arXiv subjects

Camille Laurent

Publications and source records attributed to Camille Laurent.

At least 19 recordsLinked to original sources

arXiv2026

Periodic solutions for weakly damped systems

In this article, we investigate the existence and properties of time-periodic solutions for damped evolutionary partial differential equations subject to periodic forcing. Particular emphasis is placed on configurations where the energy decay of the corresponding free equation is non-uniform. We link the speed of decay and the existence of periodic solutions for the forced equation. Furthermore, we characterize the relationship between the resolvent growth and the associated loss of regularity. The theoretical framework is illustrated through several examples, including linear and nonlinear damped wave equations and coupled hyperbolic-parabolic systems. Finally, we provide a counterexample demonstrating the occurrence of resonance, in which regular but unbounded solutions emerge despite damping.

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arXiv2026

On scattering and profile decomposition for critical nonlinear waves outside weakly trapping obstacles

We prove scattering for the defocusing energy-critical non-linear wave equation with Dirichlet boundary conditions outside two strictly convex obstacles in dimension three. This is the first large data scattering result for such an equation in the presence of trapped trajectories. Our result is in fact more general and can be used as a black box in other geometries. More precisely, under the assumptions that the corresponding linear wave equation satisfies global Strichartz estimates, that the domain is weakly non-trapping and that trajectories do not reconcentrate, we show linear and nonlinear profile decompositions in infinite time. This implies scattering under the rigidity assumption that the only compact-flow solution is the trivial one.

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arXiv2026

Global propagation of analyticity and unique continuation for semilinear conservative PDEs

We review some recent results in which we develop a new method for proving global unique continuation for some conservative PDEs. The main tool is to prove some global propagation of analyticity. We first present some known results on the subject. Then, we sketch the abstract method we use, which relies on the property of finite determining modes. We give applications to semilinear wave, plates and Schr\''odinger equations. This note was written for the \emph{Proceedings of the Journ{é}es EDP 2025}.

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arXiv2025

On the set of non radiative solutions for the energy critical wave equation

Non radiative solutions of the energy critical non linear wave equation are global solutions $u$ that furthermore have vanishing asymptotic energy outside the lightcone at both $t \to \pm \infty$:\[ \lim_{t \to \pm \infty} \| \nabla_{t,x} u(t) \|_{L^2(|x| \ge |t|+R)} = 0, \]for some $R \> 0$. They were shown to play an important role in the analysis of long time dynamics of solutions, in particular regarding the soliton resolution: we refer to the seminal works of Duyckaerts, Kenig and Merle, see \cite{DKM:23} and the references therein.We show that the set of non radiative solutions which are small in the energy space is a manifold whose tangent space at $0$ is given by non radiative solutions to the linear equation (described in \cite{CL24}). We also construct nonlinear solutions with an arbitrary prescribed radiation field.

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arXiv2025

Exact controllability of anisotropic 1D partial differential equations in spaces of analytic functions

In this article, we prove a local controllability result for a general class of 1D partial differential equations on the interval $(0,1)$. The PDEs we consider take the form $\partial_t^N y=ζ_M \partial_{x}^{M}y+f(x , y , \partial_{x} y,\ldots, \partial_x^{M-1} y)$ where $1\le N < M$, $ζ_M\in \mathbb{C} ^*$, and $f$ is some linear or nonlinear term of lower order. In this context, we prove a local controllability result between states that are analytic functions. If some boundary conditions are prescribed, a similar local controllability result holds between analytic functions satisfying some compatibility conditions that are natural for the existence of smooth solutions of the considered PDE. The proof is performed by studying a nonlinear Cauchy problem in the spatial variable with data in some spaces of Gevrey functions and by investigating the relationship between the jet of space derivatives and the jet of time derivatives. We give various examples of applications, including the (good and bad) Boussinesq equation, the Ginzburg-Landau equation, the Kuramoto-Sivashinsky equation and the Korteweg-de Vries equation.

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arXiv2024

Indirect stabilization of semilinear coupled wave systems

In this paper, we study the indirect stabilization problem for a system of two coupled semilinear wave equations with internal damping in a bounded domain in $\mathbb{R}^3$. The nonlinearity is assumed to be subcritical, defocusing and analytic. Under geometric control condition on both coupling and damping regions, we establish the exponential energy decay rate.

