arXiv subjects

Emmanuel Risler

Publications and source records attributed to Emmanuel Risler.

13 recordsLinked to original sources

arXiv2023

Existence of a pulled or pushed travelling front invading a critical point for parabolic gradient systems

For nonlinear parabolic gradient systems of the form \[ u_t = -\nabla V(u) + u_{xx} \,, \] where the spatial domain is the whole real line, the state variable $u$ is multidimensional, and the potential function $V$ is coercive at infinity, the following result is proved: for every critical point of $V$ which is not a global minimum point, there exists a travelling front, either pushed or pulled, invading this critical point at a speed which is not smaller than its linear spreading speed. By contrast with previous existence results of the same kind, no further assumption is made (neither that the invaded critical point is a non-degenerate local minimum point, nor other assumptions ensuring pushed invasion).

math.AP↗

arXiv2023

Global relaxation of bistable solutions for gradient systems in one unbounded spatial dimension

This paper is concerned with parabolic gradient systems of the form \[ u_t=-\nabla V (u) + \mathcal{D} u_{xx}\,, \] where the spatial domain is the whole real line, the state variable $u$ is multidimensional, $\mathcal{D}$ denotes a fixed diffusion matrix, and the potential $V$ is coercive at infinity. Bistable solutions, that is solutions close at both ends of space to stable homogeneous equilibria, are considered. For a solution of this kind, it is proved that, if the equilibria approached at both ends belong to the same level set of the potential and if an appropriate (localized in space) energy remains bounded from below as time increases, then the solution approaches, as time goes to infinity, a pattern of profiles of stationary solutions homoclinic or heteroclinic to stable homogeneous equilibria, going slowly away from one another. This result provides a step towards a complete description of the global behaviour of all bistable solutions that is pursued in a companion paper. Some consequences are derived, and applications to some examples are given.

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arXiv2023

Global behaviour of bistable solutions for gradient systems in one unbounded spatial dimension

This paper is concerned with parabolic gradient systems of the form \[ u_t=-\nabla V (u) + u_{xx}\,, \] where the spatial domain is the whole real line, the state variable $u$ is multidimensional, and the potential $V$ is coercive at infinity. For such systems, under generic assumptions on the potential, the asymptotic behaviour of every bistable solution (that is, every solution close at both ends of space to stable homogeneous equilibria) is described. Every such solution approaches, far to the left in space a stacked family of bistable fronts travelling to the left, far to the right in space a stacked family of bistable fronts travelling to the right, and in between a pattern of profiles of stationary solutions homoclinic or heteroclinic to stable homogeneous equilibria, going slowly away from one another. This result pushes one step further the program initiated in the late seventies by Fife and McLeod about the global asymptotic behaviour of bistable solutions, by extending their results to the case of systems. In the absence of maximum principle, the arguments are purely variational, and call upon previous results obtained in companion papers.

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arXiv2023

Global behaviour of bistable solutions for hyperbolic gradient systems in one unbounded spatial dimension

This paper is concerned with damped hyperbolic gradient systems of the form \[ αu_{tt} + u_t = -\nabla V(u) + u_{xx}\,, \] where the spatial domain is the whole real line, the state variable $u$ is multidimensional, $α$ is a positive quantity, and the potential $V$ is coercive at infinity. For such systems, under generic assumptions on the potential, the asymptotic behaviour of every bistable solution (that is, every solution close at both ends of space to stable homogeneous equilibria) is described. Every such solution approaches, far to the left in space a stacked family of bistable fronts travelling to the left, far to the right in space a stacked family of bistable fronts travelling to the right, and in between a pattern of profiles of stationary solutions homoclinic or heteroclinic to stable homogeneous equilibria, going slowly away from one another. In the absence of maximum principle, the arguments are purely variational. This extends previous results obtained in companion papers for damped wave equations or parabolic gradient systems, in the spirit of the program initiated in the late seventies by Fife and McLeod on the global asymptotic behaviour of bistable solutions for parabolic equations.

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arXiv2023

Global behaviour of radially symmetric solutions stable at infinity for gradient systems

This paper is concerned with radially symmetric solutions of systems of the form \[ u_t = -\nabla V(u) + Δ_x u \] where space variable $x$ and and state-parameter $u$ are multidimensional, and the potential $V$ is coercive at infinity. For such systems, under generic assumptions on the potential, the asymptotic behaviour of solutions "stable at infinity", that is approaching a spatially homogeneous equilibrium when $|x|$ approaches $+\infty$, is investigated. It is proved that every such solutions approaches a stacked family of radially symmetric bistable fronts travelling to infinity. This behaviour is similar to the one of bistable solutions for gradient systems in one unbounded spatial dimension, described in a companion paper. It is expected (but unfortunately not proved at this stage) that behind these travelling fronts the solution again behaves as in the one-dimensional case (that is, the time derivative approaches zero and the solution approaches a pattern of stationary solutions).

