Microlocal smoothing effect for the Schrödinger evolution equation in a Gevrey class

We discuss the microlocal Gevrey smoothing effect for the Schrödinger equation with variable coefficients via the propagation property of the wave front set of homogenous type. We apply the microlocal exponential estimates in a Gevrey case to prove our result.     

Nekhoroshev estimates for finitely differentiable quasi-convex Hamiltonians

A major result about perturbations of integrable Hamiltonian systems is the Nekhoroshev theorem, which gives exponential stability for all solutions provided the system is analytic and the integrable Hamiltonian is generic. In the particular but important case where the latter is quasi-convex, these exponential estimates have been generalized by Marco and Sauzin if the Hamiltonian is Gevrey regular, using a method introduced by Lochak in the analytic case. In this paper, using the same approach, we investigate the situation where the Hamiltonian is assumed to be only finitely differentiable, for which it is known that exponential stability does not hold but nevertheless we prove estimates of polynomial stability.     

Hyperbolic equations, function spaces with exponential weights and Nemytskij operators

Different problems in the theory of hyperbolic equations bases on function spaces of Gevrey type are studied. Beside the original Gevrey classes, spaces defined by the behaviour of the Fourier transform were also used to prove basic results about the well-posedness of Cauchy problems for non-linear hyperbolic systems. In these approaches only the algebra property of the function spaces was used to include analytic non-linearities. Here we will generalize this dependence. First we investigate superposition operators in spaces with exponential weights. Then we show in concrete situations how a priori estimates of strictly hyperbolic type lead to the well-posedness of certain semi-linear hyperbolic Cauchy problems in suitable function spaces with exponential weights of Gevrey type. Mathematics Subject Classification (2000) 46E35, 35L80, 35L15, 47H30     

Direct and inverse theorems of approximate methods for the solution of an abstract Cauchy problem

We consider an approximate method for the solution of the Cauchy problem for an operator differential equation based on the expansion of the exponential function in orthogonal Laguerre polynomials. For an initial value of finite smoothness with respect to the operator A, we prove direct and inverse theorems of the theory of approximation in the mean and give examples of the unimprovability of the corresponding estimates in these theorems. We establish that the rate of convergence is exponential for entire vectors of exponential type and subexponential for Gevrey classes and characterize the corresponding classes in terms of the rate of convergence of approximation in the mean. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 6, pp. 838–852, June, 2007.     

Remarks On The Regularity Of Weak Solutions Of The Navier–Stokes Equations

In this paper we extend the results of Foias–Guillopé–Temam on the regularity and a priori estimates for the weak solutions of the Navier–Stokes equations. More specifically, we obtain upperbounds for the temporal averages of the Gevrey class norm for the weak solutions of the equations. The estimates are obtained first by getting integrated version of Foias–Temam's local in time estimate for Gevrey class norms of strong solutions and next by an induction procedure. We also strengthen slightly the local in time Gevrey class regularization of strong solutions.     

Propagation des singularités Gevrey pour l’équation de Cauchy-Riemann dégénérée

In this paper we study the degenerate Cauchy-Riemann equation in Gevrey classes. We first prove the local solvability in Gevrey classes of functions and ultra-distributions. Using microlocal techniques with Fourier integral operators of infinite order and microlocal energy estimates, we prove a result of propagation of singularities along one dimensional bicharacteristics.      

Sub-exponential decay of operator kernels for functions of generalized Schrödinger operators

We study the decay at large distances of operator kernels of functions of generalized Schrödinger operators. We prove sub-exponential decay for functions in Gevrey classes and exponential decay for real analytic functions.

     

Exponential averaging for traveling wave solutions in rapidly varying periodic media

Reaction diffusion systems on cylindrical domains with terms that vary rapidly and periodically in the unbounded direction can be analyzed by averaging techniques. Here, using iterated normal form transformations and Gevrey regularity of bounded solutions, we prove a result on exponential averaging for such systems, i.e., we show that traveling wave solutions can be described by a spatially homogenous equation and exponentially small remainders. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Global $$L^q$$-Gevrey Functions and Their Applications

The goal of this paper is to introduce a class of \(C^\infty \) functions derivatives of which satisfy quantitative size estimates. The estimates, called global \(L^q\) Gevrey estimates, first arose in the work of Boggess and Raich ( J. Fourier Anal. Appl. 19:180–224, 2013) when they investigated how to capture a particular type of exponential decay through estimates on the Fourier transform. In the present work, we refine the notion of global \(L^q\)-Gevrey functions and include a discussion of the function theory as well as the relationship to Gevrey classes and known function spaces. In addition, we present explicit examples of global \(L^q\)-Gevrey functions and ways to generate new global \(L^q\)-Gevrey functions from old ones. We conclude with three applications: The first is solving a Carleman-type problem for constructing functions derivatives of which are a given sequence of global \(L^q\)-Gevrey functions. The other two applications concern extensions of a given global \(L^q\)-Gevrey function: the first is constructing an almost analytic extension, and the second is building an approximate solution to a first-order complex vector field coefficients of which are global \(L^q\)-Gevrey functions.     

On the critical index of Gevrey solvability for some linear partial differential equations

We propose precise estimates of the critical index of Gevrey solvability of some classes of linear partial differential equations with multiple characteristics.

