Mikhaël Balabane: On the linear Helmholtz equation.

Matania Ben-Artzi: Decay estimates and vanishing viscosity for viscous Hamilton-Jacobi equations.
Résumé: Sharp temporal decay estimates are established for the gradient and time derivative of solutions to the Hamilton-Jacobi equation ∂_tv+H(|∇_xv|)=εΔv, the parameter ε being either positive or zero. Special care is given to the dependence of the estimates on ε. As a by-product, we obtain convergence of the solutions , as ε→0, to a viscosity solution, the initial condition being only continuous and either bounded or non-negative. The main requirement on H is that it grows superlinearly or sublinearly at infinity, including in particular H(r)=r^p for r∈[0,∞) and p∈(0,∞), p≠1. (Joint work with S.Benachour and Ph.Laurencot.)

Jerry Bona: Nonlinear Dispersive Equations with Analytic Data.
Résumé: This lecture will be concerned with nonlinear, dispersive evolution equations and focus especially on the situation that arises when the auxiliary data is analytic. Questions of obvious interest include whether or not the analyticity persists, and if so what one can say about the radius of convergence. An extended concrete example will serve to illustrate some of the theory.

Haïm Brezis: On a conjecture of J. Serrin.
Résumé: In 1964 J. Serrin proposed the following conjecture. Let u be a weak solution (in W^1,1) of a second order elliptic equation in divergence form, with Hölder continuous coefficients, then u is a "classical" solution (i.e. u belongs to H¹). I will present a solution to this conjecture assuming even weaker conditions on u (e.g. u in BV) and on the coefficients.

Michel Chipot: Uniqueness results for anisotropic nonlinear equations.
Résumé: We would like to present some uniqueness results for equations of the type ∂_tu=∂_xi(a_i(x,t,u)|∂_xiu|^p_i-2∂_xiu)+f in ΩX(0,T), together with initial and boundary conditions (joint work with S.Antontsev).

Constantine Dafermos: Progress in hyperbolic conservation laws.
Résumé: Hyperbolic conservation laws constitute a class of nonlinear partial differential equations with an illustrious pedigree and diverse applications to physics and beyond. Despite great progress in recent years, this area is still replete with challenging open problems. The aim of the lecture is to illustrate the current state of affairs and to point out the emerging research trends in the field.

Flávio Dickstein: Blowup and global sign-changing solutions of the nonlinear heat equation.
Résumé: Given α>0 we consider the nonlinear heat equation u_t-Δu=|u|^αu for x∈Ω and t>0. We suppose that Ω is the whole space or a ball, imposing then either Dirichet or Neumann boundary conditions. It is well-known that in all three cases there exist both global in time solutions and blowing up solutions. A natural question is to characterize the set G of initial data generating global solutions. This is a delicate problem, specially for sign-changing solutions. We describe some recent results obtained in collaboration with Thierry Cazenave and Fred Weissler giving some partial answers to this question. We show in particular examples where G is not star-shaped around 0.

Miguel Escobedo: Non zero flux solutions for kinetic equations.
Résumé: We shall present results on the existence of non equilibrium solutions to homogeneous kinetic equations which do not satisfy the formal conservation laws of the equation. We shall briefly motivate the interest of these solutions in the general context of the weak turbulence theory. We will describe the general outline of the proofs, describing in particular the different situations that can arise for different, although similar, models.

Marek Fila: Slow grow-up of solutions of a supercritical parabolic equations.
Résumé: We study solutions of the Cauchy problem for a supercritical semilinear parabolic equation which converge to a singular steady state from below as t→∞. We show that the grow-up rate of such solutions depends on the spatial decay rate of initial data. In particular, the grow-up rate can be arbitrarily slow.

Alain Haraux: Sorry, just a simple ODE!
Résumé: We study the rate of decay to 0 and the oscillation properties of solutions to the scalar second order ODE: u''+ c |u'|^αu' + |u|^βu = 0 where c, α, β are positive constants. Various extensions (forced equation, system) are possible.

