The radiation condition at infinity for the high-frequency Helmholtz equation with source term: a wave packet approach

We consider the high-frequency Helmholtz equation with a given source term, and a small absorption parameter \a>0. The high-frequency (or: semi-classical) parameter is \eps>0. We let \eps and \a go to zero simultaneously. We assume that the zero energy is non-trapping for the underlying classical flow. We also assume that the classical trajectories starting from the origin satisfy a transversality condition, a generic assumption. Under these assumptions, we prove that the solution u^\eps radiates in the outgoing direction, {\bf uniformly} in \eps. In particular, the function u^\eps, when conveniently rescaled at the scale \eps close to the origin, is shown to converge towards the {\bf outgoing} solution of the Helmholtz equation, with coefficients frozen at the origin. This provides a uniform version (in \eps) of the limiting absorption principle. Writing the resolvent of the Helmholtz equation as the integral in time of the associated semi-classical Schr\"odinger propagator, our analysis relies on the following tools: (i) For very large times, we prove and use a uniform version of the Egorov Theorem to estimate the time integral; (ii) for moderate times, we prove a uniform dispersive estimate that relies on a wave-packet approach, together with the above mentioned transversality condition; (iii) for small times, we prove that the semi-classical Schr\"odinger operator with variable coefficients has the same dispersive properties as in the constant coefficients case, uniformly in \eps

On The Weak-Coupling Limit for Bosons and Fermions

In this paper we consider a large system of Bosons or Fermions. We start with an initial datum which is compatible with the Bose-Einstein, respectively Fermi-Dirac, statistics. We let the system of interacting particles evolve in a weak-coupling regime. We show that, in the limit, and up to the second order in the potential, the perturbative expansion expressing the value of the one-particle Wigner function at time $t$, agrees with the analogous expansion for the solution to the Uehling-Uhlenbeck equation. This paper follows in spirit the companion work [\rcite{BCEP}], where the authors investigated the weak-coupling limit for particles obeying the Maxwell-Boltzmann statistics: here, they proved a (much stronger) convergence result towards the solution of the Boltzmann equation

Radiation condition at infinity for the high-frequency Helmholtz equation: optimality of a non-refocusing criterion

International audienceWe consider the high frequency Helmholtz equation with a variable refraction index $n^2(x)$ ($x \in \R^d$), supplemented with a given high frequency source term supported near the origin $x=0$. A small absorption parameter $\alpha_{\varepsilon}>0$ is added, which somehow prescribes a radiation condition at infinity for the considered Helmholtz equation. The semi-classical parameter is $\varepsilon>0$. We let \eps and \a_\eps go to zero {\em simultaneaously}. We study the question whether the indirectly prescribed radiation condition at infinity is satisfied {\em uniformly} along the asymptotic process \eps \to 0, or, in other words, whether the conveniently rescaled solution to the considered equation goes to the {\em outgoing} solution to the natural limiting Helmholtz equation. This question has been previously studied by the first autor. It is proved that the radiation condition is indeed satisfied uniformly in \eps, provided the refraction index satisfies a specific {\em non-refocusing condition}, a condition that is first pointed out in this reference. The non-refocusing condition requires, in essence, that the rays of geometric optics naturally associated with the high-frequency Helmholtz operator, and that are sent from the origin $x=0$ at time $t=0$, should not refocus at some later time $t>0$ near the origin again. In the present text we show the {\em optimality} of the above mentionned non-refocusing condition, in the following sense. We exhibit a refraction index which {\em does} refocus the rays of geometric optics sent from the origin near the origin again, and, on the other hand, we completely compute the asymptotic behaviour of the solution to the associated Helmholtz equation: we show that the limiting solution {\em does not} satisfy the natural radiation condition at infinity. More precisely, we show that the limiting solution is a {\em perturbation} of the outgoing solution to the natural limiting Helmholtz equation, and that the perturbing term explicitly involves the contribution of the rays radiated from the origin which go back to the origin. This term is also conveniently modulated by a phase factor, which turns out to be the action along the above rays of the hamiltonian associated with the semiclassical Helmholtz equation

Coexistence phenomena and global bifurcation structure in a chemostat-like model with species-dependent diffusion rates

International audienceWe study the competition of two species for a single resource in a chemostat. In the simplest space-homogeneous situation, it is known that only one species survives, namely the best competitor. In order to exhibit coexistence phenomena, where the two competitors are able to survive, we consider a space dependent situation: we assume that the two species and the resource follow a di usion process in space, on top of the competition process. Besides, and in order to consider the most general case, we assume each population is associated with a distinct di usion constant. This is a key di culty in our analysis: the speci c (and classical) case where all di usion constants are equal, leads to a particular conservation law, which in turn allows to eliminate the resource in the equations, a fact that considerably simpli fies the analysis and the qualitative phenomena. Using the global bifurcation theory, we prove that the underlying 2-species, stationary, di usive, chemostat-like model, does possess coexistence solutions, where both species survive. On top of that, we identify the domain, in the space of the relevant bifurcation parameters, for which the system does have coexistence solutions

Propagation of Gevrey regularity over long times for the fully discrete Lie Trotter splitting scheme applied to the linear Schroedinger equation

International audienceIn this paper, we study the linear Schroedinger equation over the d-dimensional torus, with small values of the perturbing potential. We consider numerical approximations of the associated solutions obtained by a symplectic splitting method (to discretize the time variable) in combination with the Fast Fourier Transform algorithm (to discretize the space variable). In this fully discrete setting, we prove that the regularity of the initial datum is preserved over long times, i.e. times that are exponentially long with the time discretization parameter. We here refer to Gevrey regularity, and our estimates turn out to be uniform in the space discretization parameter. This paper extends a previous text by Dujardin and Faou, where a similar result has been obtained in the semi-discrete situation, i.e. when the mere time variable is discretized and space is kept a continuous variable

Besov estimates in the high-frequency Helmholtz equation, for a non-trapping and C2 potential

AbstractWe study the high-frequency Helmholtz equation, for a potential having C2 smoothness, and satisfying the non-trapping condition. We prove optimal Morrey–Campanato estimates that are both homogeneous in space and uniform in the frequency parameter. The homogeneity of the obtained bounds, together with the weak assumptions we require on the potential, constitute the main new result in the present text. Our result extends previous bounds obtained by Perthame and Vega, in that we do not assume the potential satisfies the virial condition, a strong form of non-trapping

Analysis of a large number of Markov chains competing for transitions

International audienceWe consider the behavior of a stochastic system composed of several identically distributed, but non independent, discrete-time absorbing Markov chains competing at each instant for a transition. The competition consists in determining at each instant, using a given probability distribution, the only Markov chain allowed to make a transition. We analyze the first time at which one of the Markov chains reaches its absorbing state. When the number of Markov chains goes to infinity, we analyze the asymptotic behavior of the system for an arbitrary probability mass function governing the competition. We give conditions for the existence of the asymptotic distribution and we show how these results apply to cluster-based distributed systems when the competition between the Markov chains is handled by using a geometric distribution