arXiv subjects

Romain Joly

Publications and source records attributed to Romain Joly.

At least 19 recordsLinked to original sources

arXiv2025

Generic balanced synchrony patterns in network dynamics

A coupled cell network is a type of ordinary differential equation $\dot x(t)=f(x(t))$, with structural constraints on the vector field $f$, encoded in a directed graph, whose cells and arrows are labeled by type. The generated dynamics can model, for example, those of neural networks or ecological systems. These systems and the synchrony patterns observed in their solutions have been intensely studied, particularly by Golubitsky, Stewart, and their coauthors. In the present article, we show that, for a generic vector field $f$, the synchrony patterns of the solutions of $\dot x(t)=f(x(t))$ are always balanced. This roughly means that for almost all $f$, the observed synchrony patterns, such as synchronization in two different cells, are inherited from the structural symmetries imposed by the graph and the cell types. Any other synchronization, not directly imposed by the geometry of the graph and the cell types, cannot occur. By doing so, we are completing the proof of several conjectures, including the rigid synchrony conjecture, the full oscillation conjecture and the observation of constant states. This article is the published version of the results stated by the second author in his PhD thesis.

math.DS↗

arXiv2025

Parabolic abstract evolution equations in cylindrical domains and uniformly local Sobolev spaces

In this article, we consider parabolic equations of the type $$\partial_t u(x,t)=\Delta u(x,t) - Bu(x,t) + F(u(x,t))$$ where $u$ is valued in a transverse Hilbert space $Y$ and $B$ is a positive self-adjoint operator on $Y$, allowing a different diffusion mechanism in the transverse direction. We aim at considering solutions with infinite energy and we study the Cauchy problem in the uniformly local spaces associated with the norm $$\|u\|_{L^2_{\text{ul}}(\mathbb{R},Y)}= \sup_{a\in\mathbb{R}^d} \|u(x)\|_{L^2(B(a,1),Y)}.$$ For the classical parabolic equation, i.e. if $Y=\mathbb{R}$, it is known that the Cauchy problem is ill-posed in the weak version of the uniformly local spaces but well-posed in a stronger version, where additional uniform continuity is required. In this paper, we show that the linear operator $\partial^2_{xx} - B$ is not necessarily a sectorial operator in any version of the uniformly local Lebesgue space, due to the possible non-density of its domain. Then, we use the theory of parabolic abstract evolution equations to set a well-posed Cauchy problem, even in the weak version of the uniformly local space. In particular, we believe that this paper offers a new perspective on the comparison between both versions of the uniformly local spaces and also provides a new natural example of differential operators with non-dense domain.

math.AP↗

arXiv2025

The Graph Geometric Control Condition

In this paper, we introduce a novel concept called the Graph Geometric Control Condition (GGCC). It turns out to be a simple, geometric rewriting of many of the frameworks in which the controllability of PDEs on graphs has been studied. We prove that (GGCC) is a necessary and sufficient condition for the exact controllability of the wave equation on metric graphs with internal controls and Dirichlet boundary conditions. We then investigate the internal exact controllability of the wave equation with mixed boundary conditions and the one of the Schr\"odinger equation, as well as the internal null-controllability of the heat equation. We show that (GGCC) provides a sufficient condition for the controllability of these equations and we provide explicit examples proving that (GGCC) is not necessary in these cases.

math.OC↗

arXiv2023

Generic transversality of travelling fronts, standing fronts, and standing pulses for parabolic gradient systems

For nonlinear parabolic systems of the form \[ \partial_t w(x,t) = \partial_{x}^2 w(x,t) - \nabla V\bigl(w(x,t)\bigr) \,, \] the following conclusions are proved to hold generically with respect to the potential $V$: every travelling front invading a minimum point of $V$ is bistable, there is no standing front, every standing pulse is stable at infinity, the profiles of these fronts and pulses approach their limits at $\pm\infty$ tangentially to the eigenspaces corresponding to the smallest eigenvalues of $D^2V$ at these points, these fronts and pulses are robust with respect to small perturbations of the potential, and the set of their profiles is discrete. These conclusions are obtained as consequences of generic transversality results for heteroclinic and homoclinic solutions of the differential systems governing the profiles of such fronts and pulses. Among these results, it is proved that, for a generic Hamiltonian system of the form \[ \ddot u=\nabla V(u) \,, \] every asymmetric homoclinic orbit is transverse and every symmetric homoclinic orbit is elementary.

