arXiv subjects
Publications and source records attributed to Iván Moyano.
Consider an eigenfunction of the Laplacian on a torus. How small can its $L^2$-norm be on small balls? We provide partial answers to this question by exploiting the distribution of integer points on spheres, basic properties of polynomials, and Nazarov--Turán type estimates for exponential polynomials. Applications to quantum limits and control theory are given.
In this paper, we consider the time-dependent Born-Oppenheimer approximation (BOA) of a classical quantum molecule involving a possibly large number of nuclei and electrons, described by a Schrödinger equation. In the spirit of Born and Oppenheimer's original idea we study quantitatively the approximation of the molecular evolution. We obtain an iterable approximation of the molecular evolution to arbitrary order and we derive an effective equation for the reduced dynamics involving the nuclei equivalent to the original Schrödinger equation and containing no electron variables. We estimate the coefficients of the new equation and find tractable approximations for the molecular dynamics going beyond the one corresponding to the original Born and Oppenheimer approximation.
It is well known that both the heat equation with Dirichlet or Neumann boundary conditions are null controlable as soon as the control acts in a non trivial domain (i.e. a set of positive measure, see [10, 11, 12, 1, 6]. In this article, we show that for any couple of initial data (u0, v0) we can achieve the null control for both equations (Dirichlet and Neumann boundary conditions respectively) simultaneously with the same control function for both equations.
In this paper we consider a compact Riemannian manifold (M, g) of class C 1 $\cap$ W 2,$\infty$ and the damped wave or Schrödinger equations on M , under the action of a damping function a = a(x). We establish the following fact: if the measure of the set {x $\in$ M ; a(x) = 0} is strictly positive, then the decay in time of the associated energy is at least logarithmic.
In this note we investigate propagation of smallness properties for solutions to heat equations. We consider spectral projector estimates for the Laplace operator with Dirichlet or Neumann boundary conditions on a Riemanian manifold with or without boundary. We show that using the new approach for the propagation of smallness from Logunov-Malinnikova [7, 6, 8] allows to extend the spectral projector type estimates from Jerison-Lebeau [3] from localisation on open set to localisation on arbitrary sets of non zero Lebesgue measure; we can actually go beyond and consider sets of non vanishing d -- $δ$ ($δ$ > 0 small enough) Hausdorf measure. We show that these new spectral projector estimates allow to extend the Logunov-Malinnikova's propagation of smallness results to solutions to heat equations. Finally we apply these results to the null controlability of heat equations with controls localised on sets of positive Lebesgue measure. A main novelty here with respect to previous results is that we can drop the constant coefficient assumptions (see [1, 2]) of the Laplace operator (or analyticity assumption, see [4]) and deal with Lipschitz coefficients. Another important novelty is that we get the first (non one dimensional) exact controlability results with controls supported on zero measure sets.
The purpose of this article is to show that the spectral projector estimates for Laplace operators can be deduced from Logunov-Malinnikova's Propagation of smallness estimates for harmonic functions [11, 10, 9]. The main point is to pass from the local estimates obtained in [4] (on a compact manifold) to global estimates. We also state classical consequences in terms of observability and control for heat equations, which are direct consequences of these spectral projector estimates..
Following Arnold's geometric interpretation, the Euler equations of an incompressible fluid moving in a domain D are known to be the optimality equation of the minimizing geodesic problem along the group of orientation and volume preserving diffeomorphisms of D. This problem admits a well-established convex relaxation which generates a set of "relaxed", "multi-stream", version of the Euler equations. However, it is unclear that such relaxed equations are appropriate for the initial value problem and the theory of turbulence, due to their lack of well-posedness for most initial data. As an attempt to get a more relevant set of relaxed Euler equations, we address the multi-stream pressure-less gravitational Euler-Poisson system as an approximate model, for which we show that the initial value problem can be stated as a concave maximization problem from which we can at least recover a large class of smooth solutions for short enough times.
