arXiv subjects

Ludovic Godard-Cadillac

Publications and source records attributed to Ludovic Godard-Cadillac.

11 recordsLinked to original sources

arXiv2026

Local well-posedness for a moving rigid region in Surface Quasi-Geostrophic equations

We introduce and analyze a class of Surface Quasi-Geostrophic (SQG) equations in the presence of moving rigid obstacles. The model is motivated both by vortex-wave type asymptotics for singular structures in active scalar equations and by geophysical phenomena exhibiting rigid-like coherent regions, such as cyclone eyes or long-lived atmospheric dust clouds. We consider the critical SQG equation in a time-dependent exterior domain generated by a prescribed rigid motion and reconstruct the velocity through a nonlocal elliptic formulation adapted to impermeability constraints. The active scalar is assumed to remain constant inside the rigid region and in a neighborhood of its boundary, yielding a plateau structure compatible with the transport dynamics. For a single moving obstacle, we establish local well-posedness of classical solutions in Sobolev spaces $H^k$, $k\geq 4$ together with uniqueness, local stability, and a blow-up criterion. The analysis relies on a reformulation in adapted coordinates reducing the problem to a fixed domain, combined with integral representations for the fractional elliptic operator, regularization procedures, and a nonlinear fixed-point argument. A central difficulty comes from the critical singularity of the SQG Biot-Savart kernel in the case $s=\frac{1}{2}$, for which the velocity reconstruction near the moving boundary requires commutator estimates. We further prove propagation of the plateau property and derive a priori estimates controlling both the support of the scalar gradient and the Sobolev norm of the solution. This work provides, to our knowledge, the first well-posedness theory for SQG equations with moving rigid obstacles and constitutes a first step toward the rigorous derivation of point-vortex type dynamics from shrinking rigid bodies in SQG flows.

math.AP↗

arXiv2025

The Spear and the Ring: Emergent Structures in Magnetic Colloidal Suspensions

We study from a mathematical point of view the nanoparticle model of a magnetic colloid, presented by G. Klughertz. Our objective is to obtain properties of stable stationary structures that arise in the long-time limit for the magnetic nanoparticles dynamics following this model. In this article, we present a detailed study of two specific structures using techniques from the calculus of variations. The first, called the spear, consists of a chain of aligned particles interacting via a Lennard-Jones potential. We establish existence and uniqueness results, derive bounds on the distances between neighboring particles, and provide a sharp asymptotic description as the number of particles tends to infinity. The second structure, the ring, features particles uniformly distributed along a circle. We prove its existence and uniqueness and derive an explicit formula for its radius.

math-ph↗

arXiv2025

Existence and Uniqueness for the SQG Vortex-Wave System when the Vorticity is Constant near the Point-Vortex

This article studies the vortex-wave system for the Surface Quasi-Geostrophic equation with parameter 0 < s < 1. We obtained local existence of classical solutions in H^4 under the standard ''plateau hypothesis'', H^2-stability of the solutions, and a blow-up criterion. In the sub-critical case s > 1/2 we established global existence of weak solutions. For the critical case s = 1/2, we introduced a weaker notion of solution (V-weak solutions) to give a meaning to the equation and prove global existence.

math.AP↗

arXiv2024

Existence and Uniqueness of Domain Walls for Notched Ferromagnetic Nanowires

In this article, we investigate a simple model of notched ferromagnetic nanowires using tools from calculus of variations and critical point theory. Specifically, we focus on the case of a single unimodal notch and establish the existence and uniqueness of the critical point of the energy. This is achieved through a lifting argument, which reduces the problem to a generalized Sturm-Liouville equation. Uniqueness is demonstrated via a Mountain-Pass argument, where the assumption of two distinct critical points leads to a contradiction. Additionally, we show that the solution corresponds to a system of magnetic spins characterized by a single domain wall localized in the vicinity of the notch. We further analyze the asymptotic decay of the solution at infinity and explore the symmetric case using rearrangement techniques.

math.AP↗

arXiv2024

On the Dynamics of Point Vortices with Positive Intensities collapsing with the boundary

In this paper, we study the point-vortex dynamics with positive intensities. We show that in the half-plane and in a disk, collapses of point vortices with the boundary in finite time are impossible, hence the solution of the dynamics is global in time. We also give some necessary conditions for the existence of collapses with the boundary in general smooth bounded domains, in particular, that the trajectory of at least one point vortex has no limit. Some minor results are obtained with unsigned intensities.

math.AP↗

arXiv2023

Hölder regularity for collapses of point vortices

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. In these models the kernel of the Biot-Savart law is a power function of exponent $-α$. It is proved that, under a standard non-degeneracy hypothesis, the trajectories of the point-vortices have a Hölder regularity up to, and including, 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 point-vortex has an accumulation point in the interior of the domain as $t\to T$, then it converges towards this point and displays the same Hölder continuity property. A partial result for point-vortices that collapse with the boundary is also established : we prove that their distance to the boundary is Hölder regular.

math.AP↗

arXiv2022

Existence of solutions for a bi-species kinetic model of a cylindrical Langmuir probe

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 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.

math.AP↗

arXiv2022

Vortex collapses for the Euler and Quasi-Geostrophic Models

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.

math.DS↗

arXiv2020

Smooth traveling-wave solutions to the inviscid surface quasi-geostrophic equations

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.

math.AP↗

arXiv2020

Co-rotating vortices with N fold symmetry for the inviscid surface quasi-geostrophic equation

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.

math.AP↗

arXiv2019

Tamped functions: A rearrangement in dimension 1

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{ó}lya-Szeg{ö} inequality. Contrary to the Schwarz rearrangement, the tamping also preserves the homogeneous Dirichlet boundary condition of a function.

math.AP↗