Gevrey Regularity Research Articles

Reformulation of some functional equations in the space of Gevrey distributions and regularity of solutions

We reformulate several well-known functional equations including Cauchy equation, Pexider equation, Jensen equation, Pompeiu equation and their generalized forms as equations for Gevrey distributions and show that every solution of each functional equation in the space of Gevrey distributions is Gevrey differentiable.

Gevrey regularity of random attractors for stochastic reaction-diffusion equations

Article Gevrey regularity of random attractors for stochastic reaction-diffusion equations was published on January 1, 2000 in the journal Random Operators and Stochastic Equations (volume 8, issue 2).

Gevrey Regularity for Nonlinear Analytic Parabolic Equations on the Sphere

The regularity of solutions to a large class of analytic nonlinear parabolic equations on the two-dimensional sphere is considered. In particular, it is shown that these solutions belong to a certain Gevrey class of functions, which is a subset of the set of real analytic functions. As a consequence it can be shown that the Galerkin schemes, based on the spherical harmonics, converge exponentially fast to the exact solutions, as the number of modes involved in the approximation tends to infinity. Furthermore, in the case that the underlying evolution equation has a global attractor, then this global attractor is contained in the space of spatially real analytic functions whose radii of analyticity are bounded uniformly from below.

Inertial manifolds of parabolic differential equations under high-order discretizations

This paper deals with the long-time behaviour of numerical discretizations of nonlinear parabolic differential equations. For various equations of mathematical physics, the dynamics are governed by a finite-dimensional inertial manifold, which attracts solutions at an exponential rate. We show that Runge-Kutta time and spectral Galerkin space discretizations possess inertial manifolds which approximate the inertial manifold of the continuous problem with the order of finite-time approximations of smooth solutions. We thus obtain estimates for the distance between the inertial manifolds of the partial differential equation and its semi- and full discretizations which show the high order of the time discretization and exponentially fast convergence of the space discretization. These results are obtained by using time analyticity and Gevrey regularity of solutions of the differential equation. As an application of the theory, the complex Ginzburg-Landau equation is considered.

The Navier-Stokes equations on the rotating 2-D sphere: Gevrey regularity and asymptotic degrees of freedom

In this article we prove a Gevrey class global regularity to the Navier-Stokes equations on the rotating two dimensional sphere, S 2 - a fundamental model that arises naturally in large scale atmospheric dynamics. As a result one concludes the exponential convergence of the spectral Galerkin numerical method, based on spherical harmonic functions. Moreover, we provide an upper bound for the number of asymptotic degrees of freedom for this system.

Gevrey regularity for nonlinear analytic parabolic equations

Gevrey regularity for nonlinear analytic parabolic equations

Gevrey regularity and approximate inertial manifolds for the derivative Ginzburg-Landau equation in two spatial dimensions

In the present paper , we show the Gevrey class regularity of solutions for the generalized Ginzburg-Landau equation in two spatial dimensions. We also introduce an approximate inertial manifold for this system.

Propagation of Gevrey Regularity for a Class of Hypoelliptic Equations

We prove results on the propagation of Gevrey and analytic wave front sets for a class of C ∞ C^\infty hypoelliptic equations with double characteristics.

Levi Conditions and Global Gevrey Regularity for the Solutions of Quasilinear Weakly Hyperbolic Equations

In the present paper the authors prove a global Gevrey regularity result for the solutions of the Cauchy problem for the quasilinear weakly hyperbolic equation with spatial degeneracy utt−(a(x, t)ux)x = f(x, t, u, ux). The basic tool is a well - posedness result (local existence and cone of dependence) in Gevrey spaces. Such a result can be proved only under the assumption of Levi conditions. Suitable energy estimates lead to the regularity of solutions. This result generalizes results from the strictly hyperbolic case.

Discrete gevrey regularity attractors and uppers–semicontinuity for a finite difference approximation to the ginzburg–landau equation

A semi-discrete spatial finite difference approximation to the complex Ginzburg-Landau equation with cubic non-linearity is considered. Using the fractional powers of a sectorial operator, discrete versions of the Sobolev spaces H 5, and Gevrey classes of regularityG, are introduced.Discrete versions of some standard Sobolev space norm inequalities are proved.

Système d'Euler incompressible et régularité microlocale analytique

In this paper we study the analytic or Gevrey regularity of the integral curves of non necessarily Lipschitzian solutions of Euler equation. We deduce that the analytic or Gevrey wave front set of theses solutions is contained in the characteristic set of the linearized operator.

Gevrey regularity of C3-functionals and stochastic calculus of variation

Using the stochastic calculus of variation, we prove an nth order integration by parts formula where we only need a product of divergences and not a composite. Then we apply this formula to obtain regularity results of the Gevrey type for the density and the join density of some additive functionals of Brownian motion

Time analyticity and gevrey regularity for solutions of a class of dissipative partial differential equations

Time analyticity and gevrey regularity for solutions of a class of dissipative partial differential equations

Relation between Growth and Regularity of Solutions of Hypoelliptic Equations

For a class of linear partial differential equations with variable coefficients, it is shown that the Gevrey regularity of solutions depends on their growth at infinity.

Relation between growth and regularity of solutions of hypoelliptic equations

For a class of linear partial differential equations with variable coefficients, it is shown that the Gevrey regularity of solutions depends on their growth at infinity.

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.

Some remarks on the propagation of Gevrey regularity for solutions of linear partial differential equations with constant coefficients

LetP(D) be a linear partial differential operator onR n with constant coefficients and letV be a closed cone ofR n . Some cases are discussed where or, more generally, where, for ϱ>σ, Here Γ(σ) (Ω), σ≥1, Ω an open set ofR n , denotes the Gevrey space of type σ and γ ′ (R n ) the space of Beurling ultradistributions of type ϱ.

Powers and Gevrey's regularity for a system of differential operators

Powers and Gevrey's regularity for a system of differential operators

Gevrey regularity for a class of dissipative equations with analytic nonlinearity

In this paper, we establish Gevrey class regularity of solutions to a class of dissipative equations with an analytic nonlinearity in the whole space. This generalizes the results of Ferrari and Titi in the periodic space case with initial data in $L^2-$based Sobolev spaces to the $L^p$ setting and in the whole space. Our generalization also includes considering rougher initial data, in negative Sobolev spaces in some cases including the Navier-Stokes and the subcritical quasi-geostrophic equations, and allowing the dissipation operator to be a fractional Laplacian. Moreover, we derive global (in time) estimates in Gevrey norms which yields decay of higher order derivatives which are optimal. Applications include (temporal) decay of solutions in higher Sobolev norms for a large class of equations including the Navier-Stokes equations, the subcritical quasi-geostrophic equations, nonlinear heat equations with fractional dissipation, a variant of the Burgers' equation with a cubic or higher order nonlinearity, and the generalized Cahn-Hilliard equation. The decay results for the last three cases seem to be new while our approach provides an alternate proof for the recently obtained $L^p\, (1