Section: New Results

mathematical analysis of multiscale partial differential equations

Collision of almost parallel vortex filaments

In [3], we investigate the occurrence of collisions in the evolution of vortex filaments through a system introduced by Klein, Majda and Damodaran and Zakharov. We first establish rigorously the existence of a pair of almost parallel vortex filaments, with opposite circulation, colliding at some point in finite time. The collision mechanism is based on the one of the self-similar solutions of the model, described in a previous work. In the second part of this paper we extend this construction to the case of an arbitrary number of filaments, with polygonial symmetry, that are perturbations of a configuration of parallel vortex filaments forming a polygon, with or without its center, rotating with constant angular velocity.

Free vibrations of axisymmetric shells: parabolic and elliptic cases

In [9], approximate eigenpairs (quasimodes) of axisymmetric thin elastic domains with laterally clamped boundary conditions (Lamé system) are determined by an asymptotic analysis as the thickness ($2\epsilon$) tends to zero. The departing point is the Koiter shell model that we reduce by asymptotic analysis to a scalar model that depends on two parameters: the angular frequency $k$ and the half-thickness $\epsilon$. Optimizing $k$ for each chosen $\epsilon$, we find power laws for $k$ in function of $\epsilon$ that provide the smallest eigenvalues of the scalar reductions. Corresponding eigenpairs generate quasimodes for the 3D Lamé system by means of several reconstruction operators, including boundary layer terms. Numerical experiments demonstrate that in many cases the constructed eigenpair corresponds to the first eigenpair of the Lamé system. Geometrical conditions are necessary to this approach: The Gaussian curvature has to be nonnegative and the azimuthal curvature has to dominate the meridian curvature in any point of the midsurface. In this case, the first eigenvector admits progressively larger oscillation in the angular variable as $\epsilon$ tends to 0 . Its angular frequency exhibits a power law relation of the form $k=\gamma {\epsilon }^{\beta }$ with $\beta =\frac{1}{4}$ in the parabolic case (cylinders and trimmed cones), and the various $\beta$'s ($\frac{2}{5},\frac{3}{7}$ and $\frac{1}{3}$ in the elliptic case. For these cases where the mathematical analysis is applicable, numerical examples that illustrate the theoretical results are presented.

High frequency oscillations of first eigenmodes in axisymmetric shells as the thickness tends to zero

In [24], the lowest eigenmode of thin axisymmetric shells is investigated for two physical models (acoustics and elasticity) as the shell thickness ($2\epsilon$) tends to zero. Using a novel asymptotic expansion we determine the behavior of the eigenvalue $\lambda \left(\epsilon \right)$ and the eigenvector angular frequency $k\left(\epsilon \right)$ for shells with Dirichlet boundary conditions along the lateral boundary, and natural boundary conditions on the other parts. First, the scalar Laplace operator for acoustics is addressed, for which $k\left(\epsilon \right)$ is always zero. In contrast to it, for the Lamé system of linear elasticity several different types of shells are defined, characterized by their geometry, for which $k\left(\epsilon \right)$ tends to infinity as $\epsilon$ tends to zero. For two families of shells: cylinders and elliptical barrels we explicitly provide $\lambda \left(\epsilon \right)$ and $k\left(\epsilon \right)$ and demonstrate by numerical examples the different behavior as $\epsilon$ tends to zero.

Semiclassical Sobolev constants for the electro-magnetic Robin Laplacian

This paper [15] is devoted to the asymptotic analysis of the optimal Sobolev constants in the semiclassical limit and in any dimension. We combine semiclassical arguments and concentration-compactness estimates to tackle the case when an electromagnetic field is added as well as a smooth boundary carrying a Robin condition. As a byproduct of the semiclassical strategy, we also get exponentially weighted localization estimates of the minimizers.