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arXiv2023

Global behaviour of solutions stable at infinity for gradient systems in higher space dimension: the no invasion case

This paper is concerned with parabolic gradient systems of the form \[ u_t = -\nabla V(u) + Δ_x u \,, \] where the space variable $x$ and the state variable $u$ are multidimensional, and the potential $V$ is coercive at infinity. For such systems, the asymptotic behaviour of solutions stable at infinity, that is approaching a stable homogeneous equilibrium as $|x|$ goes to $+\infty$, is investigated. A partial description of the global asymptotic behaviour of such a solution is provided, depending on the mean speed of growth of the spatial domain where the solution is not close to this equilibrium, in relation with the asymptotic energy of the solution. If this mean speed is zero, then the asymptotic energy is nonnegative, and the time derivative $u_t$ goes to $0$ uniformly in space. If conversely the mean speed is nonzero, then the asymptotic energy equals $-\infty$. This result is called upon in a companion paper where the global behaviour of radially symmetric solutions stable at infinity is described. The proof relies mainly on energy estimates in the laboratory frame and in frames travelling at a small nonzero velocity.

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arXiv2023

Global convergence towards pushed travelling fronts for parabolic gradient systems

This article addresses the issue of global convergence towards pushed travelling fronts for solutions of parabolic systems of the form \[ u_t = - \nabla V(u) + u_{xx} \,, \] where the potential $V$ is coercive at infinity. It is proved that, if an initial condition $x\mapsto u(x,t=0)$ approaches, rapidly enough, a critical point $e$ of $V$ to the right end of space, and if, for some speed $c_0$ greater than the linear spreading speed associated with $e$, the energy of this initial condition in a frame travelling at the speed $c_0$ is negative $\unicode{x2013}$ with symbols, \[ \int_{\mathbb{R}} e^{c_0 x}\left(\frac{1}{2} u_x(x,0)^2 + V\bigl(u(x,0)\bigr)- V(e)\right)\, dx < 0 \,, \] then the corresponding solution invades $e$ at a speed $c$ greater than $c_0$, and approaches, around the leading edge and as time goes to $+\infty$, profiles of pushed fronts (in most cases a single one) travelling at the speed $c$. A necessary and sufficient condition for the existence of pushed fronts invading a critical point at a speed greater than its linear spreading speed follows as a corollary. In the absence of maximum principle, the arguments are purely variational. The key ingredient is a Poincaré inequality showing that, in frames travelling at speeds exceeding the linear spreading speed, the variational landscape does not differ much from the case where the invaded equilibrium $e$ is stable. The proof is notably inspired by ideas and techniques introduced by Th. Gallay and R. Joly, and subsequently used by C. Luo, in the setting of nonlinear damped wave equations.

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arXiv2023

Generic transversality of radially symmetric stationary solutions stable at infinity for parabolic gradient systems

This paper is devoted to the generic transversality of radially symmetric stationary solutions of nonlinear parabolic systems of the form \[ \partial_t w(x,t) = -\nabla V\bigl(w((x,t))\bigr) + Δ_x w(x,t) \,, \] where the space variable $x$ is multidimensional and unbounded. It is proved that, generically with respect to the potential $V$, radially symmetric stationary solutions that are stable at infinity (in other words, that approach a minimum point of $V$ at infinity in space) are transverse; as a consequence, the set of such solutions is discrete. This result can be viewed as the extension to higher space dimensions of the generic elementarity of symmetric standing pulses, proved in a companion paper. It justifies the generic character of the discreteness hypothesis concerning this set of stationary solutions, made in another companion paper devoted to the global behaviour of (time dependent) radially symmetric solutions stable at infinity for such systems.

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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.

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arXiv2017

Competition between stable equilibria in reaction-diffusion systems: the influence of mobility on dominance

This paper is concerned with reaction-diffusion systems of two symmetric species in spatial dimension one, having two stable symmetric equilibria connected by a symmetric standing front. The first order variation of the speed of this front when the symmetry is broken through a small perturbation of the diffusion coefficients is computed. This elementary computation relates to the question, arising from population dynamics, of the influence of mobility on dominance, in reaction-diffusion systems modelling the interaction of two competing species. It is applied to two examples. First a toy example, where it is shown that, depending on the value of a parameter, an increase of the mobility of one of the species may be either advantageous or disadvantageous for this species. Then the Lotka-Volterra competition model, in the bistable regime close to the onset of bistability, where it is shown that an increase of mobility is advantageous. Geometric interpretations of these results are given.

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arXiv2017

Retracting fronts for the nonlinear complex heat equation

The "nonlinear complex heat equation" $A_t=i|A|^2A+A_{xx}$ was introduced by P. Coullet and L. Kramer as a model equation exhibiting travelling fronts induced by non-variational effects, called "retracting fronts". In this paper we study the existence of such fronts. They go by one-parameter families, bounded at one end by the slowest and "steepest" front among the family, a situation presenting striking analogies with front propagation into unstable states.

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arXiv2015

Uniqueness of the pendent drop of infinite length

We prove the uniqueness of the infinite length axisymmetric solution to the capillary equation. We observe that capillary equation can be viewed, at large depth, as a perturbation of an integrable two-dimensional differential system. Uniqueness is then proved by an elementary perturbation argument.

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arXiv2006

A variational proof of global stability for bistable travelling waves

We give a variational proof of global stability for bistable travelling waves of scalar reaction-diffusion equations on the real line. In particular, we recover some of the classical results by P. Fife and J.B. McLeod without any use of the maximum principle. The method that is illustrated here in the simplest possible setting has been successfully applied to more general parabolic or hyperbolic gradient-like systems.

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