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arXiv2024

Global propagation of analyticity and unique continuation for semilinear waves

In this article, we develop a new method to prove both global propagation of analyticity and unique continuation in finite time for solutions of semilinear wave-type equations with analytic nonlinearity. It combines control theory techniques and Galerkin approximation, inspired by Hale-Raugel, to prove that analyticity in time can be propagated for the nonlinear equation from a zone where linear observability holds towards the full space.For semilinear wave equations with Dirichlet boundary condition on a bounded domain, this implies that analyticity can be propagated to the entire domain from a subset $ω$ that satisfies the geometric control condition. It also implies the unique continuation when the solution is assumed to be zero on $ω$. When the nonlinearity is assumed to be subcritical and defocusing, we also obtain observability estimates in the optimal time of the geometric control condition.For semilinear plate equations, similar propagation of analyticity is achieved by assuming the controllability of the linear Schr\''odinger equation.

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arXiv2024

Unique continuation for Schr{ö}dinger operators with partially Gevrey coefficients

We prove a local unique continuation result for Schr\''odinger operators with time independent Lipschitz metrics and lower order terms which are Gevrey 2 in time and bounded in space. This implies global unique continuation from any open set in a connected Riemannian manifold. These results relax in the same geometric setting the analyticity assumption in time of the Tataru-Robbiano-Zuily-H\''ormander theorem for these operators. The proof is based on (i) a Tataru-Robbiano-Zuily-H\''ormander type Carleman estimate with a nonlocal weight adapted to the anisotropy of the Schr\''odinger operator and (ii) the description of the conjugation of the Schr\''odinger operator with Gevrey coefficients by this nonlocal weight.

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arXiv2024

A scattering operator for some nonlinear elliptic equations

We consider non linear elliptic equations of the form $Δu = f(u,\nabla u)$ for suitable analytic nonlinearity $f$, in the vinicity of infinity in $\mathbb{R}^d$, that is on the complement of a compact set.We show that there is a \emph{one-to-one correspondence} between the non linear solution $u$ defined there, and the linear solution $u\_L$ to the Laplace equation, such that, in an adequate space, $u - u\_L\to 0$ as $|x|\to +\infty$. This is a kind of scattering operator.Our results apply in particular for the energy critical and supercritical pure power elliptic equation and for the 2d (energy critical) harmonic maps and the $H$-system. Similar results are derived for solution defined on the neighborhood of a point in $\mathbb{R}^d$. The proofs are based on a conformal change of variables, and studied as an evolution equation (with the radial direction playing the role of time) in spaces with analytic regularity on spheres (the directions orthogonal to the radial direction).

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arXiv2023

Lectures on unique continuation for waves

These notes are intended as an introduction to the question of unique continuation for the wave operator, and some of its applications. The general question is whether a solution to a wave equation in a domain, vanishing on a subdomain has to vanish everywhere. We state and prove two of the main results in the field. We first give a proof of the classical local H{ö}rmander theorem in this context which holds under a pseudoconvexity condition. We then specialize to the case of wave operators with time-independent coefficients and prove the Tataru theorem: local unique continuation holds across any non-characteristic hypersurface. This local result implies a global unique continuation statement which can be interpreted as a converse to finite propagation speed. We finally give an application to approximate controllability, and present without proofs the associated quantitative estimates.

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arXiv2022

On uniform controllability of 1D transport equations in the vanishing viscosity limit

We consider a one dimensional transport equation with varying vector field and a small viscosity coefficient, controlled by one endpoint of the interval. We give upper and lower bounds on the minimal time needed to control to zero, uniformly in the vanishing viscosity limit. We assume that the vector field varies on the whole interval except at one point. The upper/lower estimates we obtain depend on geometric quantities such as an Agmon distance and the spectral gap of an associated semiclassical Schr{ö}dinger operator. They improve, in this particular situation, the results obtained in the companion paper [LL21]. The proofs rely on a reformulation of the problem as a uniform observability question for the semiclassical heat equation together with a fine analysis of localization of eigenfunctions both in the semiclassically allowed and forbidden regions [LL22], together with estimates on the spectral gap [HS84, All98]. Along the proofs, we provide with a construction of biorthogonal families with fine explicit bounds, which we believe is of independent interest.

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arXiv2022

Uniform observation of semiclassical Schr{ö}dinger eigenfunctions on an interval

We consider eigenfunctions of a semiclassical Schr{ö}dinger operator on an interval, with a single-well type potential and Dirichlet boundary conditions. We give upper/lower bounds on the L^2 density of the eigenfunctions that are uniform in both semiclassical and high energy limits. These bounds are optimal and are used in an essential way in a companion paper in application to a controllability problem. The proofs rely on Agmon estimates and a Gronwall type argument in the classically forbidden region, and on the description of semiclassical measures for boundary value problems in the classically allowed region. Limited regularity for the potential is assumed.

