Smoothness of the trajectories of ideal fluid particles with Yudovich vorticities in a planar bounded domain

We consider the incompressible Euler equations in a (possibly multiply connected) bounded domain Ω of R², for flows with bounded vorticity, for which Yudovich (1963) proved in [29] global existence and uniqueness of the solution. We prove that if the boundary ∂Ω of the domain is C^∞ (respectively Gevrey of order M?1) then the trajectories of the fluid particles are C^∞ (respectively Gevrey of order M+2). Our results also cover the case of “slightly unbounded” vorticities for which Yudovich (1995) extended his analysis in [30]. Moreover if in addition the initial vorticity is Hölder continuous on a part of Ω then this Hölder regularity propagates smoothly along the flow lines. Finally we observe that if the vorticity is constant in a neighborhood of the boundary, the smoothness of the boundary is not necessary for these results to hold.     

Existence of Gevrey approximate solutions for certain systems of linear vector fields applied to involutive systems of first-order nonlinear pdes

Given a Gs-involutive structure, (M,V), a Gevrey submanifold X⊂M which is maximally real and a Gevrey function u₀ on X we construct a Gevrey function u which extends u₀ and is a Gevrey approximate solution for V. We then use our construction to study Gevrey micro-local regularity of solutions, u∈C²(RN), of a system of nonlinear pdes of the form

     

Hypoellipticity and Local Solvability in Gevrey Classes

Let P be a linear partial differential operator with coefficients in the Gevrey class G^s. We prove first that if P is s‐hypoelliptic then its transposed operator ^tP is s‐locally solvable, thus extending to the Gevrey classes the well‐known analogous result in the C^∞class. We prove also that if P is s‐hypoelliptic then its null space is finite dimensional and its range is closed; this implies an index theorem for s‐hypoelliptic operators. Generalizations of these results to other classes of functions are also considered.     

Global hypoellipticity and 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 that if P is s-globally hypoelliptic in Tⁿ then its transposed operator tP is s-globally solvable in Tⁿ, thus extending to the Gevrey classes the well-known analogous result in the corresponding C^∞ class.     

FBI transforms in Gevrey classes

In this work we develop the FBI Transform tools in Gevrey classes. Our goal is to extend to a Gevrey-s obstacle withs < 3 the localization of poles result obtained by Sjöstrand [10] in the analytic class. In that work, the author proved that the pole-free zone is controlled by a constantC _0,a (which was only implicit in Bardos-Lebeau-Rauch [1]), improving the constantC _0,∞ of the results of Hargé-Lebeau [13] and Sjöstrand-Zworski [13] valid in C^∞ The works [3], [13] and [10] feature an adapted complex scaling for convex obstacles, but in [10] there is the addition of a small complex “G³ deformation”. The study of such Gevrey deformations for operators with symbols in Gevrey classes is the central point of this work.     

Almost periodic pseudodifferential operators and Gevrey classes

We study almost periodic pseudodifferential operators acting on almost periodic functions ${G_{\rm ap}^s(\mathbb {R}^d)}$ of Gevrey regularity index s ≥ 1. We prove that almost periodic operators with symbols of H?rmander type ${S_{\rho,\delta}^m}$ satisfying an s-Gevrey condition are continuous on ${G_{\rm ap}^s(\mathbb {R}^d)}$ provided 0 < ρ ≤ 1, δ?=?0 and s ρ ≥ 1. A calculus is developed for symbols and operators using a notion of regularizing operator adapted to almost periodic Gevrey functions and its duality. We apply the results to show a regularity result in this context for a class of hypoelliptic operators.     

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)     

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.     

On universal formal power series

The point source of this work is Seleznev's theorem which asserts the existence of a power series which satisfies universal approximation properties in C^∗. The paper deals with a strengthened version of this result. We establish a double approximation theorem on formal power series using a weighted backward shift operator. Moreover we give strong conditions that guarantee the existence of common universal series of an uncountable family of weighted backward shift with respect to the simultaneous approximation. Finally we obtain results on admissible growth of universal formal power series. We especially prove that you cannot control the defect of analyticity of such a series even if there exist universal series in the well-known intersection of formal Gevrey classes.     

Surface waves on steady perfect‐fluid flows with vorticity

This is a theory of two‐dimensional steady periodic surface waves on flows under gravity in which the given data are three quantities that are independent of time in the corresponding evolution problem: the volume of fluid per period, the circulation per period on the free stream line, and the rearrangement class (equivalently, the distribution function) of the vorticity field. A minimizer of the total energy per period among flows satisfying these three constraints is shown to be a weak solution of the surface wave problem for which the vorticity is a decreasing function of the stream function. This decreasing function can be thought of as an infinite‐dimensional Lagrange multiplier corresponding to the vorticity rearrangement class being specified in the minimization problem. (Note that functional dependence of vorticity on the stream function was not specified a priori but is part of the solution to the problem and ensures the flow is steady.) To illustrate the idea with a minimum of technical difficulties, the existence of nontrivial waves on the surface of a fluid flowing with a prescribed distribution of vorticity and confined beneath an elastic sheet is proved. The theory applies equally to irrotational flows and to flows with locally square‐integrable vorticity. © 2011 Wiley Periodicals, Inc.     

