Movable Singularities of Solutions of Difference Equations in Relation to Solvability and a Study of a Superstable Fixed Point

We review applications of exponential asymptotics and analyzable function theory to difference equations in defining an analogue of the Painlevé property for them, and we sketch the conclusions about the solvability properties of first-order autonomous difference equations. If the Painlevé property is present, the equations are explicitly solvable; otherwise, under additional assumptions, the integrals of motion develop singularity barriers. We apply the method to the logistic map x _n+1=ax _n (1–x _n ), where we find that the only cases with the Painlevé property are a=–2,0,2, and 4, for which explicit solutions indeed exist; otherwise, an associated conjugation map develops singularity barriers.     

Large fluctuations and growth rates of linear Volterra summation equations

This paper concerns the asymptotic behaviour of solutions of a linear convolution Volterra summation equation with an unbounded forcing term. In particular, we suppose the kernel is summable and ascribe growth bounds to the exogenous perturbation. If the forcing term grows at a geometric rate asymptotically, or is bounded by a geometric sequence, then the solution (appropriately scaled) omits a convenient asymptotic representation. Moreover, this representation is used to show that additional growth properties of the perturbation are preserved in the solution. If the forcing term fluctuates asymptotically, we prove that fluctuations of the same magnitude will be present in the solution and we also connect the finiteness of time averages of the solution with those of the perturbation. Our results, and corollaries thereof, apply to stochastic as well as deterministic equations, and we demonstrate this by studying some representative classes of examples.     

q-Borel-Laplace summation for q–difference equations with two slopes

In this paper, we consider linear q-difference systems with coefficients that are germs of meromorphic functions, with Newton polygon that has two slopes. Then, we explain how to compute similar meromorphic gauge transformations than those appearing in the work of Bugeaud, using a q-analogue of the Borel–Laplace summation.     

On singular solutions of time-periodic and steady Stokes problems in a power cusp domain

The time-periodic and steady Stokes problems with the boundary value having a nonzero flux are considered in the power cusp domains. The asymptotic expansion near the singularity point is constructed in order to reduce the problem to the case where the energy solution exists. The solution of the problem is found then as the sum of the asymptotic expansion and the term with finite dissipation of energy.     

Harnack's inequality for cooperative weakly coupled elliptic systems

Résumé

On étudie le comportement pour les grands temps des solutions de l'équation de Navier–Stokes dans la bande R ² × (0, 1). Après reformulation du problème à l'aide de variables auto-similaires, on calcule un développement asymptotique en temps de la vorticité jusqu'au second ordre, en supposant que la vorticité initiale est suffisamment petite et décro??t de manire polyno?miale à l'infini. Dans un deuxième temps, sans cette hypothèse de petitesse sur la donnée initiale, on prouve que, de nouveau, le comportement asymptotique des solutions globales est régi par l'équation de Navier–Stokes bidimensionnelle. En particulier, on montre que de telles solutions convergent vers le tourbillon d'Oseen.

Abstract

We study the long-time behavior of solutions of the Navier–Stokes equation in R ² × (0, 1). After introducing self-similar variables, we compute the long-time asymptotics of the vorticity up to second order, assuming that the initial vorticity is sufficiently small and has polynomial decay at infinity. Afterwards, we relax this smallness assumption and we prove again that the long-time behavior of global bounded solutions is governed by the two-dimensional Navier–Stokes equation. In particular, we show that solutions converge towards Oseen vortices.     

Eigenvalue Asymptotics of the Stokes Operator for Fractal Domains

The leading asymptotic term for the function that counts theeigenvalues of the Stokes operator is determined for fairlygeneral underlying bounded domains. Moreover, the remainderis estimated in terms of the fractality of the boundary of thedomain. The results obtained resemble corresponding ones forthe Dirichlet Laplacian. 1991 Mathematics Subject Classification:35P20.     

On the smallest eigenvalue of the Stokes operator in a domain with fine-grained random boundary

This article deals with a problem arising in localization of the principal eigenvalue (PE) of the Stokes operator under the Dirichlet condition on the fine-grained random boundary of a domain contained in a cube of size t ? 1. The random microstructure is assumed identically distributed in distinct unit cubic cells and, in essence, independent. In this setting, the asymptotic behavior of the PE as t → ∞ is deterministic: it proves possible to find nonrandom upper and lower bounds on the PE which apply with probability that converges to 1. It was proved earlier that in two dimensions the nonrandom unilateral bounds on the PE can be chosen asymptotically equivalent, which implies the convergence in probability to a nonrandom limit of the appropriately normalized PE. The present article extends this result to higher dimensions.     

