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G. Velo

Publications and source records attributed to G. Velo.

At least 19 recordsLinked to original sources
arXiv2013

Modified wave operators without loss of regularity for some long range Hartree equations. II

We continue the study of the theory of scattering for some long range Hartree equations with potential |x|^-gamma, performed in a previous paper, denoted as I, in the range 1/2 < gamma < 1. Here we extend the results to the range 1/3 < gamma < 1/2. More precisely, we study the local Cauchy problem with infinite initial time, which is the main step in the construction of the modified wave operators. We solve that problem without loss of regularity between the asymptotic state and the solution, as in I, but in contrast to I, we are no longer able to cover the entire subcritical range of regularity of the solutions. The method is an extension of that of I, using a better asymptotic form of the solutions, obtained as the next step of a natural procedure of successive approximations.

math.AP↗
arXiv2012

Modified wave operators without loss of regularity for some long range Hartree equations. I

We reconsider the theory of scattering for some long range Hartree equations with potential |x|^-gamma with 1/2 < gamma < 1. More precisely we study the local Cauchy problem with infinite initial time, which is the main step in the construction of the modified wave operators. We solve that problem in the whole subcritical range without loss of regularity between the asymptotic state and the solution, thereby recovering a result of Nakanishi. Our method starts from a different parametrization of the solutions, already used in our previous papers. This reduces the proofs to energy estimates and avoids delicate phase estimates.

math.AP↗
arXiv2008

Quadratic Morawetz inequalities and asymptotic completeness in the energy space for nonlinear Schr"odinger and Hartree equations

Recently several authors have developed multilinear and in particular quadratic extensions of the classical Morawetz inequality. Those extensions provide (among other results) an easy proof of asymptotic completeness in the energy space for nonlinear Schr"odinger equations in arbitrary space dimension and for Hartree equations in space dimension greater than two in the noncritical cases. We give a pedagogical review of the latter results.

math.AP↗
arXiv2007

Uniqueness at infinity in time for the Maxwell-Schr"odinger system with arbitrarily large asymptotic data

We prove the uniqueness of solutions of the Maxwell-Schr"odinger system with given asymptotic behaviour at infinity in time. The assumptions include suitable restrictions on the growth of solutions for large time and on the accuracy of their asymptotics, but no restriction on their size. The result applies to the solutions with prescribed asymptotics constructed in a previous paper.

math.AP↗
arXiv2007

Long Range Scattering for the Modified Schr"odinger Map in two space dimensions

We study the asymptotic behaviour in time of solutions and the theory of scattering for the modified Schr"odinger map in two space dimensions. We solve the Cauchy problem with large finite initial time, up to infinity in time, and we determine the asymptotic behaviour in time of the solutions thereby obtained. As a byproduct, we obtain global existence for small data in H^k inter FH^k with k > 1. We also solve the Cauchy problem with infinite initial time, namely we construct solutions defined in a neighborhood of infinity in time, with prescribed asymptotic behaviour of the previous type.

math.AP↗
arXiv2006

Long Range Scattering and Modified Wave Operators for the Maxwell-Schr"odinger System II. The general case

We study the theory of scattering for the Maxwell-Schr"odinger system in space dimension 3, in the Coulomb gauge. We prove the existence of modified wave operators for that system with no size restriction on the Schr"odinger and Maxwell asymptotic data and we determine the asymptotic behaviour in time of solutions in the range of the wave operators. The method consists in partially solving the Maxwell equations for the potentials, substituting the result into the Schr"odinger equation, which then becomes both nonlinear and nonlocal in time. The Schr"odinger function is then parametrized in terms of an amplitude and a phase satisfying a suitable auxiliary system, and the Cauchy problem for that system, with prescribed asymptotic behaviour determined by the asymptotic data, is solved by an energy method, thereby leading to solutions of the original system with prescribed asymptotic behaviour in time. This paper is the generalization of a previous paper with the same title. However it is entirely selfcontained and can be read without any previous knowledge of the latter.

math.AP↗
arXiv2005

Scattering theory for the Zakharov system

We study the theory of scattering for the Zakharov system in space dimension 3. We prove in particular the existence of wave operators for that system with no size restriction on the data in larger spaces and for more general asymptotic states than were previously considered, and we determine convergence rates in time of solutions in the range of the wave operators to the solutions of the underlying linear system. We also consider the same system in space dimension 2, where we prove the existence of wave operators in the special case of vanishing asymptotic data for the wave field.

math.AP↗
arXiv2005

Long Range Scattering and Modified Wave Operators for the Wave-Schr"odinger System III

We continue the study of scattering theory for the system consisting of a Schr"odinger equation and a wave equation with a Yukawa type coupling in space dimension 3. In previous papers, we proved the existence of modified wave operators for that system with no size restriction on the data and we determined the asymptotic behaviour in time of the solutions in the range of the wave operators, first under a support condition on the Schr"odinger asymptotic state and then without that condition, but for solutions of relatively low regularity. Here we extend the latter result to the case of more regular solutions.

math.AP↗
arXiv2004

Long range scattering for some Schr"odinger related nonlinear systems

We review some recent results on the theory of scattering and more precisely on the local Cauchy problem at infinity in time for some long range nonlinear systems including some form of the Schr"odinger equation. We consider in particular the Wave-Schr"odinger system in space dimension 3, the Maxwell-Schr"odinger system in space dimension 3, the Klein-Gordon-Schr"odinger system in space dimension 2 and the Zakharov system in space dimensions 2 and 3. By the use of a direct method which is intrinsically restricted to the case of small Schr"odinger data and to the borderline long range case, one can prove the existence of solutions defined for large times and with prescribed asymptotic behaviour in time, without any size restriction on the Wave, Maxwell or Klein-Gordon data. Furthermore one obtains convergence rates of the solutions to their asymptotic forms as negative powers of t in suitable norms.