Hiroshi Matano: En attente.
Résumé:

Lambertus A. Peletier: The dynamics of protein binding.
Résumé: When a drug enters the blood stream, on its way to a pharmaceutical target, it finds many proteins on its way which are eager to bind it and thus prevent it from reaching its destination. Whilst this may first adversely affect the beneficial effect of the drug, the drug bound to the proteins is not lost. It forms a buffer, which eventually may be released back into the blood stream and thus still reach its target. In this lecture we discuss a model proposed in order to study the dynamics of this process and determine the amount of drug that reaches its target over a given time span, say 24 hours. Mathematically, this results into the analysis of a sequence of singular perturbation problems involving systems of nonlinear ordinary differential equations.

Jean-Claude Saut: Remarks on the Cauchy problem for dispersive equations.
Résumé:

Ralph Showalter: Homogenization of Pseudo-parabolic PDE.
Résumé: (Authors: Ralph Showalter and Malgorzata Peszynska) Recent models of partially saturated flow through porous media include dynamic effects in the capillary pressure curves. These lead to partial differential equations of pseudo-parabolic type with multiple nonlinearity and degeneracy. We describe properties and numerical computation of solutions of appropriate initial-boundary-value problems and the upscaling from various types of heterogeneous media.

Walter Strauss: Steady rotational water waves.
Résumé: Consider a classical 2D gravity wave (studied by Euler, Poisson, Cauchy, Airy, Stokes, Levi-Civita,...) with an arbitrary vorticity function. Consider such a wave traveling at a constant speed over a flat bed. Using local and global bifurcation theory and topological degree, one can prove that there exist many such waves of large amplitude. I will outline the existence proof (joint with Adrian Constantin) and also exhibit some recent computations (joint with Joy Ko) of the waves using numerical continuation. The computations illustrate certain relationships between the amplitude, energy and mass flux of the waves. If the vorticity is sufficiently large, the first stagnation point of the wave occurs not at the crest (as with the much-studied irrotational flows) but on the bed directly below the crest or else in the interior of the fluid.

Slim Tayachi: Behavior of solutions for parabolic equations with nonlinear gradient terms.
Résumé: In this talk we discuss the existence of singular solutions at the origin for the Chipot-Weissler parabolic equation∂_tu=Δu-|u|^p-1u+b|∇u|^q,  t>0, x∈R^N      (1)where b>0, p>1, q>1. It is known that for b=0, p>1+(2/N ), the solutions of the nonlinear heat equation with absorption: ∂_tu=Δu-|u|^p-1u,  t>0, x∈R^N          (2)cannot develop any singularity at the origin. This differs from the linear heat equation which has a solution with initial data the Dirac delta function δ. We prove that the equation (1) with p>1+(2/N ), q>(N+2)/(N+1) has no solutions with initial data the Dirac delta function δ. We also show that the equation (1) for b>0 large, p>1+(2/N ), q=2p/(p+1) the equation (1) has solutions singular at the origin. This shows that the equation (1) has a behavior as t→0 different from (2) and from the linear heat equation. To prove the existence of such solutions, we construct forward self-similar solutions with exponentially decaying profiles which do not exist for (2) with p>1+(2/N ). We also present some results for the influence of the gradient term in the behavior of some global solutions of (1) as t→∞.

Laurent Véron: Boundary singularities of solutions of non-monotone nonlinear elliptic equations.
Résumé: Assume Ω∈R^N is a smooth domain, x₀∈∂Ω and q≥(N+1)/(N-1). We study the behavior near x₀ of any positive solution of (E) -Δu=u^q in Ω which coincides with some ζ∈C²(∂Ω)$ on ∂Ω\{0}. We prove that, if (N+1)/(N-1)< q<(N+2)/(N-2), u satisfies u(x)≤C|x-x₀|^-2/(q-1) and we give the limit of |x-x₀|^-2/(q-1)u(x) as x→x₀. In the case q=(N+1)/(N-1), u satisfies u(x)≤C|x-x₀|^1-N(ln(1/|x|))^(1-N)/2 and a corresponding precise asymptotic is obtained. We also study some existence and uniqueness questions for related equations on spherical domains.

David M. Stonner: Introduction to the special session.

Henry A. Warchall: The funding of research and higher education in mathematics: USA.

Pascal Auscher: The funding of research and higher education in mathematics: France