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arXiv2021

Concentration close to the cone for linear waves

We are concerned with solutions to the linear wave equation. We give an asymptotic formula for large time, valid in the energy space, via an operator related to the Radon transform. This allows us to show that the energy is concentrated near the light cone. This allows to derive further expressions the exterior energy (outside a shifted light cone). We in particular generalize the formulas of [CKS14] obtained in the radial setting. In odd dimension, we study the discrepancy of the exterior energy regarding initial energy, and prove in the general case the results of [KLLS15] (which were restricted to radial data).

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arXiv2021

On uniform observability of gradient flows in the vanishing viscosity limit

We consider a transport equation by a gradient vector field with a small viscous perturbation --$εΔ_g$. We study uniform observability (resp. controllability) properties in the (singular) vanishing viscosity limit $ε\rightarrow 0^+$, that is, the possibility of having a uniformly bounded observation constant (resp. control cost). We prove with a series of examples that in general, the minimal time for uniform observability may be much larger than the minimal time needed for the observability of the limit equation $ε= 0$. We also prove that the two minimal times coincides for positive solutions. The proofs rely on a semiclassical reformulation of the problem together with (a) Agmon estimates concerning decay of eigenfunctions in the classically forbidden region [HS84] (b) fine estimates of the kernel of the semiclassical heat equation [LY86].

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arXiv2020

Logarithmic decay for damped hypoelliptic wave and Schr{ö}dinger equations

We consider damped wave (resp. Schr{ö}dinger and plate) equations driven by a hypoelliptic "sum of squares" operator L on a compact manifold and a damping function b(x). We assume the Chow-Rashevski-H{ö}rmander condition at rank k (at most k Lie brackets needed to span the tangent space) together with analyticity of M and the coefficients of L. We prove decay of the energy at rate $log(t)^{-1/k}$ (resp. $log(t)^{-2/k}$ ) for data in the domain of the generator of the associated group. We show that this decay is optimal on a family of Grushin-type operators. This result follows from a perturbative argument (of independent interest) showing, in a general abstract setting, that quantitative approximate observability/controllability results for wave-type equations imply a priori decay rates for associated damped wave, Schr{ö}dinger and plate equations. The adapted quantitative approximate observability/controllability theorem for hypoelliptic waves is obtained by the authors in [LL19, LL17].

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arXiv2020

On the Observability Inequality of Coupled Wave Equations: the Case without Boundary

In this paper, we study the observability and controllability of wave equations coupled by first or zero order terms on a compact manifold. We adopt the approach in Dehman-Lebeau's paper \cite{DehmanLebeau09} to prove that: the weak observability inequality holds for wave equations coupled by first order terms on compact manifold without boundary if and only if a class of ordinary differential equations related to the symbol of the first order terms along the Hamiltonian flow are exactly controllable. We also compute the higher order part of the observability constant and the observation time. By duality, we obtain the controllability of the dual control system in a finite co-dimensional space. This gives the full controllability under the assumption of unique continuation of eigenfunctions. Moreover, these results can be applied to the systems of wave equations coupled by zero order terms of cascade structure after an appropriate change of unknowns and spaces. Finally, we provide some concrete examples as applications where the unique continuation property indeed holds.

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arXiv2019

Decay of semilinear damped wave equations:cases without geometric control condition

We consider the semilinear damped wave equation $\partial_{tt}^2 u(x,t)+γ(x)\partial_t u(x,t)=Δu(x,t)-αu(x,t)-f(x,u(x,t))$. In this article, we obtain the first results concerning the stabilization of this semilinear equation in cases where $γ$ does not satisfy the geometric control condition. When some of the geodesic rays are trapped, the stabilization of the linear semigroup is semi-uniform in the sense that $\|e^{At}A^{-1}\|\leq h(t)$ for some function $h$ with $h(t)\rightarrow 0$ when $t\rightarrow +\infty$. We provide general tools to deal with the semilinear stabilization problem in the case where $h(t)$ has a sufficiently fast decay.

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arXiv2018

Exact Controllability of Nonlinear Heat Equations in Spaces of Analytic Functions

It is by now well known that the use of Carleman estimates allows to establish the control-lability to trajectories of nonlinear parabolic equations. However, by this approach, it is not clear how to decide whether a given function is indeed reachable. In this paper, we pursue the study of the reachable states of parabolic equations based on a direct approach using control inputs in Gevrey spaces by considering a nonlinear heat equation in dimension one. The nonlinear part is assumed to be an analytic function of the spatial variable x, the unknown y, and its derivative $\partial$ x y. By investigating carefully a nonlinear Cauchy problem in the spatial variable and the relationship between the jet of space derivatives and the jet of time derivatives, we derive an exact controllability result for small initial and final data that can be extended as analytic functions on some ball of the complex plane. 2010 Mathematics Subject Classification: 35K40, 93B05

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