Olivier Goubet

Université de Lille · Laboratoire Paul Painlevé

We consider here semilinear Schrödinger equations with a non standard dispersion that is discontinuous at x=0. We first establish the existence and uniqueness of standing wave solutions for these equations. We then study the orbital stability of these standing wave into a subspace of the energy space that where classical methods as the concentratio...

In this article we address some issues related to the initial value problems for a rotating shallow water hyperbolic system of equations and the diffusive regularization of this system. For initial data close to the solution at rest, we establish the local existence and the uniqueness of a solution to the hyperbolic system, as well as the global ex...

This article partakes of the PEGASE project the goal of which is a better understanding of the mechanisms explaining the behaviour of species living in a network of forest patches linked by ecological corridors (hedges for instance). Actually we plan to study the effect of the fragmentation of the habitat on biodiversity. A simple neutral model for...

We construct positive singular solutions for the problem \(-\Delta u=\lambda \exp (e^u)\) in \(B_1\subset {\mathbb {R}}^n\) (\(n\ge 3\)), \(u=0\) on \(\partial B_1\), having a prescribed behaviour around the origin. Our study extends the one in Miyamoto (J Differ Equ 264:2684–2707, 2018) for such nonlinearities. Our approach is then carried out to...

We construct positive singular solutions for the problem $-\Delta u=\lambda \exp (e^u)$ in $B_1\subset \mathbb{R}^n$ ($n\geq 3$), $u=0$ on $\partial B_1$, having a prescribed behaviour around the origin. Our study extends the one in Y. Miyamoto [Y. Miyamoto, A limit equation and bifurcation diagrams of semilinear elliptic equations with general sup...

We show that solutions of the periodic KdV equations $$\begin{aligned} u_t+\gamma u +u_{xxx}+uu_x=f, \end{aligned}$$are asymptotically determined by their values at three points. That is if there exists \(x_1,x_2,x_3\) such that \(0< x_3-x_2<<x_3-x_1<<1\) and $$\begin{aligned} \lim _{t\rightarrow +\infty } |u_1(t,x_j)-u_2(t,x_j)|=0, \; \mathrm{for}...

In this short note, we discuss some results concerning the long-time behaviour of solutions to dissipative sine-Gordon equation in a bounded domain of Rn. We first prove that without forcing term and under some assumptions on the domain then the global attractor reduces to {0}. We then rewrite the sine-Gordon equation as a Schrödinger equation with...

We consider the nonlinear integrodifferential Benjamin-Bona-Mahony equation $$ u_t - u_{txx} + u_x - \int_0^\infty g(s) u_{xx}(t-s) {\rm d} s + u u_x = f $$ where the dissipation is entirely contributed by the memory term. Under a suitable smallness assumption on the external force $f$, we show that the related solution semigroup possesses the glob...

We consider here a weakly damped forced periodic KdV equation. We prove that if the forcing term is analytic in space, then the global attractor is also contained into a space of analytic functions. This result was conjectured in [23].

We consider a semi-discrete in time Crank-Nicolson scheme to discretize a weakly damped forced nonlinear fractional Schrödinger equation ut-i(-Δ)αu+i|u|²u+γu=f for α∈(12,1) considered in the the whole space R. We prove that such semi-discrete equation provides a discrete infinite-dimensional dynamical system in Hα(R) that possesses a global attract...

We consider a weakly damped forced nonlinear fractional Schrödinger equation \(u_t -i(-\Delta )^\alpha u +i |u|^2u+\gamma u=f\) for a given \(\alpha \in \left( \frac{1}{2}, 1\right) \) considered in the whole space \(\mathbb {R}\). We prove that this equation provides an infinite dimensional dynamical system in \(H^{\alpha }(\mathbb R)\) that posse...

In this article, we address the generalized BBM equation with white noise dispersion which reads $$\begin{aligned} du-du_{xx}+u_x \circ dW+ u^pu_xdt=0, \end{aligned}$$in the Stratonovich formulation, where W(t) is a standard real valued Brownian motion. We first investigate the well-posedness of the initial value problem for this equation. We then...

In this paper, we study the following water wave model with a nonlocal viscous term: u_{t}+u_{x}+\beta u_{xxx}+\frac{\sqrt{\nu}}{\sqrt{\pi}}\frac{\partial}{% \partial t}\int_{0}^{t}\frac{u(s)}{\sqrt{t-s}}\,ds+uu_{x}=\nu u_{xx}, where {\frac{1}{\sqrt{\pi}}\frac{\partial}{\partial t}\int_{0}^{t}\frac{u(s)}{\sqrt{% t-s}}\,ds} is the Riemann–Liouville...

In this paper, we consider the critical nonlinear Schrodinger equations in R2 with an oscillating nonlinearity, in a radial geometry. We numerically investigate the inuence of the oscillations on the time of existence for the corresponding solution, on the spirit of the recent result of Cazenave and Scialom. It can be observed that the solution con...

