Some Notes on the Multi-Level Gevrey Solutions of Singularly Perturbed Linear Partial Differential Equations

2018

Results Math

We study a family of nonlinear initial value partial differential equations in the complex domain under the action of two asymmetric time variables. Different Gevrey bounds and multisummability results are obtain depending on each element of the family, providing a more complete picture on the asymptotic behavior of the solutions of PDEs in the complex domain in several complex variables.The main results lean on a fixed point argument in certain Banach space in the Borel plane, together with a Borel summability procedure and the action of different Ramis-Sibuya type theorems. variable coefficients (see [5,6,21]). The second author [22], both authors [13] and the two authors and J. Sanz [17] have also contributed in this theory. Recently, multisummability of formal solutions of PDEs has also been put forward in different works. W. Balser [4] described a multisummability phenomenon in certain PDEs with constant coefficients. S. Ouchi [23] constructed multisummable formal solutions of nonlinear PDEs, coming from perturbation of ordinary differential equations. H. Tahara and H. Yamazawa [24] have made progresses on general linear PDEs with non constant coefficients under entire initial data. In [20], G. Lysik constructs summable formal solutions of the one dimensional Burgers equation by means of the Cole-Hopf transform. O. Costin and S. Tanveer [8] construct summable formal power series in time variable to 3D Navier Stokes equations. The authors have obtained results in this direction [14, 15].A recent overview on summability and multisummability techniques under different points of view is displayed in [18].The purpose of the present work is to study the solutions of a family of singularly perturbed partial differential equations from the asymptotic point of view. More precisely, we consider a problem of the formThe elements which conform the nonlinear part P 1 , P 2 are polynomials in their second variable with coefficients being holomorphic functions defined on some neighborhood of the origin, say D(0, 0 ), continuous up to their boundary.Here, D(0, 0 ) stands for the open disc in the complex plane centered at 0, and with positive radius 0 > 0. We write D(0, 0 ) for its closure. Moreover, P stands for some polynomial of six variables, with complex coefficients, and the forcing term f (t 1 , t 2 , z, ) is a holomorphic and bounded function in D(0, ρ) 2 × H β × D(0, 0 ), for some ρ > 0, and where H β stands for the horizontal stripfor some β > 0.The precise configuration of the elements involved in the problem is stated and described in Section 2.2.This paper provides a step beyond in the study of the asymptotic behavior of the solutions of a subfamily of singularly perturbed partial differential equations of the form (1). We first recall some previous advances made in this respect, which motivate the present framework.In [13], we studied under the asymptotic point of view the solutions of certain family of PDEs of the formwhere the elements involved in the problem depend only on one time variable t. Our next aim w...

Section: Introductionsupporting

confidence: 56%

“…We divide by and multiply by T k 2 +1 2 at both sides of (15). Under the assumptions displayed in (6) one may apply (18) and (19) in order to rewrite equation (15). This step is important to exhibit the equations as an expression where some operators algebraically well-behaved with respect to Laplace transform appear.…”

Section: Layout Of the Main Problemmentioning

confidence: 99%

On Parametric Gevrey Asymptotics for Some Initial Value Problems in Two Asymmetric Complex Time Variables

Lastra

2018

Results Math

“…In Section 3, we remind the reader of basic statements concerning -Borel-Laplace transforms, a version of the classical Borel-Laplace maps already used in previous works [3,22,23] and Fourier transforms acting on exponentially flat functions.…”

Section: ̂(mentioning

confidence: 99%

“…This novel version has already been used in works such as [3,22] when studying Cauchy problems under the presence of a small perturbation parameter. We remind also the reader of the definition of Fourier inverse transform acting on functions with exponential decay.…”

Section: Borel-laplace and Fourier Transformsmentioning

confidence: 99%

On Singular Solutions to PDEs with Turning Point Involving a Quadratic Nonlinearity

Abstract and Applied Analysis

2017

Self Cite

We study a singularly perturbed PDE with quadratic nonlinearity depending on a complex perturbation parameter . The problem involves an irregular singularity in time, as in a recent work of the author and A. Lastra, but possesses also, as a new feature, a turning point at the origin in C. We construct a family of sectorial meromorphic solutions obtained as a small perturbation in of a slow curve of the equation in some time scale. We show that the nonsingular parts of these solutions share common formal power series (that generally diverge) in as Gevrey asymptotic expansion of some order depending on data arising both from the turning point and from the irregular singular point of the main problem.

“…The case of a complex perturbation parameter has led to results in which the nature of the singularities arising from the problem describe different types of singularities. We refer to the work by M. Canalis-Durand, J. Mozo-Fernández and R. Schäfke [3], the second author [18], the authors [7] and the authors and J. Sanz [14]. In [8], the authors study the parametric multi-level Gevrey solutions coming from a splitting of the equation which generates two Gevrey levels.…”

Section: Introductionmentioning

confidence: 99%

On multiscale Gevrey and Gevrey asymptotics for some linear difference differential initial value Cauchy problems

Lastra

Journal of Difference Equations and Applications

2017

We study the asymptotic behavior of the solutions related to a singularly perturbed q-differencedifferential problem in the complex domain. The analytic solution can be splitted according to the nature of the equation and its geometry so that both, Gevrey and q−Gevrey asymptotic phenomena are observed and can be distinguished, relating the analytic and the formal solution.The proof leans on a two level novel version of Ramis-Sibuya theorem under Gevrey and q-Gevrey orders.