Continuity and analyticity for a cross-coupled Camassa–Holm equation with waltzing peakons and compacton pairs

In this paper, we mainly study the well-posedness in the sense of Hadamard, non-uniform dependence, Hölder continuity and analyticity of the data-to-solution map for a cross-coupled Camassa–Holm equation with waltzing peakons and compacton pairs on both the periodic and the nonperiodic case. Using a Galerkin-type approximation scheme, it is shown that this equation is well-posed in Sobolev spaces \(H^{s} \times H^{s},s>5/2\) in the sense of Hadamard, that is, the data-to-solution mapis continuous. In conjunction with the well-posedness estimate, it is also proved that this dependence is sharp by showing that the solution map is not uniformly continuous. Furthermore, the Hölder continuous in the \(H^r \times H^r\) topology when \(0\le r< s\) with Hölder exponent \(\alpha \) depending on both s and r are shown. Finally, applying generalized Ovsyannikov type theorem and the basic properties of Sobolev-Gevrey spaces, we prove the Gevrey regularity and analyticity of the CCCH system. Moreover, we obtain a lower bound of the lifespan and the continuity of the data-to-solution map     

Analyticity of the streamlines and of the free surface for periodic equatorial gravity water flows with vorticity

We prove analyticity of the streamlines beneath the surface and the smoothness of the free surface for geophysical equatorial water flows with a general Hölder continuously differentiable underlying vorticity distribution under the assumption of no stagnation points in the flow. Moreover, we prove that the real-analyticity of the vorticity function implies the real-analyticity of the free surface.     

Smooth global Lagrangian flow for the 2D Euler and second-grade fluid equations

We present a very simple proof of the global existence of a C^∞ Lagrangian flow map for the 2D Euler and second-grade fluid equations (on a compact Riemannian manifold with boundary) which has C^∞ dependence on initial data u₀ in the class of H^s divergence-free vector fields for s > 2.     

Similarity solutions for flow and heat transfer over a permeable surface with convective boundary condition

The steady laminar boundary layer flow over a permeable flat plate in a uniform free stream, with the bottom surface of the plate is heated by convection from a hot fluid is considered. Similarity solutions for the flow and thermal fields are possible if the mass transpiration rate at the surface and the convective heat transfer from the hot fluid on the lower surface of the plate vary like x^−1/2, where x is the distance from the leading edge of the solid surface. The governing partial differential equations are first transformed into ordinary differential equations, before being solved numerically. The effects of the governing parameters on the flow and thermal fields are thoroughly examined and discussed.     

Gevrey vectors in involutive tube structures and Gevrey regularity for the solutions of certain classes of semilinear systems

In this work, we introduce the notion of s-Gevrey vectors in locally integrable structures of tube type. Under the hypothesis of analytic hypoellipticity, we study the Gevrey regularity of such vectors and also show how this notion can be applied to the study of the Gevrey regularity of solutions of certain classes of semilinear equations.     

Gevrey regularity with weight for incompressible euler equation in the half plane

In this work we prove the weighted Gevrey regularity of solutions to the incompressible Euler equation with initial data decaying polynomially at infinity. This is motivated by the well-posedness problem of vertical boundary layer equation for fast rotating fluid. The method presented here is based on the basic weighted L²-estimate, and the main difficulty arises from the estimate on the pressure term due to the appearance of weight function.     

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.     

Quasi-linear weakly hyperbolic equations with Gevrey-Levi conditions

This paper is devoted to the study of the analytic regularity for real solutions of a non-linear weakly hyperbolic equation of the form: $$\sum\limits_{m - (1 - e)r< |a| \leqslant m} {a_a (y,u^{(\beta )} } )\partial _y^a u = g(y,u^{(\beta )} ){\text{, |}}\beta | \leqslant m - (1 - e)r,$$ wherea _α andg are analytic functions of their arguments,r≥2 denotes the largest multiplicity of the characteristic roots of (*) and ?∈(0,(r?1)/r). Assuming that 026 the characteristic roots are of constant multiplicity and that the generalized Levi’s conditions related to the index ? are satisfied, we prove that, ifu is a solution of (*) 026 and belongs to a Gevrey class of index σ<1/?, then the analyticity of Cauchy data propagates according to the geometry of the influence domains of the equation.     

一维广义Ginzburg-Landau方程解的正则性及线性与非线性Galerkin近似

本文得到了一维广义Ginzburs－Landau方程解的Gevrey类正则性和关于时间的解析性．在此基础上得到线性Galerkin近似和非线性Galerkin近似的收敛性．