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International audience; To study the temperature in a gas subjected to electromagnetic radiations, one may use the Radiative Transfer equations coupled with the Navier-Stokes equations. The problem has 7 dimensions; however with minimal simplifications it is equivalent to a small number of integrodifferential equations in 3 dimensions. We present the method and a numerical implementation using an H-matrix compression scheme. The result is a very fast: 50K physical points, all directions of radiation and 680 frequencies require less than 5 minutes on an Apple M1 Laptop. The method is capable of handling variable absorptioN and scattering functionS of spatial positions and frequencies. The implementation is done using htool 1 , a matrix compression library interfaced with the PDE solverfreefem++. Applications to the temperature in the French Chamonix valley is presented at different hours of the day with and without snow / clouds and with a variable absorption taken from the Gemini measurements. The result is precise enough to assert temperature differences due to increased absorption in the vibrational frequency subrange of greenhouse gasses.; Pour étudier la température dans un gaz soumis à des rayonnements électromagnétiques, on peut utiliser les équations du transfert radiatif couplées aux équations de Navier-Stokes. Le problème a 7 dimensions ; cependant, avec des simplifications minimales,on peut le réduire à un petit nombre d'équations intégro-différentielles en 3 dimensions. Nous présentons la méthode et une implémentation numérique utilisant un schéma de compression de matrice H. Le résultat est très rapide : 50K points physiques, toutes les directions de rayonnement et 680 fréquences nécessitent moins de 5 minutes sur un ordinateur portable Apple M1. Le procédé est capable de gérer des coefficients d'absorption et de diffusion dépendants des positions spatiales et des fréquences. L'implémentation se fait à l'aide de la bibliothèque htool , pour la compression de matrice à diagonales dominante, interfacée avec le solveur d'EDP freefem++. Les applications à la température dans la vallée française de Chamonix sont présentées à différentes heures de la journée avec et sans neige/nuage ​​et avec une absorption variable tirée des mesures Gemini. Le résultat est suffisamment précis pour obtenir les différences de température dues à une absorption accrue dans une sous-gamme de fréquence vibratoire des gaz à effet de serre.

As a sequel to our previous analysis in [9] arXiv:2202.09411 on the Gross-Pitaevskii equation on the product space $\mathbb{R} \times \mathbb{T}$, we construct a branch of finite energy travelling waves as minimizers of the Ginzburg-Landau energy at fixed momentum. We deduce that minimizers are precisely the planar dark solitons when the length of the transverse direction is less than a critical value, and that they are genuinely two-dimensional solutions otherwise. The proof of the existence of minimizers is based on the compactness of minimizing sequences, relying on a new symmetrization argument that is well-suited to the periodic setting.

We study the Gross-Pitaevskii equation in dimension two with periodic conditions in one direction, or equivalently on the product space $\mathbb{R} \times \mathbb{T}_L$ where $L > 0$ and $\mathbb{T}_L = \mathbb{R} / L \mathbb{Z}.$ We focus on the variational problem consisting in minimizing the Ginzburg-Landau energy under a fixed momentum constraint. We prove that there exists a threshold value for $L$ below which minimizers are the one-dimensional dark solitons, and above which no minimizer can be one-dimensional.

We introduce a functional framework taylored to investigate the minimality and stability properties of the Ginzburg-Landau vortex of degree one on the whole plane. We prove that a renormalized Ginzburg-Landau energy is well-defined in that framework and that the vortex is its unique global minimizer up to the invariances by translation and phase shift. Our main result is a nonlinear coercivity estimate for the renormalized energy around the vortex, from which we can deduce its orbital stability as a solution to the Gross-Pitaevskii equation, the natural Hamiltonian evolution equation associated to the Ginzburg-Landau energy.

We provide a variational construction of special solutions to the generalized surface quasi-geostrophic equations. These solutions take the form of N vortex patches with N-fold symmetry , which are steady in a uniformly rotating frame. Moreover, we investigate their asymptotic properties when the size of the corresponding patches vanishes. In this limit, we prove these solutions to be a desingularization of N Dirac masses with the same intensity, located on the N vertices of a regular polygon rotating at a constant angular velocity.

We will present a rigorous derivation of the leapfrogging mechanism for the 3D axisymmetric Gross-Pitaveskii equation (joint with R.L. Jerrard), and point out the difficulties that arise in the related context of the incompressible Euler equation.

We derive a precise motion law for fronts of solutions to scalar one-dimensional reaction-diffusion equations with equal depth multiple-wells, in the case the second derivative of the potential vanishes at its minimizers. We show that, renormalizing time in an algebraic way, the motion of fronts is governed by a simple system of ordinary differential equations of nearest neighbor interaction type. These interactions may be either attractive or repulsive. Our results are not constrained by the possible occurrence of collisions nor splittings. They present substantial differences with the results obtained in the case the second derivative does not vanish at the wells, a case which has been extensively studied in the literature, and where fronts have been showed to move at exponentially small speed, with motion laws which are not renormalizable., 68 pages

We prove an estimate for the difference of two solutions of the Schr��dinger map equation for maps from $T^1$ to $S^2.$ This estimate yields some continuity properties of the flow map for the topology of $L^2(T^1,S^2)$, provided one takes its quotient by the continuous group action of $T^1$ given by translations. We also prove that without taking this quotient, for any $t>0$ the flow map at time $t$ is discontinuous as a map from $\mathcal{C}^\infty(T^1,S^2)$, equipped with the weak topology of $H^{1/2},$ to the space of distributions $(\mathcal{C}^\infty(T^1,\R^3))^*.$, 44 pages, no figures

Leapfrogging motion of vortex rings sharing the same axis of symmetry was first predicted by Helmholtz in his famous work on the Euler equation for incompressible fluids. Its justification in that framework remains an open question to date. In this paper, we rigorously derive the corresponding leapfrogging motion for the axially symmetric three-dimensional Gross-Pitaevskii equation., Comment: 39 pages, 2 figures