math.AP↗

arXiv2022

Control of the Schr\"odinger equation by slow deformations of the domain

The aim of this work is to study the controllability of the Schr\"odinger equation \begin{equation}\label{eq_abstract} i\partial_t u(t)=-\Delta u(t)~~~~~\text{ on }\Omega(t) \tag{$\ast$} \end{equation} with Dirichlet boundary conditions, where $\Omega(t)\subset\mathbb{R}^N$ is a time-varying domain. We prove the global approximate controllability of \eqref{eq_abstract} in $L^2(\Omega)$, via an adiabatic deformation $\Omega(t)\subset\mathbb{R}$ ($t\in[0,T]$) such that $\Omega(0)=\Omega(T)=\Omega$. This control is strongly based on the Hamiltonian structure of \eqref{eq_abstract} provided by [18], which enables the use of adiabatic motions. We also discuss several explicit interesting controls that we perform in the specific framework of rectangular domains.

math.AP↗

arXiv2021

Schrödinger equation in moving domains

We consider the Schr\''odinger equation \begin{equation}\label{eq_abstract} i\partial_t u(t)=-Δu(t)~~~~~\text{ on }Ω(t) \tag{$\ast$} \end{equation}where $Ω(t)\subset\mathbb{R}$ is a moving domain depending on the time $t\in [0,T]$. The aim of this work is to provide a meaning to the solutions of such an equation. We use the existence of a bounded reference domain $Ω_0$ and a specific family of unitary maps $h^\sharp(t): L^2(Ω(t),\mathbb{C})\longrightarrow L^2(Ω_0,\mathbb{C})$. We show that the conjugation by $h^\sharp$ provides a newequation of the form \begin{equation}\label{eq_abstract2}i\partial_t v= h^\sharp(t)H(t)h_\sharp(t) v~~~~~\text{ on }Ω_0\tag{$\ast\ast$} \end{equation} where $h_\sharp=(h^\sharp)^{-1}$. The Hamiltonian $H(t)$ is a magnetic Laplacian operator of the form$$H(t)=-(div+iA)\circ(grad+iA)-|A|^2$$where $A$ is an explicit magnetic potential depending on the deformation of the domain $Ω(t)$. The formulation \eqref{eq_abstract2} enables to ensure the existence of weak and strong solutions of the initial problem \eqref{eq_abstract} on $Ω(t)$ endowed with Dirichlet boundary conditions. In addition, it also indicates that the correct Neumann type boundary conditions for \eqref{eq_abstract} are not the homogeneous but the magnetic ones$$\partial_νu(t)+i\langleν| A\rangle u(t)=0,$$even though \eqref{eq_abstract} has no magnetic term. All the previous results are also studied in presence of diffusion coefficients as well as magnetic and electric potentials. Finally, we prove some associated byproducts as an adiabatic result for slow deformations of the domain and atime-dependent version of the so-called ``Moser's trick''. We use this outcome in order to simplify Equation \eqref{eq_abstract2} and to guarantee the well-posedness for slightly less regular deformations of $Ω(t)$.

math.AP↗

arXiv2020

Asymptotic self-similarity in diffusion equations with nonconstant radial limits at infinity

