Ludovic Godard-Cadillac | Université de Nantes (original) (raw)

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Papers by Ludovic Godard-Cadillac

In this article we study quasi-geostrophic point-vortex systems in a very general setting called ... more In this article we study quasi-geostrophic point-vortex systems in a very general setting called alpha point-vortex. We study a particular case of vortex collapses called "mono-scale collapses" and this study gives the hölder regularity for the 3-vortex problem under standard non-degeneracy assumption. In a last part, we improve a previous result concerning the improbability of vortex collapses for the quasi-geostrophic model.Dans cet article nous étudions les systèmes de points-vortex quasi-géostrophiques dans un cadre très général appelé les points-vortex alpha. Nous étudions un cas particulier de collisions de vortex appelées "collisions mono-échelles" et cette étude nous donne la régularité Höldérienne du problème à 3 vortex sous des hypothèses standards de non-dégénérescence. Dans une dernière partie, nous améliorons un précédent résultat sur l'improbabilité des collisions de vortex pour le modèle quasi-géostrophique

In this article, we study a collisionless kinetic model for plasmas in the neighborhood of a cyli... more In this article, we study a collisionless kinetic model for plasmas in the neighborhood of a cylindrical metallic Langmuir probe. This model consists in a bi-species Vlasov-Poisson equation in a domain contained between two cylinders with prescribed boundary conditions. The interior cylinder models the probe while the exterior cylinder models the interaction with the plasma core. We prove the existence of a weak-strong solution for this model in the sense that we get a weak solution for the 2 Vlasov equations and a strong solution for the Poisson equation. The first parts of the article are devoted to explain the model and proceed to a detailed study of the Vlasov equations. This study then leads to a reformulation of the Poisson equation as a 1D non-linear and non-local equation and we prove it admits a strong solution using an iterative fixed-point procedure.

The first part of this article studies the collapses of point-vortices for the Euler equation in ... more The first part of this article studies the collapses of point-vortices for the Euler equation in the plane and for surface quasi-geostrophic equations in the general setting of α models. This consists in a Biot-Savart law with a kernel being a power function of exponent −α. It is proved that, under a standard non-degeneracy hypothesis, the trajectories of the vorticies have a regularity Hölder at T the time of collapse. The Hölder exponent obtained is 1/(α+ 1) and this exponent is proved to be optimal for all α by exhibiting an example of a 3-vortex collapse. The same question is then addressed for the Euler point-vortex system in smooth bounded connected domains. It is proved that if a given vortex has an adherence point in the interior of the domain as t → T , then it converges towards this point and show the same Hölder continuity property.

We provide a variational construction of special solutions to the generalized surface quasi-geost... more 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 define a new rearrangement, called rearrangement by tamping, for non-negative measurable funct... more We define a new rearrangement, called rearrangement by tamping, for non-negative measurable functions defined on R+. This rearrangement has many properties in common with the well-known Schwarz non-increasing rearrangement such as the Pólya-Szegö inequality. Contrary to the Schwarz rearrangement, the tamping also preserves the homogeneous Dirichlet boundary condition of a function.

This article studies point-vortex models for the Euler and surface quasi-geostrophic equations. I... more This article studies point-vortex models for the Euler and surface quasi-geostrophic equations. In the case of an inviscid fluid with planar motion, the point-vortex model gives account of dynamics where the vorticity profile is sharply concentrated around some points and approximated by Dirac masses. This article contains three main results with several links between each other. In the first part, we provide two uniform bounds on the trajectories for Euler and quasi-geostrophic vortices related to the non-neutral cluster hypothesis. In a second part we focus on the Euler point-vortex model and under the non-neutral cluster hypothesis we prove a convergence result. The third part is devoted to the generalization of a classical result by Marchioro and Pulvirenti concerning the improbability of collapses and the extension of this result to the quasi-geostrophic case.

arXiv: Analysis of PDEs, 2020

We provide a variational construction of special solutions to the generalized surface quasi-geost... more 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.

