Abstract. In recent years, the theory of Borel summability or multisummability of divergent power series of one variable has been established and it has been proved that every formal solution of an ordinary differential equation with irregular singular point is multisummable. For partial differential equations the summability problem for divergent solutions has not been studied so well, and in this paper we shall try to develop the Borel summability of divergent solutions of the Cauchy problem of the complex heat equation, since the heat equation is a typical and an important equation where we meet diveregent solutions. In conclusion, the Borel summability of a formal solution is characterized by an analytic continuation property together with its growth condition of Cauchy data to infinity along a stripe domain, and the Borel sum is nothing but the solution given by the integral expression by the heat kernel. We also give new ways to get the heat kernel from the Borel sum by taking a special Cauchy data. §0. Introduction and main resultsWe consider the following "characteristic" Cauchy problem for the complex heat equationwhere τ = t + iη and z = x + iy are complex variables and ϕ(z) is assumed to be analytic in a domain D, which we assume to contain a neighborhood of the origin without loss of generality.As easily seen, this Cauchy problem has the unique formal solution. This formal solution diverges for general Cauchy data ϕ(z), as is shown by the example ϕ(z) = (1 − z) −1 , where By Cauchy's integral formula, for any r > 0 such that {|z| ≤ r} ⊂ D, we have the following estimate for {u n (z)} ∞ n=0 by taking some positive constants A and B, max |z|≤r |u n (z)| ≤ AB n n!, for all n = 0, 1, 2, . . . . (0.3) Such a formal power series is called Gevrey of order one in the variable τ .In the one variable case of ordinary differential equations, after the fundamental work of J. Ecalle, great progress has been made in the theory of summability for Gevrey type power series by many authors. As an application, they proved that every formal solution of an analytic ordinary differential equation is multi-summable (cf. on an asymptotic interpretation of formal solutions. Our purpose in this paper is to improve the asymptotic results by discussing the summation of formal solutions and under which conditions this is possible or not possible. While some of the results could possibly be extended to more general differential equations, we choose to consider only the heat equation since in this simple case we obtain both necessary and sufficient conditions.It is natural to ask the following questions about the formal solution u(τ, z) constructed above:[1] under what conditions on ϕ(z) does the formal solution (0.2) converge?[2] if (0.2) diverges, is it an asymptotic expansion of actual solutions as τ → 0 in sectorial domains?[3] if asymptotic solutions exist, when are they unique and how can they be constructed?As regards question [1], an easy way to get a sufficient condition for (0.2) to converges is to assume the Cauchy data...
We consider (not necessarily conservative) perturbations of a one phase Hamiltonian system written with action-angle variables ẋ = 0, φ = ω(x), where ω c > 0 is real analytic, of the form ẋ = εf (x, ϕ, ε), φ = ω(x) + εg(x, ϕ, ε), ( * ) where f, g are real analytic in all the variables and 2π-periodic in ϕ. More generally we consider systems similar to system ( * ) with x ∈ R m . It is well known that, using an iterated averaging process, it is posssible to eliminate 'formally in ε' the phase ϕ by a formal transformation tangent to the identitywhere Û, V are 2π-periodic in ψ.Then one obtains a formal autonomous system( * * ) Fixing the normalization Û(y, 0, ε) = 0, V (y, 0, ε) = 0, we prove that the transform T is Gevrey 1 in ε. As an application, we give a new proof of a result of Neishstadt: it is possible to represent the formal transformation T by an actual transformation T (admitting T as its asymptotic expansion) such that the transformation T reduces the system ( * ) to a system which is autonomous up to perturbations which are exponentially small in ε. It is possible to use a cut-off 'at the smallest term' like Neishstadt but we prefer to use an incomplete Laplace transform. Then we obtain for T a nice dependence in ε and we improve Neishstadt's result. We will also give similar improvements for the basic adiabatic invariants theory.Our main statement generalizes a theorem of D Sauzin conjectured by P Lochak.
International audienceIn this work, we consider systems of differential equations that are doubly singular, i.e. that are both singularly perturbed and exhibit an irregular singular point. Under a condition that also guanrantees the existence of a unique formal solution, we show that this formal solution is monomially summable, i.e. summable with respect to the monomial in the independent variable and in the parameter in a (new) sense that will be defined. As a preparation, Poincaré asymptotics and Gevrey asymptotics in a monomial are studied
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