L2

Irreversible processes are accompanied by an increase in the internal entropy of a continuum, and as such the entropy production function is fundamental in determining the overall state of the system. In this talk, it will be shown that the entropy production function can be utilized for a variational analysis of certain dissipative continua in two different ways. Firstly, a novel unified Lagrangian-Hamiltonian formalism is constructed giving phase space extra structure, and applied to the study of fluid flow and brittle fracture.  Secondly, a maximum entropy production principle is presented for simple bodies and its implications to the study of fluid flow discussed. 

Second-order regularity results are established for solutions to elliptic equations and systems with the principal part having a Uhlenbeck structure and square-integrable right-hand sides. Both local and global estimates are obtained. The latter apply to solutions to homogeneous Dirichlet problems under minimal regularity assumptions on the boundary of the domain. In particular, if the domain is convex, no regularity of its boundary is needed. A critical step in the approach is a sharp pointwise inequality for the involved elliptic operator. This talk is based on joint investigations with A.Kh.Balci, L.Diening, and V.Maz'ya.

Here is the program.

Junior Strings is a seminar series where DPhil students present topics of common interest that do not necessarily overlap with their own research area. This is primarily aimed at PhD students and post-docs but everyone is welcome

 "In this talk, we will discuss invariant measures techniques to establish probabilistic global well-posedness for PDEs. We will go over the limitations that the Gibbs measures and the so-called fluctuation-dissipation measures encounter in the context of energy-supercritical PDEs. Then, we will present a new approach combining the two aforementioned methods and apply it to the energy supercritical Schrödinger equations. We will point out other applications as well."

I will introduce a class of mean-field games under forward performance and for general risk preferences. Players interact through competition in fund management, driven by relative performance concerns in an asset diversification setting. This results in a common-noise mean field game. I will present the value and the optimal policies of such games, as well as some concrete examples. I will also discuss the partial information case, i.e.. when the risk premium is not directly observed.