On the controllability of the Vlasov-Poisson system

In this paper, we study the controllability of the Vlasov-Poisson system in a periodic domain, by means of an interior control located in an spatial subdomain.The first result proves the local exact zero controllability in the two-dimensional torus between two small acceptable distribution functions, with an arbitrary control zone.A second result establishes the global exact controllability in arbitrary dimension, provided the control zone satisfies the condition that it contains a hyperplane of the torus.     

ON THE ASYMPTOTIC DYNAMICS OF THE VLASOV-YUKAWA-BOLTZMANN SYSTEM NEAR VACUUM

In this paper, we study uniform L1-stability and asymptotic completeness of the Vlasov-Yukawa-Boltzmann(V-Y-B) system. For a sufficiently small and smooth initial data, we show that classical solutions exist globally and satisfy dispersion estimates, uniform L1-stability with respect to initial data and scattering type estimate. We show that the short range nature of interactions due to the Yukawa potential enables us to construct robust Lyapunov type functional to derive scattering states. In the zero mass limit of force carrier particles, we also show that the classical solutions to the V-Y-B system converge to that of the Vlasov-Poisson-Boltzmann(V-P-B) system in any finite time interval.     

Global small amplitude solutions for two-dimensional nonlinear Klein-Gordon systems in the presence of mass resonance

We consider a nonlinear system of two-dimensional Klein-Gordon equations with masses m₁, m₂ satisfying the resonance relation m₂=2m₁>0. We introduce a structural condition on the nonlinearities under which the solution exists globally in time and decays at the rate O(|t|⁻¹) as t→±∞ in L^∞. In particular, our new condition includes the Yukawa type interaction, which has been excluded from the null condition in the sense of J.-M. Delort, D. Fang and R. Xue [J.-M. Delort, D. Fang, R. Xue, Global existence of small solutions for quadratic quasilinear Klein-Gordon systems in two space dimensions, J. Funct. Anal. 211 (2004) 288-323].     

Classical solutions to the Vlasov-Poisson system in an accelerating cosmological setting

In this paper, we study the Vlasov-Poisson system in an accelerating cosmological setting. A homogeneous and isotropic background solution, which describes an expanding universe in an acceleration, is constructed explicitly, and then its perturbed solutions are investigated to obtain global existence of classical solutions and the uniform L¹-stability estimates.     

On the temporal decay for the Hall-magnetohydrodynamic equations

We establish temporal decay estimates for weak solutions to the Hall-magnetohydrodynamic equations. With these estimates in hand we obtain algebraic time decay for higher order Sobolev norms of small initial data solutions.     

A non-autonomous two-phase flow model with oscillating external force and its global attractor

In this article, we consider a non-autonomous diffuse interface model for an isothermal incompressible two-phase flow in a two-dimensional bounded domain. Assuming that the external force is singularly oscillating and depends on a small parameter ?, we prove the existence of the uniform global attractor A^?. Furthermore, using the method similar to that of Chepyzhov and Vishik (2007) [22] in the case of the two-dimensional Navier-Stokes systems, we study the convergence of A^? as ? goes to zero. Let us mention that the nonlinearity involved in the model considered in this article is slightly stronger than the one in the two-dimensional Navier-Stokes system studied in Chepyzhov and Vishik (2007) [22].     

Convergence of Galerkin approximation for two-dimensional Vlasov-Poisson equation

We give explicit estimates on the rate of convergence of the solutions of finite dimensional truncations (by means of Fourier-Hermite expansion) of Vlasov-Poisson equation in a two-dimensional flat torus.

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Zusammmenfassung Wir geben eine explizite Abschätzung der Konvergenzgeschwindigkeit einer endlichdimensionalen Fourier-Hermite-Entwicklung gegen die Lösung der Vlasov-Poisson-Gleichung für den räumlich 2-dimensionalen periodischen Fall an.

     

On the Boussinesq/Full dispersion systems and Boussinesq/Boussinesq systems for internal waves

In this paper, we consider some asymptotic models for internal waves in the small amplitude/small amplitude regime, which were derived recently by Bona, Lannes and Saut. We first prove that the Boussinesq/Full dispersion systems and the Boussinesq/Boussinesq systems can be derived from the Full dispersion/Full dispersion systems. Then using a contraction-mapping argument and the energy method, we will prove that the derived systems that are linearly well-posed are in fact locally nonlinearly well-posed in suitable Sobolev classes. In particular, we improve and extend some known results on the well-posedness of Boussinesq systems for surface waves.     

Parabolic equations with BMO coefficients in Lipschitz domains

In this paper, we are concerned with certain natural Sobolev-type estimates for weak solutions of inhomogeneous problems for second-order parabolic equations in divergence form. The geometric setting is that of time-independent cylinders having a space intersection assumed to be locally given by graphs with small Lipschitz coefficients, the constants of the operator being uniformly parabolic. We prove the relevant L^p estimates, assuming that the coefficients are in parabolic bounded mean oscillation (BMO) and that their parabolic BMO semi-norms are small enough.     

