Gevrey character of formal solutions of differential—difference equations with polynomials coefficients

In this paper we prove that formal factorial series as well as formal power series in 1/x solutions of a differential—difference equation with polynomial coefficients are Gevrey of some order which can be determined from a suitable Newton polygon     

Gevrey properties of the asymptotic critical wave speed in a family of scalar reaction–diffusion equations

We consider front propagation in a family of scalar reaction–diffusion equations in the asymptotic limit where the polynomial degree of the potential function tends to infinity. We investigate the Gevrey properties of the corresponding critical propagation speed, proving that the formal series expansion for that speed is Gevrey-1 with respect to the inverse of the degree. Moreover, we discuss the question of optimal truncation. Finally, we present a reliable numerical algorithm for evaluating the coefficients in the expansion with arbitrary precision and to any desired order, and we illustrate that algorithm by calculating explicitly the first ten coefficients. Our analysis builds on results obtained previously in [F. Dumortier, N. Popovi?, T.J. Kaper, The asymptotic critical wave speed in a family of scalar reaction–diffusion equations, J. Math. Anal. Appl. 326 (2) (2007) 1007–1023], and makes use of the blow-up technique in combination with geometric singular perturbation theory and complex analysis, while the numerical evaluation of the coefficients in the expansion for the critical speed is based on rigorous interval arithmetic.     

Gevrey Regularity for the Attractor of the 3D Navier–Stokes–Voight Equations

Recently, the Navier–Stokes–Voight (NSV) model of viscoelastic incompressible fluid has been proposed as a regularization of the 3D Navier–Stokes equations for the purpose of direct numerical simulations. In this work, we prove that the global attractor of the 3D NSV equations, driven by an analytic forcing, consists of analytic functions. A consequence of this result is that the spectrum of the solutions of the 3D NSV system, lying on the global attractor, have exponentially decaying tail, despite the fact that the equations behave like a damped hyperbolic system, rather than the parabolic one. This result provides additional evidence that the 3D NSV with the small regularization parameter enjoys similar statistical properties as the 3D Navier–Stokes equations. Finally, we calculate a lower bound for the exponential decaying scale—the scale at which the spectrum of the solution start to decay exponentially, and establish a similar bound for the steady state solutions of the 3D NSV and 3D Navier–Stokes equations. Our estimate coincides with the known bounds for the smallest length scale of the solutions of the 3D Navier–Stokes equations, established earlier by Doering and Titi.      

完全非线性偏微分方程解的Gevrey微局部正则性

本文中，我们首先简要回顾了Gevrey类中的仿微分运算，然后考察了相关的完全非线性偏微分方程的象征的一些性质。作为应用，我们得到解在椭圆点附近的Gevrey微局部正则性。     

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.     

Gevrey空间中抽象柯西-柯瓦列夫斯卡娅定理:能量方法

本文研究了非线性柯西问题的适定性问题.利用经典的能量法和抽象柯西-柯瓦列夫斯卡娅定理,得到非线性柯西问题在Gevrey空间中是适定的.推广了已有文献在非线性柯西问题适定性方面的研究.     

线性微分方程的非正规性与Gevrey类解(英文)

本文简要介绍了超函数(hyperfunction)的一个重要子类——Gevrey类超分布(ultra-distribution of Gevrey class),在此基础上介绍了线性微分算子的一些有关结果,并证明了线性双曲型方程 P(x,)u=f(x)的Cauchy问题,当算子P(x,)适合非正规性条件(irregularity condition)时,若定解资料属于Gevrey类,则它的解亦必属于该Gevrey类。     

Gevrey函数类和超分布中的Paley—Wiener型定理及其应用

众所周知,Paley-Wiener定理深刻地刻画了具紧支集的无穷可微函数及具紧支集的分布同它们的Fourier-Laplace变换之间的关系,建立了具紧支集的无穷可微函数或分布同指数型整函数之间的关系。正因为如此,Paley-Wiener定理在数学中、特别是在偏微分方程的C~∞理论中起着相当重要的作用。本文在具紧支集的Gevrey函数类及具紧支集的超分布中考虑同样类型的定理,称之为Paley-Wiener型定理。     

一类双曲型全特征算子的Cauchy问题的Gevrey类适定性

本文利用能量估计方法,研究了一类双曲型全特征算子Cauehy问题的Gevrey类适定性。本文主要结果如下: 在条件(Ⅰ)—(Ⅴ)下,对任意的s≥1,Cauehy问题(1.1)在B([0,T],G_(L~2)~2(R~n)内均为适定的。     

On the existence in Gevrey classes of local solutions to the Cauchy problem for nonlinear hyperbolic systems with Hölder continuous coefficients

Abstract In this paper we shall solve locally in time the solutions to the Cauchy problem for first order quasilinear hyperbolic systems of which coefficients of principal part and of lower order terms are μ- H?lder and - H?lder continuous in time variable respectively and in Gevrey class of index s with respect to space variables under the assumption , where ν denotes the maximal muliplicity of characteristics of systems. Keywords: Nonlinear hyperbolic systems, Cauchy problem, Gevrey classes     

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.     

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.     

具定重特征的一类双曲型全特征算子的Cauchy问题的Gevrey类适定性

本文研究了全特征Cauchy问题(1.1)的Gevrey类适定性。得到如下的两个主要结果: 1.在条件(Ⅰ)—(Ⅵ)下,对任意的s≥1,全特征Cauchy问题(1.1)均在B([O,T],G_(L~2)~s(R~n))内适定。 2.在条件(Ⅰ)—(Ⅴ)及(Ⅶ)下,若1≤s<θ~(-1)(θ由(1.10)式定义),则全特征Cauchy问题(1.1)在B([O,T],G_(L~3)~8(R~n))内适定;若s=θ~(-1),则存在s>0充分小,使得(1.1)在B([O,s],G_(L~3)~(θ-1)(R~n)内有唯一解。     

丢番图逼近、环面T2上的Gevrey亚椭圆性与几乎周期运动

文中考虑一类最多可以指数地有理逼近的无理数,并指出它们自然地存在于偏微分方程解析理论及动力系统中．