).

In this talk we will present some recent results on the long time

asymptotics of the generalized (non-Markovian) Langevin equation (gLE). In particular,

we will discuss about the ergodic properties of the gLE and present estimates on the rate of convergence to equilibrium, we will present

a homogenization result (invariance principle) and we will discuss

about the convergence of the gLE dynamics to the (Markovian) Langevin

dynamics, in some appropriate asymptotic limit. The analysis is based on the approximation of the gLE by a

high (and possibly infinite) dimensional degenerate Markovian system,

and on the analysis of the spectrum of the generator of this Markov

process. This is joint work with M. Ottobre and K. Pravda-Starov.

A basic example of  shear flow wasintroduced  by DiPerna and Majda to study the weaklimit of oscillatory solutions of the Eulerequations of incompressible ideal fluids. Inparticular, they proved by means of this examplethat weak limit of solutions of Euler equationsmay, in some cases, fail to be a solution of Eulerequations. We use this shear flow example toprovide non-generic, yet nontrivial, examplesconcerning the immediate loss of smoothness andill-posedness of solutions of the three-dimensionalEuler equations, for initial data that do notbelong to $C^{1,\alpha}$. Moreover, we show bymeans of this shear flow example the existence ofweak solutions for the three-dimensional Eulerequations with vorticity that is  having anontrivial density concentrated on non-smoothsurface. This is very different from what has beenproven for the two-dimensional Kelvin-Helmholtzproblem where a minimal regularity implies the realanalyticity of the interface. Eventually, we usethis shear flow to provide explicit examples ofnon-regular solutions of the three-dimensionalEuler equations that conserve the energy, an issuewhich is related to the Onsager conjecture.

This is a joint work with Claude Bardos.

The purpose of this talk is to provide a large class of examples of large initial data which gives rise to a global smooth solution. We shall explain what we mean by large initial data. Then we shall explain the concept of slowly varying function and give some flavor of the proofs of global existence.

We consider the Relativistic Vlasov-Maxwell system of equations which

describes the evolution of a collisionless plasma. We show that under

rather general conditions, one can test for linear instability by

checking the spectral properties of Schrodinger-type operators that

act only on the spatial variable, not the full phase space. This

extends previous results that show linear and nonlinear stability and

instability in more restrictive settings.

We will describe some joint work with V. G. Maz’ya and I. E. Verbitsky, concerning homogeneous quasilinear differential operators. The model operator under consideration is:

\[ L(u) = - \Delta_p u - \sigma |u|^{p-2} u. \]

Here $\Delta_p$ is the p-Laplacian operator and $\sigma$ is a signed measure, or more generally a distribution. We will discuss an approach to studying the operator L under only necessary conditions on $\sigma$, along with applications to the characterisation of certain Sobolev inequalities with indefinite weight. Many of the results discussed are new in the classical case p = 2, when the operator L reduces to the time independent Schrödinger operator.