Global s-solvability and global s-hypoellipticity for certain perturbations of zero order of systems of constant real vector fields

The main purpose of this paper is to show, in the two-dimensional torus, a necessary and sufficient condition in order to certain perturbations of zero order of a system of constant real vector fields to be globally s-solvable. We are also interested in studying its global s-hypoellipticity. We present connections between these global concepts and a priori estimates. We also present two applications of our results for systems of operators with variable coefficients.     

Weighted Clément operator and application to the finite element discretization of the axisymmetric Stokes problem

We consider the Stokes problem in an axisymmetric three-dimensional domain with data which are axisymmetric and have angular component equal to zero. We observe that the solution is also axisymmetric and the velocity has also zero angular component, hence the solution satisfies a system of equations in the meridian domain. The weak three-dimensional problem reduces to a two-dimensional one with weighted integrals. The latter is discretized by Taylor–Hood type finite elements. A weighted Clément operator is defined and approximation results are proved. This operator is then used to derive the discrete inf–sup condition and optimal a priori error estimates.     

On space-time estimates for the Vlasov-Poisson system

The motion of a collisionless plasma is described by the Vlasov-Poisson system. A space-time identity for one-dimensional plasmas with two species of charge is here generalized to higher dimensions with spherical symmetry. Some estimates on large time behavior are obtained.     

The pointwise estimates of solutions for a nonlinear convection diffusion reaction equation

This paper studies the time asymptotic behavior of solutions for a nonlinear convection diffusion reaction equation in one dimension.First,the pointwise estimates of solutions are obtained,furthermore,we obtain the optimal L~p,1≤ p ≤ +∞,convergence rate of solutions for small initial data.Then we establish the local existence of solutions,the blow up criterion and the sufficient condition to ensure the nonnegativity of solutions for large initial data.Our approach is based on the detailed analysis of the Green function of the linearized equation and some energy estimates.     

Growth Estimates for the Solutions of the Vlasov-Poisson System in the Plasma Physics Case

In [3] new a—priori—estimates for solutions of the Vlasov-Poisson system in the plasma physics case were obtained. We show that these lead to improved estimates on the possible growth of the solutions.     

Strichartz estimates and local smoothing estimates for asymptotically flat Schrödinger equations

In this article we study global-in-time Strichartz estimates for the Schrödinger evolution corresponding to long-range perturbations of the Euclidean Laplacian. This is a natural continuation of a recent article [D. Tataru, Parametrices and dispersive estimates for Schrödinger operators with variable coefficients, Amer. J. Math. 130 (2008) 571-634] of the third author, where it is proved that local smoothing estimates imply Strichartz estimates. By [D. Tataru, Parametrices and dispersive estimates for Schrödinger operators with variable coefficients, Amer. J. Math. 130 (2008) 571-634] the local smoothing estimates are known to hold for small perturbations of the Laplacian. Here we consider the case of large perturbations in three increasingly favorable scenarios: (i) without non-trapping assumptions we prove estimates outside a compact set modulo a lower order spatially localized error term, (ii) with non-trapping assumptions we prove global estimates modulo a lower order spatially localized error term, and (iii) for time independent operators with no resonance or eigenvalue at the bottom of the spectrum we prove global estimates for the projection onto the continuous spectrum.     

On the asymptotic growth of the solutions of the vlasov–poisson system

We show the global existence of classical solutions of the Vlasov-Poisson system and improve the known growth estimates.     

Existence and uniqueness analysis of a detached shock problem for the potential flow

We study a problem for two-dimensional steady potential and isentropic Euler equations in a bounded domain, where an artificial detached shock interacts with a wedge. Using the stream function, we obtain a free boundary problem for the subsonic state and the detached artificial shock curve and we prove that such configuration admits a unique solution in certain weighted Hölder spaces. The proof is based on various Hölder and Schauder estimates for second-order elliptic equations and fixed point theorems. Moreover, we pose an energy principle and remark that the physical attached shock is the minimizer of the energy functional.     

Weak solutions of the Vlasov–Poisson initial boundary value problem

This paper deals with existence results for a Vlasov-Poisson system, equipped with an absorbing-type law for the Vlasov equation and a Dirichlet-type boundary condition for the Poisson part. Using the ideas of Lions and Perthame [21], we prove the existence of a weak solution having good L^p estimates for moment and electric field, by a good control on the higher moments of the initial data. As an application, we establish a homogenization result in the Hilbertian framework for this type of problem in non-homogeneous media, following the work by Alexandre and Hamdache [2] for general kinetic equations, and Cioranescu and Mural [11] for the Laplace problem.     

The Vlasov Poisson System with Steady Spatial Asymptotics

Abstract

A collisionless plasma is modeled by the Vlasov-Poisson system in three space dimensions. A fixed background of positive charge, which is independent of time and space, is assumed. The situation in which mobile negative ions balance the positive charge as |x|→∞ is considered. Hence the total positive charge and the total negative charge are infinite. Smooth solutions with appropriate asymptotic behavior are shown to exist locally in time. Conditions for continuation of these solutions are also established.     

Parabolic equations in time dependent Reifenberg domains

In this paper we define time dependent parabolic Reifenberg domains and study Lp estimates for weak solutions of uniformly parabolic equations in divergence form on these domains. The basic assumption is that the principal coefficients are of parabolic BMO space with small parabolic BMO seminorms. It is shown that Lp estimates hold for time dependent parabolic δ-Reifenberg domains.