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.

     

Analytic Hypoellipticity at Non-Symplectic Poisson-Treves Strata for Certain Sums of Squares of Vector Fields

We consider an operator P which is a sum of squares of vector fields with analytic coefficients. The operator has a non-symplectic characteristic manifold, but the rank of the symplectic form σ is not constant on Char P. Moreover the Hamilton foliation of the non-symplectic stratum of the Poisson-Treves stratification for P consists of closed curves in a ring-shaped open set around the origin. We prove that then P is analytic hypoelliptic on that open set. And we note explicitly that the local Gevrey hypoellipticity for P is G ^k+1 and that this is sharp.      

Polynomial normal forms with exponentially small remainder for analytic vector fields

A key tool in the study of the dynamics of vector fields near an equilibrium point is the theory of normal forms, invented by Poincaré, which gives simple forms to which a vector field can be reduced close to the equilibrium. In the class of formal vector valued vector fields the problem can be easily solved, whereas in the class of analytic vector fields divergence of the power series giving the normalizing transformation generally occurs. Nevertheless the study of the dynamics in a neighborhood of the origin can very often be carried out via a normalization up to finite order. This paper is devoted to the problem of optimal truncation of normal forms for analytic vector fields in R^m. More precisely we prove that for any vector field in R^m admitting the origin as a fixed point with a semi-simple linearization, the order of the normal form can be optimized so that the remainder is exponentially small. We also give several examples of non-semi-simple linearization for which this result is still true.     

齐次群上带漂移项亚椭圆算子的整体Sobolev-Morrey 估计

设G是一个齐次群,X0,X1,X2,...,Xp0为G上满足Hormander秩条件的实左不变向量场且X1,X2,...,Xp0是1次齐次的,X0是2次齐次的.在本文中,我们研究如下带有漂移项的算子：L=∑p0i,j=1aijXiXj＋a0X0,其中（aij）是一个常数矩阵且满足椭圆条件,a0∈R／{0}.对算子L,通过建立齐型空间上的奇异积分Morrey有界性和关于此向量场的插值不等式,我们在群G上获得了整体Sobolev-Morrey估计.     

A Meyers type regularity result for approximations of second order elliptic operators by P1 finite elements

We prove a Meyers type regularity estimate for approximate solutions of second order elliptic equations obtained by P1 finite elements. The proofs rely on interpolation results for Sobolev spaces on graphs. Estimates for second order elliptic operators on rather general graphs are also obtained.     

Analyticity of the sine family with L p -maximal regularity for second order Cauchy problems

If the second order problem u（t） ＋ Bu（t） ＋ Au（t） = f（t）, u（0） =u（0） = 0 has L^p-maximal regularity, 1 〈 p 〈 ∞, the analyticity of the corresponding propagator of the sine type is shown by obtaining the estimates of ‖λ（λ^2 ＋ λB ＋ A）^-1‖ and ‖B（λ^2 ＋ λB ＋ A）^-1‖ for λ∈ C with Reλ 〉 ω, where the constant ω≥ 0.     

A note on temporal regularity of second order difference equations

This work deals with the study of maximal ?^p‐regularity of a pair (A,B) of bounded linear operators on a complex Banach space associated to the second order difference equation u_{n + 2} = Bu_{n + 1} + Au_n + f_n, where f is a given sequence on . We obtain results of characterization based on spectral analysis of the discrete sine family, which is the resolvent family of this equation.     