We study the long-time behavior of localized solutions to linear or semilinear parabolic equations in the whole space $\mathbb{R}^n$, where $n \ge 2$, assuming that the diffusion matrix depends on the space variable $x$ and has a finite limit along any ray as $|x| \to \infty$. Under suitable smallness conditions in the nonlinear case, we prove convergence to a self-similar solution whose profile is entirely determined by the asymptotic diffusion matrix. Examples are given which show that the profile can be a rather general Gaussian-like function, and that the approach to the self-similar solution can be arbitrarily slow depending on the continuity and coercivity properties of the asymptotic matrix. The proof of our results relies on appropriate energy estimates for the diffusion equation in self-similar variables. The new ingredient consists in estimating not only the difference $w$ between the solution and the self-similar profile, but also an antiderivative $W$ obtained by solving a linear elliptic problem which involves $w$ as a source term. Hence, a good part of our analysis is devoted to the study of linear elliptic equations whose coefficients are homogeneous of degree zero.

math.AP↗

arXiv2019

Permuting quantum eigenmodes by a quasi-adiabatic motion of a potential wall

We study the Schrödinger equation $i\partial_tψ=-Δψ+Vψ$ on $L^2((0,1),\mathbb{C})$ where $V$ is a very high and localized potential wall. We aim to perform permutations of the eigenmodes and to control the solution of the equation. We consider the process where the position and the height of the potential wall change as follows. First, the potential increases from zero to a very large value, so a narrow potential wall is formed that almost splits the interval into two parts; then the wall moves to a different position, after which the height of the wall decays to zero again. We show that even though the rate of the variation of the potential's parameters can be arbitrarily slow, this process alternates adiabatic and non-adiabatic dynamics, leading to a non-trivial permutation of the eigenstates. Furthermore, we consider potentials with several narrow walls and we show how an arbitrarily slow motion of the walls can lead the system from any given state to an arbitrarily small neighborhood of any other state, thus proving the approximate controllability of the above Schrödinger equation by means of a soft, quasi-adiabatic variation of the potential.

math.OC↗

arXiv2019

Generic transversality of heteroclinic and homoclinic orbits for scalar parabolic equations

In this paper, we consider the scalar reaction-diffusion equations $\partial_t u=Δu + f(x,u,\nabla u)$ on a bounded domain $Ω\subset\mathbb{R}^d$ of class $C^2$. We show that the heteroclinic and homoclinic orbits connecting hyperbolic equilibria and hyperbolic periodic orbits are transverse, generically with respect to f. One of the main ingredients of the proof is an accurate study of the singular nodal set of solutions of linear parabolic equations. Our main result is a first step for proving the genericity of Kupka-Smale property, the generic hyperbolicity of periodic orbits remaining unproved.

math.AP↗

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.

math.AP↗

arXiv2018

A striking correspondence between the dynamics generated by the vector fields and by the scalar parabolic equations

The purpose of this paper is to enhance a correspondence between the dynamics of the differential equations $\dot y(t)=g(y(t))$ on $\mathbb{R}^d$ and those of the parabolic equations $\dot u=Δu +f(x,u,\nabla u)$ on a bounded domain $Ω$. We give details on the similarities of these dynamics in the cases $d=1$, $d=2$ and $d\geq 3$ and in the corresponding cases $Ω=(0,1)$, $Ω=\mathbb{T}^1$ and dim($Ω$)$\geq 2$ respectively. In addition to the beauty of such a correspondence, this could serve as a guideline for future research on the dynamics of parabolic equations.

math.AP↗

arXiv2017

Energy decay and diffusion phenomenon for the asymptotically periodic damped wave equation

We prove local and global energy decay for the asymptotically periodic damped wave equation on the Euclidean space. Since the behavior of high frequencies is already mostly understood, this paper is mainly about the contribution of low frequencies. We show in particular that the damped wave behaves like a solution of a heat equation which depends on the H-limit of the metric and the mean value of the absorption index.

math-ph↗

arXiv2015

Exponential decay for the damped wave equation in unbounded domains

We study the decay of the semigroup generated by the damped wave equation in an unbounded domain. We first prove under the natural geometric control condition the exponential decay of the semigroup. Then we prove under a weaker condition the logarithmic decay of the solutions (assuming that the initial data are smoother). As corollaries, we obtain several extensions of previous results of stabilisation and control.

