Fronts and InteRfaces in Science and Technology

Initial Training Network of the European Commission


The Energy of Mathematics:

Two Days in the Occasion of the 70th Anniversary of S.N. Antontsev

November 11-12, 2013






N. Chemetov (Lisbon, Portugal)

The motion of the rigid body in a viscous fluid with collisions

We consider the problem of the motion of a rigid body in an incrompressible viscous fluid, filling a bounded domain. This problem was studied by many authors. They considered classical non-slip boundary conditions, which gave them very PARADOXICAL result of no collisions of the body with the boundary of the domain.

In this work we study when Navier slip conditions are prescribed on the boundary of the body (instead of non-slip conditions). We prove for this model the GLOBAL existence of weak solution, which permit COLLISIONS with the boundary of the domain.


M. Chipot (Zurich, Switzerland)

Asymptotics of eigenstates of elliptic problems with mixed boundary data on domains tending to infinity


J.I. Díaz (Madrid, Spain)

On the free boundary for quenching type parabolic problems via local energy methods

In his pioneering paper, Some problems with free boundaries for the degenerating equations of gas dynamics. (Dinamika Sploshnoi Sredy, Novosibirsk, 1973, Vyp. 13, pp. 5-17. (Russian)) Stanislav Antontsev proposed the idea of a general method to prove the existence and location of free boundaries for quasilinear parabolic equations of degenerate type. Since then, many other paper developed this clever idea giving rise to a general methodology which today is applicable to nonlinear partial differential equations on any type (elliptic, parabolic and hyperbolic) and of any order (higher order too) and systems (see, e.g. the monograph: S.N. Antontsev, J.I, Diaz and S.I. Shmarev, Energy Methods for Free Boundary Problems: Applications to Nonlinear PDEs and Fluid Mechanics, Birkhäuser, Boston, 2001.

The main goal of this lecture is to present the application of this kind of energy methods to the case of equations involving terms with negative exponents (the so called quenching problems).


G. Galiano (Oviedo, Spain)

Applying Energy

We show how the Energy Method is used to deduce the existence of free boundaries and some of their properties in two problems expressed in terms of systems of PDE's. One is a classical problem in fluid dynamics, the Boussinesq-Oberbeck system. The other comes from the modeling of soil salinization.


S.L. Gavrilyuk (Marseille, France)

Hyperbolic model of hyperelasticity and applications to high impact problems

We consider the equations of hyperelasticity for isotropic solids in the Eulerian coordinates in a special case where the specific stored energy is a sum of two functions. The first one, the hydrodynamic part of the energy, depends only on the solid density and the entropy. The second one, the shear energy, depends on the invariants of the Finger tensor in such a way that it is unaffected by the volume change. A new sufficient criterion of hyperbolicity for such a system containing 14 unknowns is formulated : if the hydrodynamic sound velocity is real and a symmetric 3x3 matrix defined in terms of the shear energy is positive definite on a one-parameter family of surfaces of the unit-determinant deformation gradient, the equations are hyperbolic.

A thermodynamically consistent multi-phase extension of the hyperelasticity is proposed for solving fluid – structure interaction problems with irreversible elasto-plastic transformations. Some applications are presented : fracture formation and spallation under high velocity impact.


S. Kamin (Tel Aviv, Israel)

Prescribed conditions at infinity for parabolic equations

A. Meirmanov (Almaty, Kazakhstan) and S. Mukhambetzhanov (Almaty, Kazakhstan)

Mathematical models of a wormhole formation

In the present talk we consider a free boundary problem describing the wormhole formation in the porous media at the microscopic level. The mathematical model consists of the Stokes system for the velocity and pressure of the liquid and the diffusion-convection equation for the concentration of the acid. Using the two-scale convergent method for non-periodic structures we derive homogenized system with unknown coefficients depending on the structure of the pore space. In turn, this structure is defined by the unknown concentration of the acid.


H. de Oliveira (Algarve, Portugal)

Analysis of the existence for the steady Navier-Stokes equations with anisotropic diffusion (joint work with S.N. Antontsev)

The boundary-value problem for the generalized Navier-Stokes equations with anisotropic diffusion is considered in this talk. For this problem we prove the existence of weak solutions in the sense that solutions and test functions are considered in the same admissible function space. We prove also the existence of very weak solutions, i.e., solutions for which the test functions have more regularity. By exploiting several examples we show, in the case of dimension 3, that these existence results improve its isotropic versions in almost all directions or for particular choices of all the diffusion coefficients.


V. Shelukhin (Novosibirsk, Russia)

Homogenization of time harmonic Maxwell equations and the frequency dispersion effect

We perform homogenization of the time-harmonic Maxwell equations in order to determine the effective dielectric permittivity and effective electric conductivity. We prove that both these effective parameters depend on pulsation; this phenomenon is known as the frequency dispersion effect. Moreover, the macroscopic Maxwell equations also depend on pulsation; they are different for small and large values of pulsation.


S. Shmarev (Oviedo, Spain)

Localization and blow-up of solutions to evolution PDEs with nonstandard growth conditions



G. Vallet (Pau, France)

On the stochastic ∆p problem

In this talk, we will be interested in a result of existence of a solution to a p-Laplace problem with a stochastic force, when p is a function of variables t and x.


J.L. Vázquez (Madrid, Spain)

Scaling methods, energies and entropies in nonlinear diffusion

The long time behavior of nonlinear evolution equations is a classical topic of the analysis of PDEs. A number of methods have allowed to make progress in the description of the asymptotic behaviour of the solutions for equations of parabolic type. I will present three methods and mention some recent progress.


L. Veron (Tours, France)

Initial trace of positive solutions of weakly superlinear parabolic equations