Comptes Rendus. Mathématique

In a recent article by Gravejat and Smets [7], it is built smooth solutions to the inviscid surfa... more In a recent article by Gravejat and Smets [7], it is built smooth solutions to the inviscid surface quasigeostrophic equation that have the form of a traveling wave. In this article we work back on their construction to provide similar solutions to a more general class of quasi-geostrophic equation where the half-laplacian is replaced by any fractional laplacian. Funding. The author acknowledges grants from the Agence nationale de la Recherche, for project "Ondes Dispersives Aléatoires" (ANR-18-CE40-0020-01).

We define a new rearrangement, called rearrangement by tamping, for non-negative measurable funct... more We define a new rearrangement, called rearrangement by tamping, for non-negative measurable functions defined on R+. This rearrangement has many properties in common with the well-known Schwarz non-increasing rearrangement such as the P{\'o}lya-Szeg{\"o} inequality. Contrary to the Schwarz rearrangement, the tamping also preserves the homogeneous Dirichlet boundary condition of a function.

arXiv: Analysis of PDEs, 2020

In a recent article by Gravejat and Smets, it is built smooth solutions to the inviscid surface q... more In a recent article by Gravejat and Smets, it is built smooth solutions to the inviscid surface quasi-geostrophic equation that have the form of a traveling wave. In this article we work back on their construction to provide solution to a more general class of quasi-geostrophic equation where the half-laplacian is replaced by any fractional laplacian.

ESAIM: Mathematical Modelling and Numerical Analysis

We construct a stationary gyrokinetic variational model for sheaths close to the metallic wall of... more We construct a stationary gyrokinetic variational model for sheaths close to the metallic wall of a magnetized plasma, following a physical extremalization principle for the natural energy. By considering a reduced set of parameters we show that our model has a unique minimal solution, and that the resulting electric potential has an infinite number of oscillations as it propagates towards the core of the plasma. We prove this result for the non linear problem and also provide a simpler analysis for a linearized problem, based on the construction of exact solutions. Some numerical illustrations show the well-posedness of the model after numerical discretization. They also exhibit the oscillating behavior.

In this article we study quasi-geostrophic point-vortex systems in a very general setting called ... more In this article we study quasi-geostrophic point-vortex systems in a very general setting called alpha point-vortex. We study a particular case of vortex collapses called "mono-scale collapses" and this study gives the hölder regularity for the 3-vortex problem under standard non-degeneracy assumption. In a last part, we improve a previous result concerning the improbability of vortex collapses for the quasi-geostrophic model.Dans cet article nous étudions les systèmes de points-vortex quasi-géostrophiques dans un cadre très général appelé les points-vortex alpha. Nous étudions un cas particulier de collisions de vortex appelées "collisions mono-échelles" et cette étude nous donne la régularité Höldérienne du problème à 3 vortex sous des hypothèses standards de non-dégénérescence. Dans une dernière partie, nous améliorons un précédent résultat sur l'improbabilité des collisions de vortex pour le modèle quasi-géostrophique

In this article, we study a collisionless kinetic model for plasmas in the neighborhood of a cyli... more In this article, we study a collisionless kinetic model for plasmas in the neighborhood of a cylindrical metallic Langmuir probe. This model consists in a bi-species Vlasov-Poisson equation in a domain contained between two cylinders with prescribed boundary conditions. The interior cylinder models the probe while the exterior cylinder models the interaction with the plasma core. We prove the existence of a weak-strong solution for this model in the sense that we get a weak solution for the 2 Vlasov equations and a strong solution for the Poisson equation. The first parts of the article are devoted to explain the model and proceed to a detailed study of the Vlasov equations. This study then leads to a reformulation of the Poisson equation as a 1D non-linear and non-local equation and we prove it admits a strong solution using an iterative fixed-point procedure.

