Gevrey hypoellipticity and solvability on the multidimensional torus of some classes of linear partial differential operators

Abstract In this paper we consider the problem of global analytic and Gevrey hypoellipticity and solvability for a class of partial differential operators on a torus. We prove that global analytic and Gevrey hypoellipticity and solvability on the torus is equivalent to certain Diophantine approximation properties. Keywords: Global hypoellipticity, Global solvability, Gevrey classes, Diophantine approximation property Mathematics Subject Classification (2000): 35D05, 46E10, 46F05, 58J99     

Global analytic and Gevrey hypoellipticity of sublaplacians under Diophantine conditions

In this paper we consider the problem of global Gevrey and analytic regularity for a class of partial differential operators on a torus in the form of a sum of squares of vector fields, which may not satisfy the bracket condition. We show that these operators are globally Gevrey or analytic hypoelliptic on the torus if and only if the coefficients satisfy certain Diophantine approximation properties.

     

On Gevrey solvability and regularity

In this paper we study global C^∞ and Gevrey solvability for a class of sublaplacian defined on the torus T ³. We also prove Gevrey regularity for a class of solutions of certain operators that are globally C^∞ hypoelliptic in the N ‐dimensional torus (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On global analytic and Gevrey hypoellipticity on the torus and the Métivier inequality

We obtain a global version in the N-dimensional torus of the Métivier inequality for analytic and Gevrey hypoellipticity, and based on it we introduce a class of globally analytic hypoelliptic operators which remain so after suitable lower order perturbations. We also introduce a new class of analytic (pseudodifferential) operators on the torus whose calculus allows us to study the corresponding perturbation problem in a far more general context.     

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.     

Regularity and solvability of linear differential operators in Gevrey spaces

We are interested in the following question: when regularity properties of a linear differential operator imply solvability of its transpose in the sense of Gevrey ultradistributions? This question is studied for a class of abstract operators that contains the usual differential operators with real‐analytic coefficients. We obtain a new proof of a global result on compact manifolds (global Gevrey hypoellipticity implying global solvability of the transpose), as well as some results in the non‐compact case by means of the so‐called property of non‐confinement of singularities. We provide applications to Hörmander operators, to operators of constant strength and to locally integrable systems of vector fields. We also analyze a conjecture stated in a recent paper of Malaspina and Nicola, which asserts that, in differential complexes naturally arising from locally integrable structures, local solvability in the sense of ultradistributions implies local solvability in the sense of distributions. We establish the validity of the conjecture when the cotangent structure bundle is spanned by the differential of a single first integral.     

Gevrey global solvability of non-singular real first-order differential operators

In this article we deal with Gevrey global solvability of non-singular first-order operators defined on an n-dimensional s-Gevrey manifold, s > 1. As done by Duistermaat and Hörmander in the C ^∞ framework, we show that Gevrey global solvability is equivalent the existence of a global cross section.     

Global regularity in ultradifferentiable classes

We study ω-regularity of the solutions of certain operators that are globally C ^∞-hypoelliptic in the N-dimensional torus. We also apply these results to prove the global ω-regularity for some classes of sublaplacians. In this way, we extend previous work in the setting of analytic and Gevrey classes. Different examples on local and global ω-hypoellipticity are also given.     

On and Gevrey regularity of sublaplacians

In this paper we consider zero order perturbations of a class of sublaplacians on the two-dimensional torus and give sufficient conditions for global regularity to persist. In the case of analytic coefficients, we prove Gevrey regularity for a general class of sublaplacians when the finite type condition holds.

     

Regularity of a class of subLaplacians on the 3-dimensional torus

In this paper we present a necessary and sufficient condition for a family of sums of squares operators to be globally hypoelliptic on a torus. This condition says that either a Diophantine condition is satisfied or there exists a point of finite type. Also, we describe the analytic and Gevrey versions of this result. The proof is based on L²-estimates and microlocal analysis.     

On the global solvability in Gevrey classes on the n-dimensional torus

Let P be a linear partial differential operator with coefficients in the Gevrey class G^s(Tⁿ), where Tⁿ is the n-dimensional torus and s?1. We prove a necessary condition for the s-global solvability of P on Tⁿ. We also apply this result to give a complete characterization for the s-global solvability for a class of formally self-adjoint operators with nonconstant coefficients.     

Well-posedness of fractional Ginzburg–Landau equation in Sobolev spaces

This article is concerned with real fractional Ginzburg–Landau equation. Existence and uniqueness of local and global mild solution for both whole space case and flat torus case are obtained by contraction semigroup method, and Gevrey regularity of mild solution for flat torus case is discussed.     

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.      

Global solvability on the torus for certain classes of formally self-adjoint operators

In this paper we prove a necessary and sufficient condition for global solvability on the torus for two classes of formally self-adjoint operators. For the first class of operators we prove that global solvability is equivalent to an algebraic condition involving Liouville vectors and simultaneous approximability. For the second class of operators, when the coefficients are not identically zero, an independence condition on the coefficients is shown to be necessary and sufficient for global solvability. Received: 21 June 1999 / Revised version: 8 May 2000     

Globally analytic hypoelliptic vector fields on compact surfaces

We present a characterization of the global analytic hypoellipticity of a complex, non-singular, real analytic vector field defined on a compact, connected, orientable, two-dimensional, real analytic manifold.

In particular, we show that such vector fields exist only on the torus.

     

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.     

Global Solvability and Simultaneously Approximable Vectors

We consider a class of sum of squares operators on a torus and we prove that global solvability is equivalent to an algebraic condition involving simultaneously approximable vectors.     

The torus related Riemann problem

The Riemann jump problem is solved for analytic functions of several complex variables with the unit torus as the jump manifold. A well-posed formulation is given which does not demand any solvability conditions. The higher dimensional Plemelj-Sokhotzki formula for analytic functions in torus domains is established. The canonical functions of the Riemann problem for torus domains are represented and applied in order to construct solutions for both of the homogeneous and inhomogeneous problems. Thus contrary to earlier research the results are similar to the respective ones for just one variable. A connection between the Riemann and the Riemann-Hilbert boundary value problem for the unit polydisc is explained.     

Global analytic regularity for sums of squares of vector fields

We consider a class of operators in the form of a sum of squares of vector fields with real analytic coefficients on the torus and we show that the zero order term may influence their global analytic hypoellipticity. Also we extend a result of Cordaro-Himonas.

     

Inertial Manifolds and Gevrey Regularity for the Moore-Greitzer Model of an Axial-Flow Compressor

Summary. In this paper, we study the regularity and long-time behavior of the solutions to the Moore-Greitzer model of an axial-flow compressor. In particular, we prove that this dissipative system of evolution equations possesses a global invariant inertial manifold, and therefore its underlying long-time dynamics reduces to that of an ordinary differential system. Furthermore, we show that the solutions of this model belong to a Gevrey class of regularity (real analytic in the spatial variables). As a result, one can show the exponentially fast convergence of the Galerkin approximation method to the exact solution, an evidence of the reliability of the Galerkin method as a computational scheme in this case. The rigorous results presented here justify the readily available low-dimensional numerical experiments and control designs for stabilizing certain states and traveling wave solutions for this model.