We study the large time behavior of Fujita-Kato type solutions to the Vlasov-Navier-Stokes system set on $\mathbb{T}^3 \times \mathbb{R}^3$. Under the assumption that the initial so-called modulated energy is small enough, we prove that the distribution function converges to a Dirac mass in velocity, with exponential rate. The proof is based on the fine structure of the system and on a bootstrap analysis allowing to get global bounds on moments.
In this paper we deal with the so-called "spectral inequalities", which yield a sharp quantification of the unique continuation for the spectral family associated with the Schrödinger operator in $ \mathbb{R}^d$ \begin{equation*} H_{g,V} = Δ_g + V(x), \end{equation*} where $Δ_g$ is the Laplace-Beltrami operator with respect to an analytic metric $g$, which is a perturbation of the Euclidean metric, and $V(x)$ a real valued analytic potential vanishing at infinity.
We prove a uniqueness result for weak solutions to the Vlasov-Navier-Stokes system in two dimensions, both in the whole space and in the periodic case, under a mild decay condition on the initial distribution function. The main result is achieved by combining methods from optimal transportation (introduced in this context by G. Loeper) with the use of Hardy's maximal function, in order to obtain some fine Wassestein-like estimates for the difference of two solutions of the Vlasov equation.
Consider the linear Boltzmann equation of radiative transfer in a half-space, with constant scattering coefficient $σ$. Assume that, on the boundary of the half-space, the radiation intensity satisfies the Lambert (i.e. diffuse) reflection law with albedo coefficient $α$. Moreover, assume that there is a temperature gradient on the boundary of the half-space, which radiates energy in the half-space according to the Stefan-Boltzmann law. In the asymptotic regime where $σ\to+\infty$ and $1-α\sim C/σ$, we prove that the radiation pressure exerted on the boundary of the half-space is governed by a fractional diffusion equation. This result provides an example of fractional diffusion asymptotic limit of a kinetic model which is based on the harmonic extension definition of $\sqrt{-Δ}$. This fractional diffusion limit therefore differs from most of other such limits for kinetic models reported in the literature, which are based on specific properties of the equilibrium distributions (heavy tails) or of the scattering coefficient as in [U. Frisch-H. Frisch: Mon. Not. R. Astr. Not. 181 (1977), 273-280].
In this article we consider a 2-D quantum particle confined a disc whose radius can be deformed continuously in time. We study the problem of controllability of such a quantum particle via deformations of the initial disc, i.e., when we set the time-dependent radius of the disc to be control variable. We prove that the resulting system is locally controllable around some radial trajectories which are linear combinations of the first three radial eigenfunc-tions of the Laplacian in the unit disc with Dirichlet boundary conditions. We prove this result, thanks to the linearisation principle, by studying the linearised system, which leads to a moment problem that can be solved using some results from Nonharmonic Fourier series. In particular, we have to deal with fine properties of Bessel functions.
We prove a null controllability result for the Vlasov-Navier-Stokes system, which describes the interaction of a large cloud of particles immersed in a fluid. We show that one can modify both the distribution of particles and the velocity field of the fluid from any initial state to the zero steady state, by means of an internal control. Indeed, we can modify the non-linear dynamics of the system in order to absorb the particles and let the fluid at rest. The proof is achieved thanks to the return method and a Leray-Schauder fixed-point argument.
In this paper we prove an exact controllability result for the Vlasov-Stokes system in the two-dimensional torus with small data by means of an internal control. We show that one can steer, in arbitrarily small time, any initial datum of class C 1 satisfying a smallness condition in certain weighted spaces to any final state satisfying the same conditions. The proof of the main result is achieved thanks to the return method and a Leray-Schauder fixed-point argument.
We consider the degenerate equation $$\partial\_t f(t,x) - \partial\_x \left( x^α \partial\_x f \right)(t,x) =0,$$ on the unit interval $x\in(0,1)$, in the strongly degenerate case $α\in [1,2)$ with adapted boundary conditions at $x=0$ and boundary control at $x=1$. We use the flatness approach to construct explicit controls in some Gevrey classes steering the solution from any initial datum $f\_0 \in L^2(0,1)$ to zero in any time $T\textgreater{}0$.