Global solvability for a class of overdetermined systems

In this work we consider systems of two smooth vector fields on the three-dimensional torus associated to a closed 1-form. We prove that, for such systems, the global solvability in the space of smooth functions is characterized by the property of all the sublevel and superlevel sets of a certain primitive of the 1-form being connected.     

Cohomology relative to a system of closed forms on the torus

To a system of closed one‐forms on the torus, we associate a differential complex and compute the induced cohomology groups provided that a related matrix satisfies a Diophantine condition.     

On the controllability of the Vlasov-Poisson system

In this paper, we study the controllability of the Vlasov-Poisson system in a periodic domain, by means of an interior control located in an spatial subdomain.The first result proves the local exact zero controllability in the two-dimensional torus between two small acceptable distribution functions, with an arbitrary control zone.A second result establishes the global exact controllability in arbitrary dimension, provided the control zone satisfies the condition that it contains a hyperplane of the torus.     

Global s-solvability and global s-hypoellipticity for certain perturbations of zero order of systems of constant real vector fields

The main purpose of this paper is to show, in the two-dimensional torus, a necessary and sufficient condition in order to certain perturbations of zero order of a system of constant real vector fields to be globally s-solvable. We are also interested in studying its global s-hypoellipticity. We present connections between these global concepts and a priori estimates. We also present two applications of our results for systems of operators with variable coefficients.     

Existence of the universal attractor for the 3-D viscous primitive equations of large-scale moist atmosphere

In this paper, we consider the initial-boundary value problem for the three-dimensional viscous primitive equations of large-scale moist atmosphere which are used to describe the turbulent behavior of long-term weather prediction and climate changes. By obtaining the existence and uniqueness of global strong solutions for the problem and studying the long-time behavior of strong solutions, we prove the existence of the universal attractor for the dynamical system generated by the primitive equations of large-scale moist atmosphere.     

Local existence and uniqueness for the hydrostatic Euler equations on a bounded domain

We address the question of well-posedness in spaces of analytic functions for the Cauchy problem for the hydrostatic incompressible Euler equations (inviscid primitive equations) on domains with boundary. By a suitable extension of the Cauchy-Kowalewski theorem we construct a locally in time, unique, real-analytic solution and give an explicit rate of decay of the radius of real-analyticity.     

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.

     

Time analyticity and backward uniqueness of weak solutions of the Navier-Stokes equations of multidimensional compressible flow

We prove that solutions of the Navier-Stokes equations of three-dimensional compressible flow, restricted to fluid-particle trajectories, can be extended as analytic functions of complex time. As consequences we derive backward uniqueness of solutions as well as sharp rates of smoothing for higher-order Lagrangean time derivatives. The solutions under consideration are in a reasonably broad regularity class corresponding to small-energy initial data with a small degree of regularity, the latter being required for conversion to the Lagrangean coordinate system in which the analysis is carried out.     

The periodic Cauchy problem of the modified Hunter-Saxton equation

We prove that the periodic initial value problem for the modified Hunter-Saxton equation is locally well-posed for initial data in the space of continuously differentiable functions on the circle and in Sobolev spaces when s > 3/2. We also study the analytic regularity (both in space and time variables) of this problem and prove a Cauchy-Kowalevski type theorem. Our approach is to rewrite the equation and derive the estimates which permit application of o.d.e. techniques in Banach spaces. For the analytic regularity we use a contraction argument on an appropriate scale of Banach spaces to obtain analyticity in both time and space variables.     

Biasymptotic solutions of perturbed integrable Hamiltonian systems

We prove that small perturbations of a real analytic integrable Hamiltonian system ind degrees of freedom generically have biasymptotic orbits which are obtained as intersections of the stable and unstable manifolds of invariant hyperbolic tori of dimensiond–1. Hence, these solutions will be forward and backward asymptotic to such a torus and not to a periodic solution. The generic condition, which is open and dense, is given by an explicit condition on the averaged perturbation.     

Summability of divergent solutions of the n-dimensional heat equation

The Cauchy problem for n-dimensional complex heat equation is considered. The Borel summability of formal solutions is characterized in terms of analytic continuation with an appropriate growth condition of the spherical mean of the Cauchy data.     