In this article we investigate both numerically and theoretically the influence of a defect on the blow-up of radial solutions to a cubic NLS equation in dimension 2.

We consider a damped forced nonlinear Schrödinger-Poisson system in ℝ3. This provides us with an infinite-dimensional dynamical system in the energy space H1(ℝ3). We prove the existence of a finite dimensional global attractor for this dynamical system.

The water wave theory traditionally assumes the fluid to be perfect, thus neglecting all effects of the viscosity. However, the explanation of several experimental data sets requires the explicit inclusion of dissipative effects. In order to meet these practical problems, the theory of visco-potential flows has been developed (see P.-F. Liu & A. Or...

We consider here a nonlinear elliptic equation in an unbounded sectorial domain of the plane. We prove the existence of a minimal solution to this equation and study its properties. We infer from this analysis some asymptotics for the stationary solution of an equation arising in plasma physics.

We consider here a nonlinear elliptic equation in an unbounded sectorial domain of the plane. We prove the existence of a minimal solution to this equation and study its properties. We infer from this analysis some asymptotics for the stationary solution of an equation arising in plasma physics.

We consider a semi-discrete in time relaxation scheme to discretize a damped forced nonlinear Klein-Gordon Schrödinger system. This provides us with a discrete infinite-dimensional dynamical system. We prove the existence of a finite dimensional global attractor for this dynamical system.

Let Ω⊂ℝ n be a bounded smooth open set. We prove that the singular set of any extremal solution of the system -Δu=μe v ,-Δv=λe u inΩ, with u=v=0 on ∂Ω, μ,λ≥0, has Hausdorff dimension at most n-10.

We study the long-time behavior of the solutions to a nonlinear damped driven Schrödinger type equation on a strip. We prove that this behavior is described by a regular compact global attractor.

We prove here the existence of boundary blow up solutions for fully nonlinear equations in general domains, for a nonlinearity satisfying the Keller-Osserman type condition. If moreover the nonlinearity is non decreasing, we prove uniqueness for boundary blow up solutions on balls for operators related to Pucci’s operators.

We study the existence of a global attractor for a damped parametric nonlinear Schrödinger equation. We provide sufficient conditions for this attractor to have finite dimension.

We study stable and finite Morse index solutions of the equation Δ2 = eu. If the equation is posed in ℝN, we classify radial stable solutions. We then construct nonradial stable solutions and we prove that, unlike the corresponding second order problem, no Liouville-type theorem holds, unless additional information is available on the asymptotics o...

It is well-known that the Ginzburg-Landau equation on ℝ has a global attractor [15] that attracts in L loc ∞ (ℝ) all the trajectories. This attractor contains bounded trajectories that are analytical functions in space. A famous theorem due to P. Collet and JP. Eckmann asserts that the ε-entropy per unit length in L ∞ of this global attractor is fi...

The exact controllability of the second order time-dependent Maxwell equations for the electric field is addressed through the Hilbert Uniqueness Method. A two-grid preconditioned conjugate gradient algorithm is employed to inverse the H.U.M. operator and to construct the numerical control. The underlying initial value problems are discretized by L...

In this article, we study a viscous asymptotical model equation for water waves u(t) + u(x) - beta u(txx) + v(D(1/2)u + F-1(i|xi|(1/2) sgn(xi)(u) over cap(xi))) + gamma uu(x) = 0 proposed in Kakutani and Matsuuchi (1975) [6]. Theoretical questions including the existence and regularity of the solutions will be answered. Numerical simulations of its...

Given a nondecreasing nonlinearity $f$, we prove uniqueness of large solutions in the following two cases: the domain is the ball or the domain has nonnegative mean curvature and the nonlinearity is asymptotically convex.

This article deals with the long-time behavior of the solution of the two-dimensional Navier–Stokes equations. At each time step, we use finite elements to split the solution into a large-scale component and a small-scale component, and we follow both components in time. Next, considering a mixed finite element approximation of the equations, we pr...

We analyze the semilinear elliptic equation $\Delta u=\rho(x) f(u)$, $u>0$ in ${\mathbf R}^D$ $(D\ge3)$, with a particular emphasis put on the qualitative study of entire large solutions, that is, solutions $u$ such that $\lim_{|x|\rightarrow +\infty}u(x)=+\infty$. Assuming that $f$ satisfies the Keller-Osserman growth assumption and that $\rho$ de...

We apply and discuss a multilevel method to solve a scattering problem. The multilevel method belongs to the class of incremental unknowns method as proposed by P. Poullet and A. Boag [Numer. Methods Partial Differ. Equations 23, No. 6, 1396–1410 (2007; Zbl 1129.65074)]; in this work, the best performance was obtained with a coarsest grid having ro...

