Telč, Czech Republic, 2.–5. 5. 2018

~~To be announced.~~ The current conference programme is available for download (Last update on 19th April 2018.).

We present two alternative conditions which establish full regularity for the solutions to Navier-Stokes-equation (incompressible case), which are based on certain sign conditions on the pressure. The method does not rely on the well known theory of regular points or the blow-up-method. It works also for p- fluids in the sense that one gets an L^q(L^q) improvement for the solution.

We consider quasilinear parabolic equations of p-Laplacian type. We discuss sharp weighted norm inequalities for the gradient of solutions.

I will present the construction of degraded mixing solutions for the IPM system. This system models the dynamics of an incompressible and viscous fluid in a porous media and under the gravitational force. When the initial density of the fluid just take two values the existence of solutions for IPM is known as the Muskat problem. In a previous work, together with D. Córdoba and D. Faraco, we show the existence of solutions in the unstable regime which consist of the mixing of the two densities. In this talk I will skech a new construction in which we show that the solutions, in average, mix in a linear way. This is a work in collaboration with D. Faraco and F. Mengual Bretón.

It is well-known that minimizers of variational integrals, even under favourable conditions, might not be fully regular nor unique in the multi-dimensional vectorial case. However under suitable smallness conditions it is possible to regain both. In this talk we discuss such results under natural conditions on the integrand. These conditions are flexible and cover in particular the case of nonconvex variational integrals that are merely coercive on the space of maps of bounded variation. Parts of the talk is based on joint work with Judith Campos Cordero (Mexico City) and Franz Gmeineder (Bonn).

Lipschitz estimates are very often a turning point in regularity theory. For instance, when dealing with uniformly elliptic equations, they are a necessary step towards establishing gradient continuity of solutions and, eventually, to use boot-strap methods. In the case of non-uniformly elliptic problems, this is often the real focal point of regularity, as, after knowing that the gradient is bounded, uniformly elliptic theorems apply to several non-uniformly elliptic model cases. I will present a list of recent approaches to Lipschitz (and sometimes higher) continuity, for both uniformly and non-uniformly elliptic problems. The results are from joint work with Paolo Baroni (Parma), Lisa Beck (Augburg), Maria Colombo (Lausanne), Cristiana De Filippis (Oxford) and Tuomo Kuusi (Oulu).

Abstract in PDF can de downloaded here.