Prescribing analytic singularities for solutions of a class of vector fields on the torus

We consider the operator acting on distributions on the two-torus where and are real-valued, real analytic functions defined on the unit circle We prove, among other things, that when changes sign, given any subset of the set of the local extrema of the local primitives of there exists a singular solution of such that the projection of its analytic singular support is furthermore, for any and any closed subset of there exists such that and We also provide a microlocal result concerning the trace of at

     

Analytic regularity for an operator with Treves curves

We study a partial differential operator with analytic coefficients, which is of the form “sum of squares”. is hypoelliptic on any open subset of , yet possesses the following properties: (1) is not analytic hypoelliptic on any open subset of that contains 0. (2) If u is any distribution defined near with the property that is analytic near 0, then u must be analytic near 0. (3) The point 0 lies on the projection of an infinite number of Treves curves (bicharacteristics).These results are consistent with the Treves conjectures. However, it follows that the analog of Treves conjecture, in the sense of germs, is false.As far as we know, is the first example of a “sum of squares” operator which is not analytic hypoelliptic in the usual sense, yet is analytic hypoelliptic in the sense of germs.     

Continuity of Lyapunov exponent for analytic quasi-periodic cocycles on higher-dimensional torus

It is known that the Lyapunov exponent is not continuous at certain points in the space of continuous quasi-periodic cocycles. We show that the Lyapunov exponent is continuous for a higher-dimensional analytic category in this paper. It has a modulus of continuity of the form exp(−∣logt∣ ^σ ) for some 0 < σ < 1.     

Sobolev regularity of the second biharmonic problem on a rectangle

Summary It is proved that for any f &esin; L²(Ω) the weak solution of the second biharmonic problem on a rectangle satisfies u&esin; H⁴(Ω). The proof uses the decomposition of the problem into two Poisson equations and a general condition for H⁴-regularity via the eigenvalues and eigenfunctions of second order elliptic operators.     

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     

Global regularity criterion for the Navier‐Stokes equations based on the direction of vorticity

We study the regularity criterion for the Navier‐Stokes equations and show that the (β₁,β₂,β₃)‐Hölder continuity assumption in (x₁,x₂,x₃) on the direction of the vorticity ensures the regularity of the solution. This may be viewed as an extension of many previous results, since some of the β_i can be arbitrarily small.     

On the Pythagoras numbers of real analytic curves

We show that the Pythagoras number of a real analytic curve is the supremum of the Pythagoras numbers of its singularities, or that supremum plus 1. This includes cases when the Pythagoras number is infinite.      

Recurrence of singularities for second order isotropic pseudodifferential operators

Let H be a self‐adjoint isotropic elliptic pseudodifferential operator of order 2. Denote by the solution of the Schrödinger equation with initial data . If u₀ is compactly supported the solution is smooth for small , but not for all t. We determine the wavefront set of in terms of the wavefront set of u₀ and the principal and subprincipal symbol of H.     

Geometric bounds on the linearization domain and analytic dependence on parameters for families of analytic vector fields in a neighborhood of a singular point

We study families of holomorphic vector fields, holomorphically depending on parameters, in a neighborhood of an isolated singular point. When the singular point is in the Poincaré domain for every vector field of the family we prove, through a modification of classical Sternberg's linearization argument, cf. Nelson (1969) [7] too, analytic dependence on parameters of the linearizing maps and geometric bounds on the linearization domain: each vector field of the family is linearizable inside the smallest Euclidean sphere which is not transverse to the vector field, cf. Brushlinskaya (1971) [2], Ilyashenko and Yakovenko (2008) [5] for related results. We also prove, developing ideas in Martinet (1980) [6], a version of Brjuno's Theorem in the case of linearization of families of vector fields near a singular point of Siegel type, and apply it to study some 1-parameter families of vector fields in two dimensions.     

Global regularity estimates for multidimensional equations of compressible non-Newtonian fluids

The present paper is devoted to the problem of global existence of sufficiently regular solutions to two- and three-dimensional equations of a compressible non-Newtonian fluid. In the case of the potential stress tensor, we develop a technique for deriving energy identities that do not contain derivatives of density. On the basis of these identities, in the case of sufficiently rapidly increasing potentials, we obtain an extended system ofa priori estimates for the equations mentioned above. We also study the related problem of estimating solutions to the nonlinear elliptic system generated by the stress tensor. Translated fromMatematicheskie Zametki, Vol. 68, No. 3, pp. 360–376, September, 2000.