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.

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.

 

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].

 

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 R² and a nontrivial solution u of Δu=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.