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Sunto Si propongono stime precise dell’indice critico della risolubilità di Gevrey di alcune classi di equazioni lineari differenziali con caratteristiche multiple.

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To the memory of L. Cattabriga

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Research partially supported by a Coordinated Research Project of the University of Cagliari and by INDAM-GNAFA, Italy.     

On Landau damping

Going beyond the linearized study has been a longstanding problem in the theory of Landau damping. In this paper we establish exponential Landau damping in analytic regularity. The damping phenomenon is reinterpreted in terms of transfer of regularity between kinetic and spatial variables, rather than exchanges of energy; phase mixing is the driving mechanism. The analysis involves new families of analytic norms, measuring regularity by comparison with solutions of the free transport equation; new functional inequalities; a control of non-linear echoes; sharp “deflection” estimates; and a Newton approximation scheme. Our results hold for any potential no more singular than Coulomb or Newton interaction; the limit cases are included with specific technical effort. As a side result, the stability of homogeneous equilibria of the non-linear Vlasov equation is established under sharp assumptions. We point out the strong analogy with the KAM theory, and discuss physical implications. Finally, we extend these results to some Gevrey (non-analytic) distribution functions.     

Gevrey Class Regularity and Exponential Decay Property for Navier-Stokes-α Equations

The Navier-Stokes-α equations subject to the periodic boundary conditions are considered.An-alyticity in time for a class of solutions taking values in a Gevrey class of functions is proven.Exponentialdecay of the spatial Fourier spectrum for the analytic solutions and the lower bounds on the rate defined by theexponential decay are also obtained.     

On the periodic KdV equation in weighted Sobolev spaces

We prove well-posedness results for the initial value problem of the periodic KdV equation as well as Kam type results in classes of high regularity solutions. More precisely, we consider the problem in weighted Sobolev spaces, which comprise classical Sobolev spaces, Gevrey spaces, and analytic spaces. We show that the initial value problem is well posed in all spaces with subexponential decay of Fourier coefficients, and ‘almost well posed’ in spaces with exponential decay of Fourier coefficients.     

Approximation of an Inverse Initial Problem for a Biparabolic Equation

In this paper, we consider the problem of finding the initial distribution for the linear inhomogeneous and nonlinear biparabolic equation. The problem is severely ill-posed in the sense of Hadamard. First, we apply a general filter method to regularize the linear nonhomogeneous problem. Then, we also give a regularized solution and consider the convergence between the regularized solution and the sought solution. Under the a priori assumption on the exact solution belonging to a Gevrey space, we consider a generalized nonlinear problem by using the Fourier truncation method to obtain rigorous convergence estimates in the norms on Hilbert space and Hilbert scale space.     

Rapidly convergent iteration method for simultaneous normal forms of commuting maps

We propose a new rapidly convergent iteration scheme under weak simultaneous arithmetic conditions for overdetermined systems related to simultaneous conjugation of commuting maps. The first application is for commuting holomorphic maps in fixing the origin, not necessarily diffeomorphisms if . The second one is for orientation preserving Gevrey circle mappings by means of nonlinear superposition estimates. Received January 7, 1998; in final form July 28, 1998     

Temps de vie des solutions d'un problème de Cauchy non linéaire

We solve the global Cauchy problem, with small initial data, in the space of the holomorphic functions with respect to t and Gevrey class with respect to x. We establish the existence and the stability of the solution to Cauchy problem with nul initial data without hyperbolicity hypothesis. In the stationary case, we give estimates of life span of the solutions with respect to size of the initial data.     

Gevrey properties of real planar singularly perturbed systems

By applying geometric techniques to real analytic singularly perturbed vector fields on the plane, we develop a way to give a bound on the Gevrey type of the Taylor development of canard manifolds at degenerate planar turning points. By blowing up the phase space at the turning point, we find asymptotic estimates even when such expansions w.r.t. traditional phase space variables do not exist. The asymptotic estimates are then used to give a sufficient and necessary condition on the existence of (local) canard solutions.     

Gevrey Smoothness of Families of Invariant Curves for Analytic Area Preserving Mappings

In this paper we prove the existence of a Gevrey family of invariant curves for analytic area preserving mappings. The Gevrey smoothness is expressed by Gevrey index. We specifically obtain the Gevrey index of families of invariant curves which is related to the smoothness of area preserving mappings and the exponent of small divisors condition. Moreover, we obtain a Gevrey normal form of area preserving mappings in a neighborhood of the union of the invariant curves.     

Gain of analyticity for semilinear Schrödinger equations

We discuss gain of analyticity phenomenon of solutions to the initial value problem for semilinear Schrödinger equations with gauge invariant nonlinearity. We prove that if the initial data decays exponentially, then the solution becomes real-analytic in the space variable and a Gevrey function of order 2 in the time variable except in the initial plane. Our proof is based on the energy estimates developed in our previous work and on fine summation formulae concerned with a matrix norm.     

Global s-solvability, global s-hypoellipticity and Diophantine phenomena

In this paper we consider the problem of global Gevrey solvability for a class of sublaplacians on a toruswith coefficients in the Gevrey class Gs(TN). For this class of operators we show that global Gevrey solvability and global Gevrey hypoellipticity are both equivalent to the condition that the coefficients satisfy a Diophantine condition.