1,721,066 research outputs found
Long time behavior of certain Vlasov equations : mathematics and numerics
No full textCette thèse porte sur le comportement en temps long de solutions d’équations de type Vlasov, principalement le modèle Vlasov-HMF. On s’intéresse en particulier au phénomène d’amortissement Landau, prouvé mathématiquement dans divers cadres, pour plusieurs équations de type Vlasov, comme l’équation de Vlasov-Poisson ou le modèle Vlasov-HMF, et présentant certaines analogies avec le phénomène d’amortissement non visqueux pour l’équation d’Euler 2D. Les résultats qui y sont décrits sont les suivants. Le premier est un théorème d’amortissement Landau pour des solutions numériques du modèle Vlasov-HMF, obtenues par discrétisation en temps de ce dernier via des méthodes de splitting. Nous prouvons en outre la convergence des schémas numériques. Le second est un théorème d’amortissment Landau pour des solutions du modéle Vlasov-HMF linéarisé autour d’états stationnaires inhomogènes. Ce théorème est accompagné de nombreuses simulations numériques destinées à étudier numériquement le cas non-linéaire, et semblant mettre en lumière de nouveaux phénomènes. Enfin, le dernier résultat porte sur la discrétisation en temps de l’équation d’Euler 2D par un intégrateur de Crouch-Grossman symplectique. Nous prouvons la convergence du schéma.This thesis concerns the long time behavior of certain Vlasov equations, mainly the Vlasov- HMF model. We are in particular interested in the celebrated phenomenon of Landau damp- ing, proved mathematically in various frameworks, foar several Vlasov equations, such as the Vlasov-Poisson equation or the Vlasov-HMF model, and exhibiting certain analogies with the inviscid damping phenomenon for the 2D Euler equation. The results described in the document are the following.The first one is a Landau damping theorem for numerical solutions of the Vlasov-HMF model, constructed by means of time-discretizations by splitting methods. We prove more- over the convergence of the schemes. The second result is a Landau damping theorem for solutions of the Vlasov-HMF model linearized around inhomogeneous stationary states. We provide moreover a quite large amount of numerical simulations, which are designed to study numerically the nonlinear case, and which seem to show new phenomenons. The last result is the convergence of a scheme that discretizes in time the 2D Euler equation by means of a symplectic Crouch-Grossmann integrator
Around Schrödinger-Langevin equation and isothermal Euler system with damping
No full textOn s’intéresse dans cette thèse à la dynamique de l’équation de Schrödinger-Langevin et son lien avec le système d’Euler-Korteweg isotherme amortie via la transformation de Madelung. L’étude des solutions particulières gaussiennes sur l’espace permets d’expliciter le comportement en temps long des solutions de cette équation. Sur le tore, on montre la stabilité asymptotique des solutions de type onde plane. L’existence de solutions dissipatives au système d’Euler est obtenue par limite visqueuse du système de Navier-Stokes-Korteweg avec amortissement et la construction d’une entropie relative adéquate. On étudie également la dynamique du système d’Euler isotherme amortie.This manuscript deals with the dynamics of the Schrödinger-Langevin equation and how it is related with the isothermal Euler-Korteweg system using the Madelung transform. The study of Gaussian solutions on the whole space highlights the long-time behaviour of the solutions of this equation. On the torus, we show the asymptotic stability of plane waves solutions. The global existence of dissipative solutions of the Euler system is obtained through the viscous limit of the damped Navier-Stokes-Korteweg system and the use of a particular relative entropy. We also look at the dynamic of the Euler system with damping
Study of some equations with many symmetries : resonances and stability
No full textCette thèse est un recueil de constructions et de résultats variés autour de problèmes de résonances et de stabilités. Premièrement, on s'intéresse à la conception et à l'analyse de méthodes numériques pour des problèmes académiques tels que le problème de Dirichlet sur un segment ou l'équation de transport associée à une rotation du plan. Ensuite, on étend l'analyse linéaire classique des équations de Vlasov-Poisson autour d'états d'équilibre homogènes pour décrire des phénomènes multidimensionnels et non linéaires. Enfin, une large partie est consacrée à l'étude d'équations de Schrödinger non linéaires en dimension 1. D'une part, on étudie l'impact d'une semi-discrétisation naturelle sur les ondes solitaires progressives et la croissance des normes de Sobolev. D'autre part, on développe une nouvelle famille de formes normales permettant de décrire la dynamique des petites solutions régulières pendant des temps très longs.This manuscript deals with many problems about resonance and stability. First, we design and analyse numerical methods for academic problems like the Dirichlet problem on a segment line or the transport equation associated with a two dimensional rotation. Then, we extend the classical linear analysis of Vlasov-Poisson equations near homogeneous equilibria to describe nonlinear and multidimensional phenomena. Finally, a large part of this thesis is devoted to nonlinear Schrödinger equations in dimension 1. On the one hand, we study the impact of a natural semi-discretisation on the solitary traveling waves and on the growth of the high order Sobolev norms. On the other hand, we develop a new family of normal forms to describe the dynamic of small and smooth solutions for very long times
