Gevrey Functions Research Articles

Irregularities of semilinear hyperbolic equations

AbstractWe consider local solvability of semilinear hyperbolic Cauchy problems for Gevrey functions. To obtain a general result, we define the notion of irregularities, and we give a criterion for the local solvability. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Fractional derivative estimates in Gevrey spaces, global regularity and decay for solutions to semilinear equations in [formula omitted

We propose a unified functional analytic approach to study the uniform analytic-Gevrey regularity and the decay of solutions to semilinear elliptic equations on R n . First, we develop a fractional calculus for nonlinear maps in Banach spaces of L p based Gevrey functions, 1< p<∞. Then we propose an abstract result on uniform analytic Gevrey regularity, which covers as particular cases solitary wave solutions to both dispersive and dissipative equations. We require a priori low H p s( R n) regularity, with s> s cr>0 depending on the nonlinearity. Next, we investigate the type of decay—polynomial or exponential—of the derivatives of solutions to semilinear elliptic equations, provided they decay a priori slowly as o(| x| − τ ), | x|→∞ for some small τ>0. The restrictions, involved in our results, are optimal. In particular, given a hyperplane L, we construct 2 d−2 strongly singular solutions (locally in H p s( R n) for s< s cr) to the semilinear Laplace equation Δ u+ cu d =0, whose singularities are concentrated on L.

Stability and instability for Gevrey quasi-convex near-integrable Hamiltonian systems

Abstract. – We prove a theorem about the stability of action variables for Gevrey quasi-convex near-integrable Hamiltonian systems and construct in that context a system with an unstable orbit whose mean speed of drift allows us to check the optimality of the stability theorem.¶Our stability result generalizes those by Lochak-Neishtadt and Poschel, which give precise exponents of stability in the Nekhoroshev Theorem for the quasi-convex case, to the situation in which the Hamiltonian function is only assumed to belong to some Gevrey class instead of being real-analytic. For n degrees of freedom and Gevrey-α Hamiltonians, α ≥ 1, we prove that one can choose a = 1/2nα as an exponent for the time of stability and b = 1/2n as an exponent for the radius of confinement of the action variables, with refinements for the orbits which start close to a resonant surface (we thus recover the result for the real-analytic case by setting α = 1).¶On the other hand, for α > 1, the existence of compact-supported Gevrey functions allows us to exhibit for each n ≥ 3 a sequence of Hamiltonian systems with wandering points, whose limit is a quasi-convex integrable system, and where the speed of drift is characterized by the exponent 1/2(n−2)α. This exponent is optimal for the kind of wandering points we consider, inasmuch as the initial condition is located close to a doubly-resonant surface and the stability result holds with precisely that exponent for such an initial condition. We also discuss the relationship between our example of instability, which relies on a specific construction of a perturbation of a discrete integrable system, and Arnold’s mechanism of instability, whose main features (partially hyperbolic tori, heteroclinic connections) are indeed present in our system.

Hörmander's type conditions for evolution in spaces

We give Hormander's type necessary and/or sufficient conditions for the existence of solutions of the Cauchy problem for (overdetermined) systems of linear partial differential operators with constant coefficients, in spaces of (small) Gevrey functions.

Oscillatory Integrals As Ultradistributions *

An oscillatory integral of the form naturally defines a distribution provided that φ is a phase function and a is a symbol in Hormander's symbol class. In this paper extending this result to the case of Gevrey class we show that the above oscillatory integral defines a Gevrey ultradistribution provided that φ is a Gevrey phase function and a is a symbol in Gevrey symbol class of infinite order. We also prove the singular support theorem and the singular spectrum theorem which are important in application

Motion planning for the heat equation

SUMMARYThe paper gives an explicit open-loop control, able to approximately steer the one-dimensional heatequationwith control onthe boundaryfromanystate to any otherstate. The controlisobtained thankstoaparametrizationofthesolutionsoftheheatequationbyaseriesinvolvinginnitelymanyderivativesofthesystem &#at output’. Copyright ( 2000 John Wiley & Sons, Ltd. KEY WORDS : heat equation; boundary control; motion planning; #atness; Gevrey functions; approximatecontrollability 1. INTRODUCTIONMotion planning, i.e. the construction of an open-loop control connecting an initial state toa nal state, is a fundamental problem of control theory both from a practical and theoreticalpointofview.Forsystemsgovernedby ordinary di!erentialequationsthenotionof- atness [1,2]providesaconstructivesolutiontothisproblem.AsnoticedinReference[2],theideaunderlyingequivalence and #atness*the existence of a one-to-one correspondence between trajectories ofsystems*can be adapted to partial di!erential equations [3}5] with boundary control.In this paper, which develops ideas introduced in [2, 6], we study in this spirit the heat equationwithonespacedimension and controlontheboundary.Wegive anexplicitparametrizationofthetrajectoriesasapower series inthespacevariablewithcoe $cients involvingtimederivatives ofthe&#at’ output. This series is convergent when the #atoutput is restricted to be a Gevrey function (i.e.asmoothfunctionwitha&not too divergent’ Taylor expansion). This parameterization

On local solvability of linear partial differential operators with multiple characteristics

Let WY D)=C,,,., c,(x)D” be a linear partial differential operator with analytic coefficients c,, D = -ia, and L2 an open subset of R”; we denote by G’“)(Q) the class of Gevrey functions of order S, 1 es < 03. The operator L is said to be @)-solvable in 52 if for every f in G:)(Q), the space of all the Gevrey functions of order s with compact support, there exists a solution U, in the space of ultradistributions Gtr(sZ), of the equation Lu =f: We shall speak similarly of (co )-solvable operators when, in the above sentence, the class CF of all the indefinitely differentiable functions with compact support takes the place of Gt’ (and 9’ of Gr”). Owing to these definitions of solvability, when an operator L is not (s)-solvable then it is also not (t)-solvable for any t, s < t < co; we can therefore obtain non-solvability results in the class of indefinitely differentiable functions from non-solvability results in some Gevrey classes. Vice versa, dealing with operators whose unsolvability in the C” category is known, we could study up to which orders s they remain (s)-unsolvable. This way of considering the problem of the solvability of differential operators was introduced by Rodino in [13] and then carried on by the author in [3]; this last work, in particular, is the starting point of the present paper (see also [4]). We shall study here necessary conditions for the local solvability (that is, in small neighborhoods of points in R”) in Gevrey classes of differential operators with analytic coefficients of the form

Pseudodifferential operators of infinite order and Gevrey classes

A class of pseufodifferential operators of infinite order acting on spaces of Gevrey functions and their duals is defined. For the corresponding symbols the rules of the classical symbolic calculus are proved. In particular for operators satisfying an “hipoellipticity condition” a result of propagation of Gevrey regularity, is obtained by proving the existence of a parametrix.

Semigroups and abstract Gevrey spaces

Conditions are found under which a closed linear operator A in a Banach space X generates a continuous semigroup in a linear topological space Y which is dense in X. The space Y is an abstract Gevrey space associated with the operator A. This is an abstract setting for some results for hyperbolic systems with data in spaces of Gevrey functions.