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Jean-Baptiste Casteras

Publications and source records attributed to Jean-Baptiste Casteras.

At least 19 recordsLinked to original sources
arXiv2025

Entropy and Fisher information in non-convex domains: one chain to rule them all

We prove that the (square root) Fisher information functional is a strong Wasserstein upper gradient of the entropy on non-convex Riemannian domains. This fills a gap in the literature by allowing one to completely dispense from $\lambda$-displacement convexity arguments. Along the way we establish a novel quantitative short-time control of the Fisher information along the Neumann heat flow, and establish an exact chain rule under stronger $AC_2$ assumptions typically satisfied by curves of measures obtained as limits of JKO schemes.

math.AP↗
arXiv2025

Construction of 2-bubbles for the energy critical bi-harmonic Schrödinger equation

We construct a blowing-up solution for the energy critical focusing biharmonic nonlinear Schrödinger equation in infinite time in dimension $N\geq 13$. Our solution is radially symmetric and converges asymptotically to the sum of two bubbles. The scale of one of the bubble is of order $1$ whereas the other one is of order $|t|^{-\frac{2}{N-12}}$. Moreover, the phase between the two bubbles form a right angle.

math.AP↗
arXiv2025

Sticky-reflecting diffusion as a Wasserstein gradient flow

In this paper we identify the Fokker-Planck equation for (reflected) Sticky Brownian Motion as a Wasserstein gradient flow in the space of probability measures. The driving functional is the relative entropy with respect to a non-standard reference measure, the sum of an absolutely continuous interior part plus a singular part supported on the boundary. Taking the small time-step limit in a minimizing movement (JKO scheme) we prove existence of weak solutions for the coupled system of PDEs satisfying in addition an Energy Dissipation Inequality.

math.AP↗
arXiv2025

Large deviations for sticky-reflecting Brownian motion with boundary diffusion

We study a Schilder-type large deviation principle for sticky-reflected Brownian motion with boundary diffusion, both at the static and sample path level in the short-time limit. A sharp transition for the rate function occurs, depending on whether the tangential boundary diffusion is faster or slower than in the interior of the domain. The resulting intrinsic distance naturally gives rise to a novel optimal transport model, where motion and kinetic energy are treated differently in the interior and along the boundary.

math.AP↗
arXiv2024

Higher order expansion for the probabilistic local well-posedness theory for a cubic nonlinear Schrödinger equation

In this paper, we study the probabilistic local well-posedness of the cubic Schrödinger equation (cubic NLS): \[ (i\partial_{t} + Δ) u = \pm |u|^{2} u \text{ on } [0,T) \times \mathbb{R}^{d}, \] with initial data being a Wiener randomization at unit scale of a given function $f$. We prove that a solution exists almost-surely locally in time provided \(f\in H^{S}_{x}(\mathbb{R}^{d})\) with \(S>\max\big(\frac{d-3}{4},\frac{d-4}{2}\big)\) for \(d\geq 3\). In particular, we establish that the local well-posedness holds for any \(S>0\) when \(d=3\). We also show that, under appropriate smallness conditions for the initial data, the solutions are global in time and scatter. The solutions are constructed as a sum of an explicit multilinear expansion of the flow in terms of the random initial data and of an additional smoother remainder term with deterministically subcritical regularity. This construction allows us to introduce a new and refined notion of graded scattering. We develop the framework of directional space-time norms to control the (probabilistic) multilinear expansion and the (deterministic) remainder term and to obtain improved bilinear probabilistic-deterministic Strichartz estimates.

math.AP↗
arXiv2023

On the existence and partial stability of standing waves for a nematic liquid crystal director field equation

In this paper, following the studies of Amorim and al. in Partial Differ.Equ. Appl. '23, we consider some new aspects of the motion of the director field of a nematic liquid crystal submitted to a magnetic field and to a laser beam. In particular, we study the existence and partial orbital stability of special standing waves, in the spirit of Cazenave and Lions and Hadj Selem and al. in Milan J. Math. '14 and we present some numerical simulations.

math.AP↗
arXiv2023

Trivariate distribution of sticky Brownian motion

In this short note we derive a closed form for the trivariate distribution (position, local time at the origin, and positive occupation time) of the one-dimensional sticky Brownian motion, thereby filling some gaps and fixing some mistakes in the literature.

math.PR↗
arXiv2023

Almost sure local well-posedness for cubic nonlinear Schrodinger equation with higher order operators