math.AP↗
arXiv2004

Scattering Theory for the Schr"odinger Equation in some external time dependent magnetic fields

We study the theory of scattering for a Schr"odinger equation in an external time dependent magnetic field in the Coulomb gauge, in space dimension 3. The magnetic vector potential is assumed to satisfy decay properties in time that are typical of solutions of the free wave equation, and even in some cases to be actually a solution of that equation. That problem appears as an intermediate step in the theory of scattering for the Maxwell-Schr"odinger (MS) system. We prove in particular the existence of wave operators and their asymptotic completeness in spaces of relatively low regularity. We also prove their existence or at least results going in that direction in spaces of higher regularity. The latter results are relevant for the MS system. As a preliminary step, we study the Cauchy problem for the original equation by energy methods, using as far as possible time derivatives instead of spaces derivatives.

math.AP↗
arXiv2004

Partially classical limit of the Nelson model

We consider the Nelson model which describes a quantum system of nonrelativistic identical particles coupled to a possibly massless scalar Bose field through a Yukawa type interaction. We study the limiting behaviour of that model in a situation where the number of Bose excitations becomes infinite while the coupling constant tends to zero in a suitable sense. In that limit the appropriately rescaled Bose field converges to a classical solution of the free wave or Klein-Gordon equation depending on whether the mass of the field is zero or not, the quantum fluctuations around that solution satisfy the wave or Klein-Gordon equation and the evolution of the nonrelativistic particles is governed by a quantum dynamics with an external potential given by the previous classical solution.

math-ph↗
arXiv2004

Long range scattering for the Wave-Schr"odinger system with large wave data and small Schr"odinger data

We study the theory of scattering for the Wave-Schr"odinger system with Yukawa type coupling in space dimension 3. We prove in particular the existence of modified wave operators for that system with no size restriction on the wave data in the framework of a direct method which requires smallnes of the Schr"odinger data, and we determine the asymptotic behaviour in time of solutions in the range of the wave operators.

math.AP↗
arXiv2004

Long range scattering for the Maxwell-Schr"odinger system with large magnetic field data and small Schr"odinger data

We study the theory of scattering for the Maxwell-Schr"odinger system in the Coulomb gauge in space dimension 3. We prove in particular the existence of modified wave operators for that system with no size restriction on the magnetic field data in the framework of a direct method which requires smallness of the Schr"odinger data, and we determine the asymptotic behaviour in time of solutions in the range of the wave operators.

math.AP↗
arXiv2003

Long Range Scattering and Modified Wave Operators for the Wave-Schr"odinger system II

We continue the study of scattering theory for the system consisting of a Schr"odinger equation and a wave equation with a Yukawa type coupling in space dimension 3. In a previous paper we proved the existence of modified wave operators for that system with no size restriction on the data and we determined the asymptotic behaviour in time of solutions in the range of the wave operators, under a support condition on the asymptotic state required by the different propagation properties of the wave and Schr"odinger equations.Here we eliminate that condition by using an improved asymptotic form for the solutions.

math.AP↗
arXiv2002

Long Range Scattering and Modified Wave Operators for the Maxwell-Schr"odinger system I.The case of vanishing asymptotic magnetic field

We study the theory of scattering for the Maxwell-Schr"odinger system in space dimension 3,in the Coulomb gauge.In the special case of vanishing asymptotic magnetic field,we prove the existence of modified wave operators for that system with no size restriction on the Schr"odinger data and we determine the asymptotic behaviour in time of solutions in the range of the wave operators.The method consists in partially solving the Maxwell equations for the potentials,substituting the result into the Schr"odinger equation,which then becomes both nonlinear and nonlocal in time,and treating the latter by the method previously used for the Hartree equation and for the Wave-Schr"odinger system.

math.AP↗
arXiv2001

Long Range Scattering and Modified Wave Operators for the Wave-Schr"odinger system

We study the theory of scattering for the system consisting of a Schr"odinger equation and a wave equation with a Yukawa type coupling,in space dimension 3.We prove in particular the existence of modified wave operators for that system with no size restriction on the data and we determine the asymptotic behaviour in time of solutions in the range of the wave operators.The method consists in solving the wave equation, substituting the result into the Schr"odinger equation,which then becomes both nonlinear and nonlocal in time,and treating the latter by the method previously used for a family of generalized Hartree equations with long range interactions.

math.AP↗
arXiv2000

Long Range Scattering and Modified Wave Operators for some Hartree Type Equations III. Gevrey spaces and low dimensions

We study the theory of scattering for a class of Hartree type equations with long range interactions in arbitrary space dimension n > or = 1, including the case of Hartree equations with time dependent potential V(t,x) = kappa t^(mu - gamma) |x|^{- mu} with 0 < gamma < or =1 and 0 < mu < n.This includes the case of potential V(x) = kappa |x|^(-gamma) and can be extended to the limiting case of nonlinear Schr"odinger equations with cubic nonlinearity kappa t^(n- gamma) u|u|^2.Using Gevrey spaces of asymptotic states and solutions,we prove the existence of modified local wave operators at infinity with no size restriction on the data and we determine the asymptotic behaviour in time of solutions in the range of the wave operators,thereby extending the results of previous papers (math.AP/9807031 and math.AP/9903073) which covered the range 0 < gamma < or = 1, but only 0 < mu < or = n-2, and were therefore restricted to space dimension n>2.

math.AP↗