On the MIT Bag Model in the Non-relativistic Limit

This paper [2] is devoted to the spectral investigation of the MIT bag model, that is, the Dirac operator on a smooth and bounded domain of ${ℝ}^3$ with certain boundary conditions. When the mass m goes to $\pm \infty$, we provide spectral asymptotic results.

Dimension reduction for dipolar Bose-Einstein condensates in the strong interaction regime

In [4], we study dimension reduction for the three-dimensional Gross-Pitaevskii equation with a long-range and anisotropic dipole-dipole interaction modeling dipolar Bose-Einstein condensation in a strong interaction regime. The cases of disk shaped condensates (confinement from dimension three to dimension two) and cigar shaped condensates (confinement to dimension one) are analyzed. In both cases, the analysis combines averaging tools and semiclassical techniques. Asymptotic models are derived, with rates of convergence in terms of two small dimensionless parameters characterizing the strength of the confinement and the strength of the interaction between atoms.

Nonlinear stability criteria for the HMF Model

In [17], we study the nonlinear stability of a large class of inhomogeneous steady state solutions to the Hamiltonian Mean Field (HMF) model. Under a specific criterion, we prove the nonlinear stability of steady states which are decreasing functions of the microscopic energy. To achieve this task, we extend to this context the strategy based on generalized rearrangement techniques which was developed recently for the gravitational Vlasov-Poisson equation. Explicit stability inequalities are established and our analysis is able to treat non compactly supported steady states to HMF, which are physically relevant in this context but induces additional difficulties, compared to the Vlasov-Poisson system.

Strong confinement limit for the nonlinear Schrödinger equation constrained on a curve

This paper [20] is devoted to the cubic nonlinear Schrödinger equation in a two dimensional waveguide with shrinking cross section. For a Cauchy data living essentially on the first mode of the transverse Laplacian, we provide a tensorial approximation of the solution in this limit, with an estimate of the approximation error, and derive a limiting nonlinear Schrödinger equation in dimension one.

Stable ground states for the HMF Poisson Model

In [36], we prove the nonlinear orbital stability of a large class of steady states solutions to the Hamiltonian Mean Field (HMF) system with a Poisson interaction potential. These steady states are obtained as minimizers of an energy functional under one, two or infinitely many constraints. The singularity of the Poisson potential prevents from a direct run of the general strategy which was based on generalized rearrangement techniques, and which has been recently extended to the case of the usual (smooth) cosine potential. Our strategy is rather based on variational techniques. However, due to the boundedness of the space domain, our variational problems do not enjoy the usual scaling invariances which are, in general, very important in the analysis of variational problems. To replace these scaling arguments, we introduce new transformations which, although specific to our context, remain somehow in the same spirit of rearrangements tools introduced in the references above. In particular, these transformations allow for the incorporation of an arbitrary number of constraints, and yield a stability result for a large class of steady states.

The quantum Liouville-BGK equation and the moment problem

This work [19] is devoted to the analysis of the quantum Liouville-BGK equation. This equation arises in the work of Degond and Ringhofer on the derivation of quantum hydrodynamical models from first principles. Their theory consists in transposing to the quantum setting the closure strategy by entropy minimization used for kinetic equations. The starting point is the quantum Liouville-BGK equation, where the collision term is defined via a so-called quantum local equilibrium, defined as a minimizer of the quantum free energy under a local density constraint. We then address three related problems: we prove new results about the regularity of these quantum equilibria; we prove that the quantum Liouville-BGK equation admits a classical solution; and we investigate the long-time behavior of the solutions. The core of the proofs is based on a fine analysis of the properties of the minimizers of the free energy.

Averaging of nonlinear Schrödinger equations with strong magnetic confinement

In [16], we consider the dynamics of nonlinear Schrödinger equations with strong constant magnetic fields. In an asymptotic scaling limit the system exhibits a purely magnetic confinement, based on the spectral properties of the Landau Hamiltonian. Using an averaging technique we derive an associated effective description via an averaged model of nonlinear Schrödinger type. In a special case this also yields a derivation of the LLL equation.