In this paper, we study the local well-posedness of the cubic Schrödinger equation: \[ (i \partial_t - \mathscr{L}) u = \pm |u|^2 u \quad \text{ on } I \times \mathbb{R}^d, \] with randomized initial data, and $\mathscr{L}$ being an operator of degree $σ\geq 2$. Using estimates in directional spaces, we improve and extend known results for the standard Schrödinger equation (i.e. $\mathscr{L} = Δ$) to any dimension and obtain results under natural assumptions for general $\mathscr{L}$.

math.AP↗
arXiv2022

Classification of radial blow-up at the first critical exponent for the Lin-Ni-Takagi problem in the ball

We investigate the behaviour of radial solutions to the Lin-Ni-Takagi problem in the ball $B_R \subset \mathbb{R}^N$ for $N \ge 3$: \begin{equation*} \left \{ \begin{aligned} - \triangle u_p + u_p & = |u_p|^{p-2}u_p & \textrm{ in } B_R, \\ \partial_νu_p & = 0 & \textrm{ on } \partial B_R, \end{aligned} \right. \end{equation*} when $p $ is close to the first critical Sobolev exponent $2^* = \frac{2N}{N-2}$. We obtain a complete classification of finite energy radial smooth blowing up solutions to this problem. We describe the conditions preventing blow-up as $p \to 2^*$, we give the necessary conditions in order for blow-up to occur and we establish their sharpness by constructing examples of blowing up sequences. Our approach allows for asymptotically supercritical values of $p$. We show in particular that, if $p \geq 2^\ast$, finite-energy radial solutions are precompact in $C^2(\bar{B_R})$ provided that $N\geq 7$. Sufficient conditions are also given in smaller dimensions if $p=2^\ast$. Finally we compare and interpret our results to the bifurcation analysis of Bonheure, Grumiau and Troestler in Nonlinear Anal. 147 (2016).

math.AP↗
arXiv2022

Hidden dissipation and convexity for Kimura equations

In this paper we establish a rigorous gradient flow structure for one-dimensional Kimura equations with respect to some Wasserstein-Shahshahani optimal transport geometry. This is achieved by first conditioning the underlying stochastic process to non-fixation in order to get rid of singularities on the boundaries, and then studying the conditioned $Q$-process from a more traditional and variational point of view. In doing so we complete the work initiated in [Chalub et Al., Gradient flow formulations of discrete and continuous evolutionary models: a unifying perspective. Acta App Math., 171(1), 1-50], where the gradient flow was identified only formally. The approach is based on the Energy Dissipation Inequality and Evolution Variational Inequality notions of metric gradient flows. Building up on some convexity of the driving entropy functional, we obtain new contraction estimates and quantitative long-time convergence towards the stationary distribution.

math.AP↗
arXiv2022

Fourth order Schrödinger equation with mixed dispersion on certain Cartan-Hadamard manifolds

We study the fourth order Schrödinger equation with mixed dispersion on an $N$-dimensional Cartan-Hadamard manifold. At first, we focus on the case of the hyperbolic space. Using the fact that there exists a Fourier transform on this space, we prove the existence of a global solution to our equation as well as scattering for small initial data. Next, we obtain weighted Strichartz estimates for radial solutions on a large class of rotationally symmetric manifolds by adapting the method of Banica and Duyckaerts (Dyn. Partial Differ. Equ., 07). Finally, we give a blow-up result for a rotationally symmetric manifold relying on a localized virial argument.

math.AP↗
arXiv2022

Non-parametric mean curvature flow with prescribed contact angle in Riemannian products

Assuming that there exists a translating soliton $u_\infty$ with speed $C$ in a domain $Ω$ and with prescribed contact angle on $\partialΩ$, we prove that a graphical solution to the mean curvature flow with the same prescribed contact angle converges to $u_\infty +Ct$ as $t\to\infty$. We also generalize the recent existence result of Gao, Ma, Wang and Weng to non-Euclidean settings under suitable bounds on convexity of $Ω$ and Ricci curvature in $Ω$.

math.DG↗
arXiv2021

Invariant measures and global well-posedness for a fractional Schrödinger equation with Moser-Trudinger type nonlinearity