math.AP↗

arXiv2013

Stabilization for the semilinear wave equation with geometric control condition

In this article, we prove the exponential stabilization of the semilinear wave equation with a damping effective in a zone satisfying the geometric control condition only. The nonlinearity is assumed to be subcritical, defocusing and analytic. The main novelty compared to previous results, is the proof of a unique continuation result in large time for some undamped equation. The idea is to use an asymptotic smoothing effect proved by Hale and Raugel in the context of dynamical systems. Then, once the analyticity in time is proved, we apply a unique continuation result with partial analyticity due to Robbiano, Zuily, Tataru and Hörmander. Some other consequences are also given for the controllability and the existence of a compact attractor.

math.AP↗

arXiv2013

A note on the global controllability of the semilinear wave equation

We study the internal controllability of the semilinear wave equation $$v_{tt}(x,t)-Δv(x,t) + f(x,v(x,t))= \Un_ω u(x,t)$$ for some nonlinearities $f$ which can produce several non-trivial steady states. One of the usual hypotheses to get global controllability, is to assume that $f(x,v)v\geq 0$. In this case, a stabilisation term $u=γ(x)v_t$ makes any solution converging to zero. The global controllability then follows from a theorem of local controllability and the time reversibility of the equation. In this paper, the nonlinearity $f$ can be more general, so that the solutions of the damped equation may converge to another equilibrium than $0$. To prove global controllability, we study the controllability inside a compact attractor and show that it is possible to travel from one equilibrium point to another by using the heteroclinic orbits.

math.AP↗

arXiv2011

How opening a hole affects the sound of a flute

In this paper, we consider an open tube of diameter $ε>0$, on the side of which a small hole of size $ε^2$ is pierced. The resonances of this tube correspond to the eigenvalues of the Laplacian operator with homogeneous Neumann condition on the inner surface of the tube and Dirichlet one the open parts of the tube. We show that this spectrum converges when $ε$ goes to 0 to the spectrum of an explicit one-dimensional operator. At a first order of approximation, the limit spectrum describes the note produced by a flute, for which one of its holes is open.

math.AP↗

arXiv2011

Observation and inverse problems in coupled cell networks

A coupled cell network is a model for many situations such as food webs in ecosystems, cellular metabolism, economical networks... It consists in a directed graph $G$, each node (or cell) representing an agent of the network and each directed arrow representing which agent acts on which one. It yields a system of differential equations $\dot x(t)=f(x(t))$, where the component $i$ of $f$ depends only on the cells $x_j(t)$ for which the arrow $j\rightarrow i$ exists in $G$. In this paper, we investigate the observation problems in coupled cell networks: can one deduce the behaviour of the whole network (oscillations, stabilisation etc.) by observing only one of the cells? We show that the natural observation properties holds for almost all the interactions $f$.

math.DS↗

arXiv2010

Generic Morse-Smale property for the parabolic equation on the circle

In this paper, we show that, for scalar reaction-diffusion equations $u_t=u_{xx}+f(x,u,u_x)$ on the circle $S^1$, the Morse-Smale property is generic with respect to the non-linearity $f$. In \cite{CR}, Czaja and Rocha have proved that any connecting orbit, which connects two hyperbolic periodic orbits, is transverse and that there does not exist any homoclinic orbit, connecting a hyperbolic periodic orbit to itself. In \cite{JR}, we have shown that, generically with respect to the non-linearity $f$, all the equilibria and periodic orbits are hyperbolic. Here we complete these results by showing that any connecting orbit between two hyperbolic equilibria with distinct Morse indices or between a hyperbolic equilibrium and a hyperbolic periodic orbit is automatically transverse. We also show that, generically with respect to $f$, there does not exist any connection between equilibria with the same Morse index. The above properties, together with the existence of a compact global attractor and the Poincaré-Bendixson property, allow us to deduce that, generically with respect to $f$, the non-wandering set consists in a finite number of hyperbolic equilibria and periodic orbits . The main tools in the proofs include the lap number property, exponential dichotomies and the Sard-Smale theorem. The proofs also require a careful analysis of the asymptotic behavior of solutions of the linearized equations along the connecting orbits.

math.AP↗