Steady flows of Jeffrey–Hamel type from the half-plane into an infinite channel. 1. Linearization on an antisymmetric solution

In this paper our objective is to provide physically reasonable solutions for the stationary Navier–Stokes equations in a two-dimensional domain with two outlets to infinity, a semi-strip Π₋ and a half-plane K. The same problem in an aperture domain, i.e. in a domain with two half-plane outlets to infinity, has been studied but only under symmetry restrictions on the data. Here, we assume that the main asymptotic term of the solution takes an antisymmetric form in K and apply the technique of weighted spaces with detached asymptotics, i.e. we use spaces where the functions have prescribed asymptotic forms in the outlets.After first showing that the corresponding Stokes problem admits a unique solution if and only if certain compatibility conditions are satisfied, we write the Navier–Stokes equations as a perturbation of the Stokes problem and the crucial compatibility condition as an algebraic equation by which the flux becomes determined. Assuming that the coefficient of the main (antisymmetric) asymptotic term of the solution in K does not vanish and that the data are sufficiently small, we use a contraction principle to solve the Navier–Stokes system coupled with the algebraic equation.Finally, we discuss the ill-posedness of the Navier–Stokes problem with prescribed flux.     

Universality and asymptotics of graph counting problems in non-orientable surfaces

Bender-Canfield showed that a plethora of graph counting problems in orientable/non-orientable surfaces involve two constants tg and pg for the orientable and the non-orientable case, respectively. T.T.Q. Le and the authors recently discovered a hidden relation between the sequence tg and a formal power series solution u(z) of the Painlevé I equation which, among other things, allows to give exact asymptotic expansion of tg to all orders in 1/g for large g. The paper introduces a formal power series solution v(z) of a Riccati equation, gives a non-linear recursion for its coefficients and an exact asymptotic expansion to all orders in g for large g, using the theory of Borel transforms. In addition, we conjecture a precise relation between the sequence pg and v(z). Our conjecture is motivated by the enumerative aspects of a quartic matrix model for real symmetric matrices, and the analytic properties of its double scaling limit. In particular, the matrix model provides a computation of the number of rooted quadrangulations in the 2-dimensional projective plane. Our conjecture implies analyticity of the O(N)- and Sp(N)-types of free energy of an arbitrary closed 3-manifold in a neighborhood of zero. Finally, we give a matrix model calculation of the Stokes constants, pose several problems that can be answered by the Riemann-Hilbert approach, and provide ample numerical evidence for our results.     

Borel equivalence relations between <Emphasis Type="Italic">ℓ</Emphasis><Subscript>1</Subscript> and <Emphasis Type="Italic">ℓ</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>

In this paper, we show that, for each p 〉 1, there are continuum many Borel equivalence relations between Rω/l1 and Rω/p ordered by ≤B which are pairwise Borel incomparable.     

Multi-resurgence and connection-to-Stokes formulæ for some linear meromorphic differential systems

In this article, we consider a linear meromorphic differential system with several levels $r_1<...<r_p$. For any k, we prove that the Borel transforms of its $r_k$-reduced formal solutions are resurgent and we give the general form of all their singularities. Next, under some convenient hypotheses on the geometric configuration of singular points, we display exact formulæ to express some Stokes multipliers of level $r_k$ of initial system in terms of connection constants in the Borel plane, generalizing thus formulæ already obtained by M. Loday-Richaud and the author for systems with a single level. As an illustration, we develop one numerical example.     

Weighted Means, Strict Ergodicity, and Uniform Distributions

We strengthen the well-known Oxtoby theorem for strictly ergodic transformations by replacing the standard Cesaro convergence by the weaker Riesz or Voronoi convergence with monotonically increasing or decreasing weight coefficients. This general result allows, in particular, to strengthen the classical Weyl theorem on the uniform distribution of fractional parts of polynomials with irrational coefficients.     

Axial Borel functions

A function from the plane to the plane is axial if it does not change one coordinate. We show that every Borel permutation of the plane is a superposition of 11 Borel axial permutations.     

Discrete spectrum in a critical coupling case of Jacobi matrices with spectral phase transitions by uniform asymptotic analysis

For a two-parameter family of Jacobi matrices exhibiting first-order spectral phase transitions, we prove discreteness of the spectrum in the positive real axis when the parameters are in one of the transition boundaries. To this end, we develop a method for obtaining uniform asymptotics, with respect to the spectral parameter, of the generalized eigenvectors. Our technique can be applied to a wide range of Jacobi matrices.