In this paper, we construct invariant measures and global-in-time solutions for a fractional Schr\" odinger equation with a Moser-Trudinger type nonlinearity $$ i\partial_t u= (-Δ)^αu+ 2βu e^{β|u|^2},\qquad\mbox{for}\qquad(x,t)\in \ M\times \mathbb{R} $$ on a compact Riemannian manifold $M$ without boundary of dimension $d\geq 2$. To do so, we use the so-called Inviscid-Infinite-dimensional limits introduced by Sy ('19) and Sy and Yu ('21). More precisely, we show that if $s>d/2$ or if $s\leq d/2$ and $s\leq 1+α$, there exists an invariant measure $μ^{s}$ and a set $Σ^s \subset H^s$ containing arbitrarily large data such that $μ^{s}(Σ^s ) =1$ and that the fractional NLS is globally well-posed on $Σ^{s}$. For strong regularities $s>d/2$ we also obtain a logarithmic upper bound on the growth of the $H^r$-norm of our solutions for $r<s$. This gives new examples of invariant measures supported in highly regular spaces in comparison with the Gibbs measure constructed by Robert ('21) for the same equation.

math.AP↗
arXiv2021

Unbounded mass radial solutions for the Keller-Segel equation in the disk

We consider the boundary value problem $$ \left\{ \begin{array}{rcll} -Δu+ u -λe^u&=&0,\ u>0 & \mathrm{in}\ B_1(0)\\ \partial_νu&=&0&\mathrm{on}\ \partial B_1(0), \end{array}\right. $$ whose solutions correspond to steady states of the Keller--Segel system for chemotaxis. Here $B_1(0)$ is the unit disk, $ν$ the outer normal to $\partial B_1(0)$, and $λ>0$ is a parameter. We show that, provided $λ$ is sufficiently small, there exists a family of radial solutions $u_λ$ to this system which blow up at the origin and concentrate on $\partial B_1(0)$, as $λ\to 0$. These solutions satisfy $$ \lim_{λ\to 0} \frac{u_λ(0)}{|\lnλ|}=0\quad \mbox{and}\quad 0<\lim_{λ\to 0} \frac{1}{|\lnλ|}\int_{B_1(0)}λe^{u_λ(x)}dx<\infty, $$ having in particular unbounded mass, as $λ\to 0$.

math.AP↗
arXiv2021

Translating solitons over Cartan-Hadamard manifolds

We prove existence results for entire graphical translators of the mean curvature flow (the so-called bowl solitons) on Cartan-Hadamard manifolds. We show that the asymptotic behaviour of entire solitons depends heavily on the curvature of the manifold, and that there exist also bounded solutions if the curvature goes to minus infinity fast enough. Moreover, it is even possible to solve the asymptotic Dirichlet problem under certain conditions.

math.DG↗
arXiv2021

Existence of traveling waves for a fourth order Schr\" odinger equation with mixed dispersion in the Helmholtz regime

In this paper, we study the existence of traveling waves for a fourth order Schr\" odinger equations with mixed dispersion, that is, solutions to $$Δ^2 u +βΔu +i V \nabla u +αu =|u|^{p-2} u,\ in\ \R^N ,\ N\geq 2.$$ We consider this equation in the Helmholtz regime, when the Fourier symbol $P$ of our operator is strictly negative at some point. Under suitable assumptions, we prove the existence of solution using the dual method of Evequoz and Weth provided that $p\in (p_1 , 2N/(N-4)_+)$. The real number $p_1$ depends on the number of principal curvature of $M$ staying bounded away from $0$, where $M$ is the hypersurface defined by the roots of $P$. We also obtained estimates on the Green function of our operator and a $L^p - L^q$ resolvent estimate which can be of independent interest and can be applied to other operators.

math.AP↗
arXiv2020

Bifurcation analysis of the Hardy-Sobolev equation

In this paper, we prove existence of multiple non-radial solutions to the Hardy-Sobolev equation $$\begin{cases} -Δu-\displaystyle\frac γ{|x|^2}u=\displaystyle\frac{1}{|x|^s}|u|^{p_s-2}u & \text{ in } \mathbb{R}^N\setminus\{0\},\\ u\geq 0, & \end{cases}$$ where $N\geq 3$, $s\in[0,2)$, $p_s=\frac{2(N-s)}{N-2}$ and $γ\in (-\infty,\frac{(N-2)^2} 4)$. We extend results of E.N. Dancer, F. Gladiali, M. Grossi, Proc. Roy. Soc. Edinburgh Sect. A 147 (2017) where only the case $s=0$ is considered. Moreover, thanks to monotonicity properties of the solutions, we separate two branches of non-radial solutions.

math.AP↗