**The 2012-2013 ****laureates**

for : "Regularity in a one-phase free boundary problem for the fractional Laplacian". Read article here

Abstract

For a one-phase free boundary problem involving a fractional Laplacian, we prove that “flat free boundaries” are C1,αC1,α. We recover the regularity results of Caffarelli for viscosity solutions of the classical Bernoulli-type free boundary problem with the standard Laplacian.

pour : "The Stokes conjecture for waves with vorticity". Read article here

Abstract

We study stagnation points of two-dimensional steady gravity free-surface water waves with vorticity.

We obtain for example that, in the case where the free surface is an injective curve, the asymptotics at any stagnation point is given either by the “Stokes corner flow” where the free surface has a *corner of* 120°, or the free surface ends in a *horizontal cusp*, or the free surface is *horizontally flat* at the stagnation point. The cusp case is a new feature in the case with vorticity, and it is not possible in the absence of vorticity.

In a second main result we exclude horizontally flat singularities in the case that the vorticity is 0 on the free surface. Here the vorticity may have infinitely many sign changes accumulating at the free surface, which makes this case particularly difficult and explains why it has been almost untouched by research so far.

Our results are based on calculations in the original variables and do not rely on structural assumptions needed in previous results such as isolated singularities, symmetry and monotonicity.

**The 2011 laureate**

for : "Two soliton collision for nonlinear Schrödinger equations in dimension 1". Read article here.

Abstract

We study the collision of two solitons for the nonlinear Schrödinger equation *i**ψ**t*=−*ψ**x**x*+*F*(^{2}|*ψ*|)*ψ*, *F*(*ξ*)=−2*ξ*+*O*(*ξ*^{2}) as *ξ*→0, in the case where one soliton is small with respect to the other. We show that in general, the two soliton structure is not preserved after the collision: while the large soliton survives, the small one splits into two outgoing waves that for sufficiently long times can be controlled by the cubic NLS: *i**ψ**t*=−*ψ**x**x*−2^{2}|*ψ*|*ψ*.

**The 2010 ****laureates**

for : "From the Klein–Gordon–Zakharov system to a singular nonlinear Schrödinger system ". Read article here.

Summary

In this article, we carry on with investigating the high frequency and subsonic limits of the Klein-Gordon-Zakharov system. Literally, the limit system is the nonlinear Schrödinger system. However, for a particular case of the parameters, we find a new model that features a singular term. The object of this paper is to give rigorous derivation of that model and to show the convergence in the space of energy.

**The 2009 ****laureate**

for: "Hydrodynamic limits: some improvements of the relative entropy method." Consult the article.

Summary

This article is dedicated to the study – by the relative entropy method – of the asymptotic characteristic of the Boltzmann equation leading to the incompressible Euler equations. It extends the result established by the author in [L. Saint-Raymond, Convergence of solutions to the Boltzmann equation in the incompressible Euler limit, Arch. Ration. Mech. Anal. 166 (2003) 47–80] for well-prepared data. The issue is to take into account the acoustic waves and the original layer of relaxation in order to obtain a convergence result under slightly restrictive hypothesis about the original data.

This article is dedicated to the study – by the relative entropy method – of the asymptotic characteristic of the Boltzmann equation leading to the incompressible Euler equations. It extends the result established by the author in [L. Saint-Raymond, Convergence of solutions to the Boltzmann equation in the incompressible Euler limit, Arch. Ration. Mech. Anal. 166 (2003) 47–80] for well-prepared data. The issue is to take into account the acoustic waves and the original layer of relaxation in order to obtain a convergence result under slightly restrictive hypothesis about the original data.

The study presented here requires in return a (non-uniform) control about the distribution of high speeds, which is satisfied, for instance, by the classical solutions build by Guo in [Y. Guo, The Vlasov–Poisson–Boltzmann system near Maxwellians, Comm. Pure Appl. Math. 55 (2002) 1104–1135].

**The 2008 ****laureates**

"On the optimality of the observability inequalities for parabolic and hyperbolic systems with potentials." Read article here.

Abstract

In this paper we prove the optimality of the observability inequality for parabolic systems with potentials in even space dimensions *n*⩾2. This inequality (derived by E. Fernández-Cara and the third author in the context of the scalar heat equation with potentials in any space dimension) asserts, roughly, that for small time, the total energy of solutions can be estimated from above in terms of the energy localized in a subdomain with an observability constant of the order of , *a* being the potential involved in the system. The problem of the optimality of the observability inequality remains open for scalar equations.

The optimality is a consequence of a construction due to V.Z. Meshkov of a complex-valued bounded potential *q*=*q*(*x*) in *R*^{2} and a nontrivial solution *u* of Δ*u*=*q*(*x*)*u* with the decay property |*u*(*x*)|⩽exp(−|*x*|^{4/3}). Meshkov's construction may be generalized to any even dimension. We give an extension to odd dimensions, which gives a sharp decay rate up to some logarithmic factor and yields a weaker optimality result in odd space-dimensions.

We address the same problem for the wave equation. In this case it is well known that, in space-dimension *n*=1, observability holds with a sharp constant of the order of . For systems in even space dimensions *n*⩾2 we prove that the best constant one can expect is of the order of for any *T*>0 and any observation domain. Based on Carleman inequalities, we show that the positive counterpart is also true when *T* is large enough and the observation is made in a neighborhood of the boundary. As in the context of the heat equation, the optimality of this estimate is open for scalar equations.

We address similar questions, for both equations, with potentials involving the first order term. We also discuss issues related with the impact of the growth rates of the nonlinearities on the controllability of semilinear equations. Some other open problems are raised.

*Last updated 06/10/2014*