We consider a weakly coupled system of nonlinear Schrödinger equations which models a Bose Einstein condensate with an impurity. The first equation is dissipative, while the second one is conservative. We consider this dynamical system into the framework of non-autonomous dynamical systems, the solution to the conservative equation being the symbol...

The authors discuss a linear viscous asymptotic model for water waves and the decay rate of solutions towards the equilibrium. KeywordsWater waves-Viscous asymptotic models-Nonlocal operators-Long-time asymptotics 2000 MR Subject Classification35Q35-35Q53-76B15

In this article, we investigate a water wave model with a nonlocal viscous term ut + ux + βuxxx + √v/√π ∫ot ut(s)/√t-s ds + uux = vuxx. The wellposedness of the equation and the decay rate of solutions are investigated theoretically and numerically.

We prove that a parametric nonlinear Schrödinger equation possesses a finite dimensional smooth global attractor in a suitable energy space.

We prove that the weakly damped nonlinear Schrödinger flow in L2(R) provides a dynamical system which possesses a global attractor. The proof relies on the continuity of the Schrödinger flow for the weak topology in L2(R).

We study the existence, uniqueness and boundary profile of nonnegative boundary blow-up solution to the cooperative system in a smooth bounded domain of RN, where f, g are nondecreasing, nonnegative C1 functions vanishing in (−∞,0] and β>0 is a parameter.

We prove that the weakly damped Davey-Stewartson system (DS), considered as an infinite-dimensional dynamical system in H 1(ℝ 2), has a compact global attractor that is actually a compact subset of H 2(ℝ 2).

To model the invasion of Prunus serotina invasion within a real forest landscape we built a spatially explicit, non-linear Markov chain which incorporated a stage-structured population matrix and dispersal functions. Sensitivity analyses were subsequently conducted to identify key processes controlling the spatial spread of the invader, testing the...

In this article we investigate the possibility of finite time blow-up in H1(R2) for solutions to critical and supercritical nonlinear Schrödinger equations with an oscillating nonlinearity. We prove that despite the oscillations some solutions blow up in finite time. Conversely, we observe that for a given initial data oscillations can extend the l...

In this article, we consider the two-dimensional dissipative Boussi-nesq systems which model surface waves in three space dimensions. The long time asymptotics of the solutions for a large class of such systems are obtained rigorously for small initial data.

We apply a semi-discrete in time relaxation scheme to a weakly damped forced nonlinear Schrodinger system. This provides us with a discrete infinite-dimensional dynamical system. We prove the existence of a global attractor for this dynamical system.

We consider a semi-discrete in time Crank-Nicolson scheme to discretize a damped forced nonlinear Schrödinger equation. This provides us with a discrete infinite-dimensional dynamical system. We prove the existence of a finite dimensional global attractor for this dynamical system.

The effect of environmental heterogeneity on spatial spread of invasive species has received little attention in the literature. Altering landscape heterogeneity may be a suitable strategy to control invaders in man-made landscapes. We use a population-based, spatially realistic matrix model to explore mechanisms underlying the observed invasion pa...

We introduce a Crank-Nicolson scheme to study numerically the long- time behavior of solutions to a one dimensional damped forced nonlinear Schrodinger equation. We prove the existence of a smooth global attractor for these discretized equations. We also provide some numerical evidences of this asymptotical smoothing effect.

In this article we prove that the global attractor for a weakly damped nonlinear Schrodinger equation is smooth i.e. it is made of smooth functions when the forcing term is smooth. The proof of this result which is well-known for other dissipative equations does not apply to dispersive equations for which the dissipation is on the low order term. A...

We consider the so-called Gross-Pitaevskii equations supplemented with non-standard boundary conditions. We prove two mathematical results concerned with the initial value problem for these equations in Zhidkov spaces.

This article is concerned with the existence, uniqueness and numerical approximation of boundary blow up solutions for elliptic PDE's as $\Delta u=f(u)$ where $f$ satisfies the so-called Keller-Osserman condition. We {\bleu characterize} existence of such solutions {\bleu for non-monotone $f$} . As an example, we construct an infinite family of bou...

Biological invasions are widely accepted as having a major impact on ecosystem functioning worldwide, giving urgency to a better understanding of the factors that control their spread. Modelling tools have been developed for this purpose but are often discrete-space, discrete-time spatial-mechanistic models that adopt a computer simulation approach...

An abstract functional framework is developed for the dual Petrov-Galerkin formulation of the initial-boundary-value problems with a third-order spatial derivative. This framework is then applied to study the wellposedness and decay properties of the KdV equation in a finite interval.

In this paper, we study various dissipative mechanics associated with the Boussinesq systems which model two-dimensional small amplitude long wavelength water waves. We will show that the decay rate for the damped one-directional model equations, such as the KdV and BBM equations, holds for some of the damped Boussinesq systems which model two-dire...

