Global well-posedness and the classical limit of the solution for the quantum Zakharov system

In this paper, we study the quantum Zakharov system, which describes the nonlinear interaction between the quantum Langmuir and quantum ion-acoustic waves. The global well-posedness result of this system in the energy and above energy spaces is obtained in the case d = 1, 2, 3. Moreover, the classical limit behavior of the quantum Zakharov system is also investigated as the quantum parameter tends to zero.     

Smooth solutions to a class of mixed type Monge–Ampère equations

We prove the existence of C ^∞ local solutions to a class of mixed type Monge–Ampère equations in the plane. More precisely, the equation changes type to finite order across two smooth curves intersecting transversely at a point. Existence of C ^∞ global solutions to a corresponding class of linear mixed type equations is also established. These results are motivated by and may be applied to the problem of prescribed Gaussian curvature for graphs, the isometric embedding problem for 2-dimensional Riemannian manifolds into Euclidean 3-space, and also transonic fluid flow.     

Noetherian varieties in definably complete structures

We prove that the zero-set of a C ^∞ function belonging to a noetherian differential ring M can be written as a finite union of C ^∞ manifolds which are definable by functions from the same ring. These manifolds can be taken to be connected under the additional assumption that every zero-dimensional regular zero-set of functions in M consists of finitely many points. These results hold not only for C ^∞ functions over the reals, but more generally for definable C ^∞ functions in a definably complete expansion of an ordered field. The class of definably complete expansions of ordered fields, whose basic properties are discussed in this paper, expands the class of real closed fields and includes o-minimal expansions of ordered fields. Finally, we provide examples of noetherian differential rings of C ^∞ functions over the reals, containing non-analytic functions.     

Coherence of some rings of functions

To say that a commutative ring $\text{R}$ with unit is coherent amounts to saying, in case $\text{R}$ has no divisors of zero, that the intersection of two finitely generated ideals in $\text{R}$ is finitely generated. We prove that the ring H^∞ of bounded analytic functions in the unit disc is coherent, while the disc algebra A is not coherent. For any positive measure μ, L^∞(μ) is coherent.     

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.     

Paralinearization of the Dirichlet to Neumann Operator,and Regularity of Three-Dimensional Water Waves

This paper is concerned with a priori C ^∞ regularity for three-dimensional doubly periodic travelling gravity waves whose fundamental domain is a symmetric diamond. The existence of such waves was a long standing open problem solved recently by Iooss and Plotnikov. The main difficulty is that, unlike conventional free boundary problems, the reduced boundary system is not elliptic for three-dimensional pure gravity waves, which leads to small divisors problems. Our main result asserts that sufficiently smooth diamond waves which satisfy a Diophantine condition are automatically C ^∞. In particular, we prove that the solutions defined by Iooss and Plotnikov are C ^∞. Two notable technical aspects are that (i) no smallness condition is required and (ii) we obtain an exact paralinearization formula for the Dirichlet to Neumann operator.     

A sufficient condition for the existence of approximate inertial manifolds containing the global attractor

Using the recently introduced concept of inertial manifold with delay we present a new method of construction of approximate inertial manifolds (AIMs). In the case when the global attractor can be embedded into a finite-dimensional C²-manifold we construct AIMs of the same dimension which contain the attractor. To cite this article: A. Rezounenko, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 1015–1020.     

Attractors for Fully Discrete Finite Difference Scheme of Dissipative Zakharov Equations

A fully discrete finite difference scheme for dissipative Zakharov equations is analyzed. On the basis of a series of the time-uniform priori estimates of the difference solutions, the stability of the difference scheme and the error bounds of optimal order of the difference solutions are obtained in L²×H¹×H² over a finite time interval (0, T]. Finally, the existence of a global attractor is proved for a discrete dynamical system associated with the fully discrete finite difference scheme.     

Global regularity in ultradifferentiable classes

We study ω-regularity of the solutions of certain operators that are globally C ^∞-hypoelliptic in the N-dimensional torus. We also apply these results to prove the global ω-regularity for some classes of sublaplacians. In this way, we extend previous work in the setting of analytic and Gevrey classes. Different examples on local and global ω-hypoellipticity are also given.     

离子声学模型Zakharov方程组的整体吸引子的存在性(英文)

In this paper,we consider the initial-boundary problem for Zakharov equations arising from ion-acoustic modes and obtain the existence of global attractor for the initialboundary problem for Zakharov equations.     

耗散的量子Zakharov方程解的渐进性行为

主要研究量子Zakharov方程.在先验估计的基础上通过标准的Galerkin逼近方法得到了量子Zakharov方程解的存在唯一性.同时,也讨论了相应的解的渐进性行为,并且构造出在不同的能量空间上弱拓扑意义下的全局吸引子.     