The first part of this article studies the collapses of point-vortices for the Euler equation in ... more The first part of this article studies the collapses of point-vortices for the Euler equation in the plane and for surface quasi-geostrophic equations in the general setting of α models. This consists in a Biot-Savart law with a kernel being a power function of exponent −α. It is proved that, under a standard non-degeneracy hypothesis, the trajectories of the vorticies have a regularity Hölder at T the time of collapse. The Hölder exponent obtained is 1/(α+ 1) and this exponent is proved to be optimal for all α by exhibiting an example of a 3-vortex collapse. The same question is then addressed for the Euler point-vortex system in smooth bounded connected domains. It is proved that if a given vortex has an adherence point in the interior of the domain as t → T , then it converges towards this point and show the same Hölder continuity property.

We provide a variational construction of special solutions to the generalized surface quasi-geost... more 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 define a new rearrangement, called rearrangement by tamping, for non-negative measurable funct... more We define a new rearrangement, called rearrangement by tamping, for non-negative measurable functions defined on R+. This rearrangement has many properties in common with the well-known Schwarz non-increasing rearrangement such as the Pólya-Szegö inequality. Contrary to the Schwarz rearrangement, the tamping also preserves the homogeneous Dirichlet boundary condition of a function.

This article studies point-vortex models for the Euler and surface quasi-geostrophic equations. I... more This article studies point-vortex models for the Euler and surface quasi-geostrophic equations. In the case of an inviscid fluid with planar motion, the point-vortex model gives account of dynamics where the vorticity profile is sharply concentrated around some points and approximated by Dirac masses. This article contains three main results with several links between each other. In the first part, we provide two uniform bounds on the trajectories for Euler and quasi-geostrophic vortices related to the non-neutral cluster hypothesis. In a second part we focus on the Euler point-vortex model and under the non-neutral cluster hypothesis we prove a convergence result. The third part is devoted to the generalization of a classical result by Marchioro and Pulvirenti concerning the improbability of collapses and the extension of this result to the quasi-geostrophic case.

arXiv: Analysis of PDEs, 2020

We provide a variational construction of special solutions to the generalized surface quasi-geost... more 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.

Comptes Rendus. Mathématique

In a recent article by Gravejat and Smets [7], it is built smooth solutions to the inviscid surfa... more In a recent article by Gravejat and Smets [7], it is built smooth solutions to the inviscid surface quasigeostrophic equation that have the form of a traveling wave. In this article we work back on their construction to provide similar solutions to a more general class of quasi-geostrophic equation where the half-laplacian is replaced by any fractional laplacian. Funding. The author acknowledges grants from the Agence nationale de la Recherche, for project "Ondes Dispersives Aléatoires" (ANR-18-CE40-0020-01).

We define a new rearrangement, called rearrangement by tamping, for non-negative measurable funct... more We define a new rearrangement, called rearrangement by tamping, for non-negative measurable functions defined on R+. This rearrangement has many properties in common with the well-known Schwarz non-increasing rearrangement such as the P{\'o}lya-Szeg{\"o} inequality. Contrary to the Schwarz rearrangement, the tamping also preserves the homogeneous Dirichlet boundary condition of a function.

arXiv: Analysis of PDEs, 2020

In a recent article by Gravejat and Smets, it is built smooth solutions to the inviscid surface q... more In a recent article by Gravejat and Smets, it is built smooth solutions to the inviscid surface quasi-geostrophic equation that have the form of a traveling wave. In this article we work back on their construction to provide solution to a more general class of quasi-geostrophic equation where the half-laplacian is replaced by any fractional laplacian.

ESAIM: Mathematical Modelling and Numerical Analysis

We construct a stationary gyrokinetic variational model for sheaths close to the metallic wall of... more We construct a stationary gyrokinetic variational model for sheaths close to the metallic wall of a magnetized plasma, following a physical extremalization principle for the natural energy. By considering a reduced set of parameters we show that our model has a unique minimal solution, and that the resulting electric potential has an infinite number of oscillations as it propagates towards the core of the plasma. We prove this result for the non linear problem and also provide a simpler analysis for a linearized problem, based on the construction of exact solutions. Some numerical illustrations show the well-posedness of the model after numerical discretization. They also exhibit the oscillating behavior.