Elasticity on a Thin Shell: Formal Series Solution
No full textThe three-dimensional equations of elasticity are posed on a domain of R^3 defining a thin shell of thickness 2ε. The traction free conditions are imposed on the upper and lower faces together with the clamped boundary conditions on the lateral boundary. After a scaling in the transverse variable, the elasticity operator admits a power series expansion in with intrinsic coefficients with respect to the mean surface of the shell. This leads to define a formal series problem in associated with the three-dimensional equations. The main result is the reduction of this problem to a formal series boundary value problem posed on the mean surface of the shell
Linearized wave turbulence convergence results for three-wave systems
Full text linkInternational audienceWe consider stochastic and deterministic three-wave semi-linear systems with bounded and almost continuous set of frequencies. Such systems can be obtained by considering nonlinear lattice dynamics or truncated partial differential equations on large periodic domains. We assume that the nonlinearity is small and that the noise is small or void and acting only in the angles of the Fourier modes (random phase forcing). We consider random initial data and assume that these systems possess natural invariant distributions corresponding to some Rayleigh-Jeans stationary solutions of the wave kinetic equation appearing in wave turbulence theory. We consider random initial modes drawn with probability laws that are perturbations of theses invariant distributions. In the stochastic case, we prove that in the asymptotic limit (small nonlinearity, continuous set of frequency and small noise), the renormalized fluctuations of the amplitudes of the Fourier modes converge in a weak sense towards the solution of the linearized wave kinetic equation around these Rayleigh-Jeans spectra. Moreover, we show that in absence of noise, the deterministic equation with the same random initial condition satisfies a generic Birkhoff reduction in a probabilistic sense, without kinetic description at least in some regime of parameters
Going Beyond Counting First Authors in Author Co-citation Analysis
Full text linkThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
Linearized wave turbulence convergence results for three-wave systems
No full textWe consider stochastic and deterministic three-wave semi-linear systems with bounded and almost continuous set of frequencies. Such systems can be obtained by considering nonlinear lattice dynamics or truncated partial differential equations on large periodic domains. We assume that the nonlinearity is small and that the noise is small or void and acting only in the angles of the Fourier modes (random phase forcing). We consider random initial data and assume that these systems possess natural invariant distributions corresponding to some Rayleigh-Jeans stationary solutions of the wave kinetic equation appearing in wave turbulence theory. We consider random initial modes drawn with probability laws that are perturbations of theses invariant distributions. In the stochastic case, we prove that in the asymptotic limit (small nonlinearity, continuous set of frequency and small noise), the renormalized fluctuations of the amplitudes of the Fourier modes converge in a weak sense towards the solution of the linearized wave kinetic equation around these Rayleigh-Jeans spectra. Moreover, we show that in absence of noise, the deterministic equation with the same random initial condition satisfies a generic Birkhoff reduction in a probabilistic sense, without kinetic description at least in some regime of parameters
Multiscale Expansions for Linear Clamped Elliptic Shells
No full textWe investigate solutions of the two-dimensional Koiter model and of the three-dimensional linear shell model in the case where the shell is clamped and its mean surface is elliptic. For smooth data, these solutions admit multiscale expansions in powers of ε^1/2 where ε denotes the (half-)thickness of the shell. Both expansions contain terms independent of ε and boundary layer terms exponentially decreasing with respect to r/√ε, with r the distance to the boundary of the mean surface. The expansion of the three-dimensional displacement contains supplementary boundary layers, exponentially decreasing with respect to r/ε like for plates. Using these expansions we obtain sharp estimates between the two models in various norms
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