International Summer School on Evolution Equations, Prague, Czech Republic, 9.–13. 7. 2012

We discuss different models for the flow of two viscous incompressible Newtonian fluids. In classical models the interface between both fluids is a two-dimensional surface (a ``sharp interface''). On the other hand in so-called ``diffuse interface models'' the macroscopically immiscible fluids are assumed to partly mix in a small interfacial region. Moreover, diffusion of both components is taken into account. We present analytic results concerning well-posedness and asymptotic behavior for both kind of models. Moreover, we discuss the relations between diffuse and sharp interface model. Moreover, we will show that the diffusion present in the diffuse interface models leads to new effects compared to the classical sharp interface models. Furthermore, we will discuss a non-classical sharp interface model, where diffusion is still present and which has analytically a quite different behavior from the classical model.

eveq-2012-abels.pdfThe aim of these lectures is to provide an overview of some of the main results and recent developments in nonlinear water waves. We will present a selection of aspects of water-wave motion that are believed to be of intrinsic mathematical interest and of physical relevance, and where a mathematical study can be pursued to an advanced stage, enabling us to derive conclusions that explain, to some extent, observed phenomena. The mathematical considerations will be preceded by a discussion of the underlying physical factors, and we discuss the physical relevance of the mathematical results that are presented. The central theme concerns wave-current interactions, more precisely the existence theory of travelling two-dimensional water waves propagating at the surface of water with a flat bed in a flow with a general vorticity distribution. The approach relies on a fruitful combination of methods specific to elliptic partial differential equations with bifurcation and topological degree theory.

eveq-2012-constantin-1.pdf , eveq-2012-constantin-2.pdf , eveq-2012-constantin-3.pdf

We will discuss some theoretical aspects of fluid-solid dynamics. Emphasis will be put on the connection between the regularity of the solids and the qualitative properties of the Navier-Stokes model (well-posedness, occurence of collisions between solids). We will eventually focus on the computation of the drag for various models of solid roughness.

The lectures will review some of the classical topics in the evolution problem for the Einstein equations, including its formulation and construction of solutions, as well as more recent results on well-posedness and stability.

In these lectures I will present the notion of Gamma-convergence of gradient flows, introduced in a joint work with Etienne Sandier. It is a notion that serves to pass to the asymptotic limit in some evolution problems which are gradient flows of energies which are themselves related by Gamma-convergence (as introduced by De Giorgi). I will present the abstract scheme, review the applications it has had, and describe in particular the application to deriving the dynamical law of vortices in the Ginzburg-Landau equation of superconductivity.

eveq-2012-serfay-a.pdf Gamma-convergence of gradient flows on Hilbert and metric spaces and applications, *Disc. Cont. Dyn. Systems A*, 31, No. 4, 2011, 1427-1451, special issue in honor of De Giorgi and Stampacchia

eveq-2012-serfay-b.pdf Gamma-convergence of gradient flows and applications to Ginzburg-Landau vortex dynamics. Topics on concentration phenomena and problems with multiple scales, 267--292, Lect. Notes Unione Mat. Ital., 2, Springer, Berlin, 2006

I will present a series of lectures on the global-in-time variational treatment of evolution equations. In particular, we shall be interested in the possible reformulation of evolution systems, either of conservative or dissipative type, in terms of minimization problems. The interest for this perspective is that of moving the successful machinery of the Calculus of Variations (direct method, gamma-convergence, relaxation) to evolutionary situations.

After reviewing some classical material, I will focus on the recently proposed Weighted-Inertia-Dissipation-Energy (WIDE) principle. This variational approach relies on the minimization of a specific parameter-dependent family of functionals plus the check of the corresponding parameter limit on minimizers. The WIDE variational program has been already successfully applied in different nonlinear contexts ranging from parabolic to the hyperbolic.

I plan to show some basics of the WIDE variational technique with specific emphasis on the prototypical cases of gradient flows and Lagrangian Mechanics. Applications to nonlinear PDEs including mean curvature flow as well as to rate-independent evolutions will be mentioned. In the case of semilinear wave equations, the WIDE principle corresponds to a long-standing conjecture by De Giorgi. I will report on the recently discovered proof of this conjecture.