A New Approach to the Parameterization Method for Lagrangian Tori of Hamiltonian Systems

We compute invariant Lagrangian tori of analytic Hamiltonian systems by the parameterization method. Under Kolmogorov’s non-degeneracy condition, we look for an invariant torus of the system carrying quasi-periodic motion with fixed frequencies. Our approach consists in replacing the invariance equation of the parameterization of the torus by three conditions which are altogether equivalent to invariance. We construct a quasi-Newton method by solving, approximately, the linearization of the functional equations defined by these three conditions around an approximate solution. Instead of dealing with the invariance error as a single source of error, we consider three different errors that take account of the Lagrangian character of the torus and the preservation of both energy and frequency. The condition of convergence reflects at which level contributes each of these errors to the total error of the parameterization. We do not require the system to be nearly integrable or to be written in action-angle variables. For nearly integrable Hamiltonians, the Lebesgue measure of the holes between invariant tori predicted by this parameterization result is of \({\mathcal {O}}(\varepsilon ^{1/2})\), where \(\varepsilon \) is the size of the perturbation. This estimate coincides with the one provided by the KAM theorem.     

From the dynamics of gaseous stars to the incompressible Euler equations

A model for the dynamics of gaseous stars is introduced and formulated by the Navier-Stokes-Poisson system for compressible, reacting gases. The combined quasineutral and inviscid limit of the Navier-Stokes-Poisson system in the torus Tn is investigated. The convergence of the Navier-Stokes-Poisson system to the incompressible Euler equations is proven for the global weak solution and for the case of general initial data.     

Analyticity of solutions for semilinear elliptic systems of second order

This paper deals with systems , , where the right hand side is a -valued, real analytic function. We prove that a solution of such a system can be continued across a straight line segment , if one prescribe certain nonlinear, mixed boundary conditions on , which are assumed to be real analytic too. This continuation will be constructed by solving certain hyperbolic initial boundary value problems, generalizing an idea of H. Lewy. We apply this result to surfaces of prescribed mean curvature and to minimal surfaces in Riemannian manifolds spanned into a regular Jordan curve : Supposing analyticity of all data, we show that both types of surfaces can be continued across . Received: 29 December 2000 / Accepted: 11 July 2001 / Published online: 29 April 2002     

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.     

Exact controllability in projections for three-dimensional Navier–Stokes equations

The paper is devoted to studying controllability properties for 3D Navier–Stokes equations in a bounded domain. We establish a sufficient condition under which the problem in question is exactly controllable in any finite-dimensional projection. Our sufficient condition is verified for any torus in R³${\mathbb{R}}^3$. The proofs are based on a development of a general approach introduced by Agrachev and Sarychev in the 2D case. As a simple consequence of the result on controllability, we show that the Cauchy problem for the 3D Navier–Stokes system has a unique strong solution for any initial function and a large class of external forces.     

On global hypoellipticity of degenerate elliptic operators

We consider a class of degenerate elliptic operators on a torus and prove that global hypoellipticity is equivalent to an algebraic condition involving Liouville vectors and simultaneous approximability. For another class of operators we show that the zero order term may influence global hypoellipticity. Received August 13, 1997     

Hyperbolicity, transversality and analytic first integrals

Let A be a (normally) hyperbolic compact invariant manifold of an analytic diffeomorphism f of an analytic manifold M. We assume that the stable and unstable manifold of A intersect transversally (in an admissible way), the dynamics on A is ergodic and the modulus of the eigenvalues associated to the stable and unstable manifold, respectively, satisfy a non-resonance condition. In the case where A is a point or a torus, we prove that the discrete dynamical system associated to f does not admit an analytic first integral. The proof is based on a triviality lemma, which is of combinatorial nature, and a geometrical lemma. The same techniques, allow us to prove analytic non-integrability of Hamiltonian systems having Arnold diffusion. In particular, using results of Xia, we prove analytic non-integrability of the elliptic restricted three-body problem, as well as the planar three-body problem.     

Analyticity of the Cauchy problem for two-component Hunter-Saxton systems

This paper is mainly concerned with the periodic Cauchy problem for a generalized two-component μ-Hunter-Saxton system with analytic initial data. The analyticity of its solutions is proved in both variables, globally in space and locally in time. The obtained result can be also applied to its special cases—the classical integrable two-component Hunter-Saxton system, the generalized μ-Hunter-Saxton equation and the classical Hunter-Saxton equation.