Sobolev bundles with abelian structure groups

We consider fundamental analytical and topological questions for Sobolev bundles in the supercritical case. We in particular solve C ^∞-equivalence, density and weak compactness problems for the case of Sobolev bundles with compact connected abelian structure groups.     

n-Widths,Faber Expansion,and Computation of Analytic Functions

LetBH^∞(Ω) be the space of analytic functionsfin the region Ω for which |f(z)| ≤ 1,z∈ Ω, and letKbe a compact subset of Ω. How can we compute the values of any functionf∈BH^∞(Ω) at an arbitrary pointz∈K? One of the approaches to this problem applies the results concerning then-widths and ϵ-entrophy of classBH^∞(Ω) in the metricC(K). In the case whenKhas a simply connected complement inC and Ω is a canonical neighbourhood ofK, the classical tools for approximation off∈BH^∞(Ω) inC(K) give the Faber series. This work is concerned with the following: the exact values of Kolmogorov and othern-widths of Hardy spacesH^p, then-widths and ϵ-entrophy of classBH^∞(Ω), the optimality of Faber approximations, and computing values of analytic functions with the help of Faber series.     

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.     

Particular features of implementation of an unsaturated numerical method for the exterior axisymmetric Neumann problem

Using Babenko’s profound ideas, we construct a fundamentally new unsaturated numerical method for solving the spectral problem for the operator of the exterior axisymmetric Neumann problem for Laplace’s equation. We estimate the deviation of the first eigenvalue of the discretized problem from the eigenvalue of the Neumann operator. More exactly, the unsaturated discretization of the spectral Neumann problem yields an algebraic problem with a good matrix, i.e., a matrix inheriting the spectral properties of the Neumann operator. Thus, its spectral portrait lacks “parasitic” eigenvalues provided that the discretization error is sufficiently small. The error estimate for the first eigenvalue involves efficiently computable parameters, which in the case of C ^∞-smooth data provides a foundation for a guaranteed success.     

A singular Monge-Ampère equation on unbounded domains

In this paper, we study the Dirichlet problem for a singular Monge-Amp`ere type equation on unbounded domains. For a few special kinds of unbounded convex domains, we find the explicit formulas of the solutions to the problem. For general unbounded convex domain ?, we prove the existence for solutions to the problem in the space C∞(?) ∩ C(?). We also obtain the local C1/2-estimate up to the ?? and the estimate for the lower bound of the solutions.     

On the closedness of the algebraic difference of closed convex sets

We characterize in a reflexive Banach space all the closed convex sets C₁ containing no lines for which the condition C₁^∞∩C₂^∞={0} ensures the closedness of the algebraic difference C₁−C₂ for all closed convex sets C₂. We also answer a closely related problem: determine all the pairs C₁, C₂ of closed convex sets containing no lines such that the algebraic difference of any sufficiently small uniform perturbations of C₁ and C₂ remains closed. As an application, we state the broadest setting for the strict separation theorem in a reflexive Banach space.     

Spectral analysis and synthesis on symmetric spaces

The problem of spectral analysis and synthesis is studied in the space C^∞ (X) of smooth functions on symmetric space X of noncompact type and of rank one. One proves that the spectral analysis and synthesis hold for those invariant subspaces of C^∞(X) whose spectrum does not contain the zeroes of the Harish-Chandra function c.     

Global solutions with shock waves to the generalized Riemann problem for a system of hyperbolic conservation laws with linear damping

It is proven that a class of the generalized Riemann problem for quasilinear hyperbolic systems of conservation laws with the uniform damping term admits a unique global piecewise C¹ solution u=u(t,x) containing only n shock waves with small amplitude on t?0 and this solution possesses a global structure similar to that of the similarity solution of the corresponding homogeneous Riemann problem. As an application of our result, we prove the existence of global shock solutions, piecewise continuous and piecewise smooth solution with shock discontinuities, of the flow equations of a model class of fluids with viscosity induced by fading memory with a single jump initial data. We also give an example to show that the uniform damping mechanism is not strong enough to prevent the formation of shock waves.     

Averaging of the planetary 3D geostrophic equations with oscillating external forces

In this article, we consider a non-autonomous three-dimensional planetary geostrophic model of the ocean with a singularly oscillating external force depending on a small parameter ?. We prove the existence of the uniform global attractor A^?. Furthermore, using the method of [11] in the case of the two-dimensional Navier-Stokes systems, we study the convergence of A^? as ? goes to zero.