Convergence des solutions formelles de certaines équations fonctionnelles

Résumé Soitq un nombre algébrique de module 1, qui ne soit pas une racine de l'unité, etP [X, Y ₀,Y ₁] un polynôme non nul. Dans cet article, nous montrons que toute solution de l'équation fonctionnelleP(z, (z), (qz))=0, qui est une série formelle (z) dansQ[[z]], a un rayon de convergence non nul.

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Summary Letq Q be an algebraic number of modulus one that is not a root of unity. LetP Q[X, Y ₀,Y ₁] be a non zero polynomial. In this paper, we show that every formal power series,(z) Q[[z]], solution of the functional equationP(z), (z), (qz)) = 0 has a non zero radius of convergence.

     

Sur les racines de certaines équations

Ohne Zusammenfassung     

Sur les racines de certaines équations

Sans résumé     

Une remarque sur les équations fonctionnelles

Sans résumé     

Sur les équations fonctionnelles auxq-différences

Summary In this paper, we study the convergence of formal power series solutions of functional equations of the formP _k(x)(^[k](x))=(x), where ^[k] (x) denotes thek-th iterate of the function.We obtain results similar to the results of Malgrange and Ramis for formal solutions of differential equations: if(0) = 0, and(0) =q is a nonzero complex number with absolute value less than one then, if(x)=a(n)x ⁿ is a divergent solution, there exists a positive real numbers such that the power seriesa(n)q ^sn(n+1)2 x ⁿ has a finite and nonzero radius of convergence.

     

Problème de Cauchy pour certaines équations du type de Schrödinger

We consider a 2-evolution operator in the sense of Petrowsky, and we assume that the characteristic roots of the principal polynomial with constant coefficients are real and of constant multiplicities. Then we give sufficient conditions so that the Cauchy problem both for the future and for the past is well-posed in the Sobolev spaces. Our conditions are analogous to the Levi conditions and to the decomposition conditions of operators in the hyperbolic case for Kowalewskian operators.     

Sur les équations fonctionnelles aux itérées

Summary We study solutions of functional equationsP(f ^[10] ,,f ^[s] ) = 0, whereP is a non zero polynomial ins + 1 variables andf ^[k] denotes thekth iterate of a functionf. We deal with three distinct cases: first,f is an entire function of a complex variable, we show then thatf is a polynomial. Second, we also prove thatf is a polynomial if it is an entire function of ap-adic variable. Third, we considerf a formal power series with coefficients in a number fieldK; subject to some apparently natural restrictions onf and onP, we find thatf is an algebraic power series over the ring of polynomials inK[x].

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Sur les équations fonctionnelles aux itérées

     

Sur l'integration de certaines équations linéaires aux dérivées partielles

Résumé Il s'agit d'équations linéaires aux dérivees partielles d'ordre p+q E_p+q, dont la solution s'obtient en intégrant une équation d'ordre p, E_p, et une équation d'ordre q, E_q, toute solution de E_p+q étant la somme d'une solution de E_p et d'une solution de E_q. Le cas de deux variables indépendantes est traité complètement; des résultats partiels sont donnés pour un nombre quelconque de variables indépendantes. à M. Enrico Bompiani pour son Jubilé scientifique.     

Solutions de type multi-soliton des équations de KdV généralisées

We consider the generalized Korteweg–de Vries equations in the subcritical and critical cases. Let R_j(t,x)=Q_cj(x?c_jt?x_j) be N soliton solutions of this equation, with corresponding speeds 0<c₁<c₂<?<c_N. In this Note, we give a sketch of the proof of the following result. Given $\text{{c}{}_{\mathrm{j}}\text{},}\text{{x}{}_{\mathrm{j}}\text{}}$, there exists one and only one solution ? of the generalized KdV equation such that 6?(t)?∑R_j(t)6_H¹→0 as t→+∞. Complete proofs will appear later. To cite this article: Y. Martel, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Sur les équations différentielles ordinaires et les équations de transport

We study in this Note ordinary differential equations for divergence-free vector-fields with a limited regularity. We first observe that it is equivalent to solve the associated transport equations (i.e. Liouville equations). Then, we show existence, uniqueness, and stability results for generic vector-fields in L¹ or for “piecewise” W^1.1 vector-fields.     

Solutions périodiques des équations d'évolution

We study here the necessary and sufficient condition for the existence of periodic solutions for evolution equations in the case of linear unbounded maximal monotonous and symmetric operators. We present also an estimate of the convergence time to the periodic states in the finite-dimensional case.     

Opérateurs de Dirac et équations de Maxwell

Sans résumé     

Cascades aléatoires et équations de Navier-Stokes

This work concerns the incompressible Navier-Stokes equation in. The non linear integral equation satisfied bx the Fourier transform of the Laplacian of the velocity field can be interpreted in terms of a branching Markov process and of a composition rule defined along the associated tree. We derive from this representation new classes where global existence and uniqueness can be proven.     

Solutions faibles d'énergie infinie pour les équations de Navier—Stokes dans ℝ3

We prove existence of suitable weak solutions for the Navier-Stokes equations on (0, ∞) × ℝ³ when the initial value is uniformly locally square integrable and vanishes at infinity.     

Approximations filaires pour les équations de Poisson et de Maxwell harmoniques

We are interested in the approximation of the Poisson and time harmonic Maxwell equations around a circular cylinder with a small radius. Numerical approximation requires a typical meshsize comparable to the radius of the cylinder, thus increasing the computational cost. In many applications (spacecraft charging computation, wire antennas modelling) the mesh size becomes too large to perform realistic computations. A mathematical study proves the convergence of the sequence $u_{\mathrm{?}}$ of Poisson equation solutions when the radius ? goes to 0 toward a limit problem with a convergence rate of $1/\ln (\mathrm{?})$. A wire approximation is proposed exhibiting a rate of convergence larger that $\sqrt{\mathrm{?}}$ without constraint of mesh refinement related to the radius size. This method is extended to the case of arbitrary wire cross section and also to the time harmonic Maxwell equations. To cite this article: F. Rogier et al., C. R. Acad. Sci. Paris, Ser. I 343 (2006).