Asymptotic expansions and summability with respect to an analytic germ

Abstract

In a previous article [CMS], monomial asymptotic expansions, Gevrey asymptotic expansions, and monomial summability were introduced and applied to certain systems of singularly perturbed differential equations. In the present work, we extend this concept, introducing (Gevrey) asymptotic expansions and summability with respect to a germ of an analytic function in several variables-this includes polynomials. The reduction theory of singularities of curves and monomialization of germs of analytic functions are crucial to establish properties of the new notions, for example a generalization of the Ramis-Sibuya theorem for the existence of Gevrey asymptotic expansions. Two examples of singular differential equations are presented for which the formal solutions are shown to be summable with respect to a polynomial: one ordinary and one partial differential equation.

Boundary layer expansions for initial value problems with two complex time variables

“…near 0 ∈ C 2 . That situation is directly linked to a summability procedure with respect to a germ of function in C 2 , as described in [19]. However, in our situation, the function Ω is meromorphic near 0, not analytic.…”

“…The function U ξ defined in (19) turns out to be an actual solution of the auxiliary problem (20) in the domainT 1 ×T 2 × R × E. Moreover, the next estimates hold…”

“…We construct inner and outer solutions of the problem and relate them to asymptotic representations via Gevrey asymptotic expansions with respect to ǫ, in adequate domains. The construction of such analytic solutions is closely related to the procedure of summation with respect to an analytic germ, put forward in [19], whilst the asymptotic representation leans on the cohomological approach determined by Ramis-Sibuya Theorem.under given initial data u(0, t 2 , z, ǫ) ≡ u(t 1 , 0, z, ǫ) ≡ 0. Here, Q(X) ∈ C[X] and P (T 1 , T 2 , Z, z, ǫ) stands for a polynomial in (T 1 , T 2 , Z) with holomorphic coefficients w.r.t.…”

Tauberian theorems for k–summability with respect to an analytic germ

“…The main purpose of the operatorsT α , T α ,T P,ℓ and T P,ℓ is to provide a characterization of x α -and P -asymptotic expansion in terms of classical asymptotic expansions in one variable, respectively. In this context, we state the following result [8,Thm. 4.9].…”

“…If this is not the case, we can use Theorem 5.5 to conclude that E(P ) +Ê(Q) is not P -k-summable, for any P, k, but it is still a formal solution of the system (10). Finally, if P and Q are polynomials, so is L j (Ê(P ) +Ê(Q)), andÊ(P ) +Ê(Q) is a solution of the polynomial differential equation ∂ N j L j (y) = 0, for an appropriate N ∈ N. We refer the reader to Examples 8.1 and 8.2 in [8] for a singular ordinary and a partial differential equation with P -1-summable formal solutions, respectively, where P is a polynomial in two variables with certain conditions. We note that due to Theorem 5.5, P -1-summability is essentially the only Q-k-summability method applicable to these formal solutions.…”