In this article we consider the Boussinesq system supplemented with some dissipation terms. These equations model the propagation of a waterwave in shallow water. We prove the existence of a global smooth attractor for the corresponding dynamical system.

We perform here some meshfree methods to inhomogeneous Laplace equations. We prove the efficiency of those methods compared with classical ones, for one- or two-dimensional case for numerics, and for one-dimensional for theoretical results. Copyright © 2006 John Wiley & Sons, Ltd.

Nous dressons dans cet article un panorama (non exhaustif) de l'´etude de l'´equation de Korteweg-de Vries, en presence d'un amortissement et d'un terme de force. Les questions qui nous interessent sont lieesa l'existence d'un attracteur global pour le systeme dynamique ainsi defini, `a sa dimension, eta sa regularite.

We prove the existence of a global attractor for a damped-forced Kadomtsev-Petviashvili equation u t +u xxx -v y +uu x -νu xx =f,u x =u y · We also establish that this equation features an asymptotic smoothing effect. We use energy estimates in conjunction with a suitable splitting of the solutions.

It is well known that the coupled Klein-Gordon-Schrödinger system possesses a compact global attractor into a suitable energy space. We prove the asymptotical smoothing effect for this system, i.e., we prove that the attractor is in fact embedded into a smaller energy space.

The existence of the global attractor of a weakly damped, forced Korteweg–de Vries equation in the phase space L2() is proved. An optimal asymptotic smoothing effect of the equation is also shown, namely, that for forces in L2(), the global attractor in the phase space L2() is actually a compact set in H3(). The energy equation method is used in co...

The existence of the global attractor of a weakly damped, forced Korteweg-de Vries equation in the phase space L (R) is proved. An optimal asymptotic smoothing eect of the equation is also shown, namely, that for forces in L (R), the global attractor in the phase space L (R) is actually a compact set in H (R). The energy equation method is used in...

We prove that the global attractor for a weakly damped two-dimensional nonlinear Schrödinger equation in the usual energy space is in fact included and compact in a more regular energy space. The method relies on a suitable approximation, when time goes to infinity, of the high-frequency modes of the solutions.

The weakly damped forced KdV equation u t +γu+u xxx +uu x =f, provides a dissipative semigroup on L x 2 . We prove that this semigroup enjoys an asymptotic smoothing effect, i.e. that all solutions convergence towards a set of smoother solutions, when time goes to infinity.

In this article we prove the existence of an attractor for a dissipative nonlinear Schrödinger equation in the critical case for a two-dimensional thin domain. Moreover we prove that this attractor is smooth, i.e., made of smooth functions when the forcing term is smooth enough. The proofs use a splitting of the Fourier series of the solutions acco...

We prove that the global attractor for a weakly damped nonlinear Schrödinger equation in a suitable energy space is in fact included and compact in a more regular energy space.

We construct several approximate inertial manifolds for a weakly damped nonlinear Schrödinger equation. For that purpose, we introduce suitable smooth approximations for the solution of the equation and for its time derivatives.

We prove that the global attractor for a weakly damped nonlinear Schrödinger equation is smooth, i.e., it is made of smooth functions when the forcing term is smooth. Our study relies on a new approach that works for dispersive equations that are weakly dissipative, i.e., for equations for which the damping is on the low-order term.

In this paper we establish that the pressure gradient and the flux, for a linear stationary Stokes problem for general periodic two-dimensional channels, are related by a simple formula, the same as that for laminar Poiseuille flows.

In this article we study the operator which appears in the Usawa algorithm classically used in the solution of the Stokes problem, in particular when finite element discretisations are used (see [1],[11]). We call this operator the Usawa operator. We consider S (and-Δ) as a self–adojoint operator in L2Ωfor several domains Ωwith a particular emphasi...

In this work we develop a number of functional tools related to the approximation of a function obtained considering its L ² -orthogonal projections onto a nested family of finite element spaces. We use these tools to prove the equivalence between a discrete norm related to this decomposition, and the usual norm. This result does not depend on the...

The aim is to describe a method for the multiscale approximation of the Stokes problem. We first use a transformation of variables to substitute for this problem two unconstrained optimization problems. We then describe a finite element multiscale approximation of these problems. It turns out that this approximation allows us to approximate the sol...

Approximate inertial manifolds are constructed for a class of dissipative evolution equations. The innovation is that these manifolds are defined as graphs on orthonormal wavelet bases.

In this paper, we study various dissipative mechanics associated with the Boussinesq systems which model two-dimensional small amplitude long wavelength water waves. We will show that the decay rate for the damped one-directional model equations, such as the KdV and BBM equations, holds for some of the damped Boussinesq systems.