Gevrey Class Of Order Research Articles

Finite-time blowup and ill-posedness in Sobolev spaces of the inviscid primitive equations with rotation

Large scale dynamics of the oceans and the atmosphere are governed by the primitive equations (PEs). It is well-known that the three-dimensional viscous PEs is globally well-posed in Sobolev spaces. On the other hand, the inviscid PEs without rotation is known to be ill-posed in Sobolev spaces, and its smooth solutions can form singularity in finite time. In this paper, we extend the above results in the presence of rotation. We construct finite-time blowup solutions to the inviscid PEs with rotation, and establish that the inviscid PEs with rotation is ill-posed in Sobolev spaces in the sense that its perturbation around a certain steady state background flow is both linearly and nonlinearly ill-posed in Sobolev spaces. Its linear instability is of the Kelvin-Helmholtz type similar to the one appears in the context of vortex sheets problem. This implies that the inviscid PEs is also linearly ill-posed in Gevrey class of order s>1, and suggests that a suitable space for the well-posedness is Gevrey class of order s=1, which is exactly the space of analytic functions.

On the Gevrey strong hyperbolicity

In this paper we are concerned with a homogeneous differential operator $p$ of order $m$ of which characteristic set of order $m$ is assumed to be a smooth manifold. We define the Gevrey strong hyperbolicity index as the largest number $s$ such that the Cauchy problem for $p+Q$ is well-posed in the Gevrey class of order $s$ for any differential operator $Q$ of order less than $m$. We study the case of the largest index and we discuss in which way the Gevrey strong hyperbolicity index relates with the geometry of bicharacteristics of $p$ near the characteristic manifold.

THE DEGREE OF HOLOMORPHIC APPROXIMATION ON A TOTALLY REAL SET

AbstractLet E be a totally real set on a Stein open set Ω on a complete noncompact Kähler manifold (M,g) with nonnegative holomorphic bisectional curvature such that (Ω,g) has bounded geometry at E. Then every function f in a Cp class with compact support on Ω and $\overline \partial $-flat on E up to order p−1,p≥2 (respectively, in a Gevrey class of order s>1, with compact support on Ω and $\overline \partial $-flat on E up to infinite order) can be approximated on compacts subsets of E by holomorphic functions fk on Ω with degree of approximation equal k−p/2 (respectively, exp (−c(s)k1/2(s−1)) ).

An example of the Cauchy problem well posed in any Gevrey class

We study the Cauchy problem for second order hyperbolic operators \(P=D_{t}^{2}-D_{x}H(t,x)D_{x}\) in the Gevrey classes where H(t,x) is given by the limit of a finite sum of functions such as a(t)b(x) with a(t) ≥ 0, b(x) ≥ 0. As a result, for any given positive integer N, we give an example H(t,x) which depends not only on t but also on x such that the Cauchy problem for P is well posed in the Gevrey class of order N.

On the Cauchy Problem for D t 2-D xa(t,x)D x in the Gevrey Class of Order s > 2

We study the Cauchy problem for second order hyperbolic equations with non negative characteristic form of two independent variables. We show that for such equations in divergence-free form, the Cauchy problem is well posed in the Gevrey class of order less than 5/2.

Propagation of singularities along a characteristic boundary for a model problem of shell theory and relation with the boundary layer

We consider the propagation of singularities for a differential system which constitutes a simplified model of thin shells with developable middle surface (parabolic case). Extensions of the solutions out of the domain allow us to consider either boundary or internal singularities. The properties of propagation of singularities and their relation with the structure of the boundary layers are given. We remove a mistake in [2], Section 6.1, concerning the analyticity of solutions (in fact they are in the Gevrey class of order 3).

Gevrey regularizing effect for the (generalized) Korteweg-de Vries equation and nonlinear Schrödinger equations

This paper is concerned with regularizing effects of solutions to the (generalized) Korteweg-de Vries equation(gKdV){∂tu+∂x3u=λup−1∂xu,(t,x)∈ℝ×ℝ,u(0)=ϕ,x∈ℝ, and nonlinear Schrödinger equations in one space dimension(NLS){i∂tu+12∂x2u=G(u,u¯),(t,x)∈ℝ×ℝ,u(0)=ψ,x∈ℝ, where p is an integer satisfying p ≥ 2, λ ∊ ℂ and G is a polynomial of (u,u¯). We prove that if the initial function ϕ is in a Gevrey class of order 3 defined in Section 1, then there exists a positive time T such that the solution of (gKdV) is analytic in space variable for t ∊ [−T, T]\\{0}, and if the initial function ψ in a Gevrey class of order 2, then there exists a positive time T such that the solution of (NLS) is analytic in space variable for t ∊ [−T, T]\\{0}.