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Cauchy problem in function spaces with asymptotic expansions with respect to time variable

Sunao Ouchi

Abstract

A system of nonlinear Cauchy problem $\partial_t u_i=f_i(t,x, U, \nabla_xU )$ $u_i(0,x)= u_{i,0}(x)$ is studied in function spaces with asymptotic expansion with respect to $t$. To be specific, it is discussed in Borel summable or multisummable function space.It is recognized that these functions are important classes in asymptotic analysis. We study equations under the condition $\{f_i(t,x, U, P)\}_{i=1}^m$ are in these function spaces with respect to $t$ and show $\{u_i(t,x)\}_{i=1}^m$ have also the same summability.

Cauchy problem in function spaces with asymptotic expansions with respect to time variable

Abstract

A system of nonlinear Cauchy problem is studied in function spaces with asymptotic expansion with respect to . To be specific, it is discussed in Borel summable or multisummable function space.It is recognized that these functions are important classes in asymptotic analysis. We study equations under the condition are in these function spaces with respect to and show have also the same summability.

Paper Structure

This paper contains 16 sections, 24 theorems, 136 equations.

Table of Contents

  1. Introduction
  2. Notations and definitions
  3. Cauchy Problem in Borel summable function spaces
  4. Preliminary
  5. Convolution equation
  6. Calculation of ${\mathcal{G}}_{r,\ell}(\xi,x)$
  7. Estimate and convergence of $\widehat{v}_r(\xi,x,\varepsilon) =\sum_{\ell=1}^{\infty} \widehat{v}_{r,\ell}(\xi,x)\varepsilon^\ell$
  8. Majorant functions
  9. Estimate
  10. Proof of Proposition \ref{['prop-main']} constants $\{M_{r,\ell}\}$
  11. Proof of Proposition \ref{['prop-main']} -- convergence
  12. Multisummable functions
  13. Acceleration operator
  14. k-Multisummable functions
  15. Cauchy problem in ${\bf k}$-multisummable function spaces
  16. ...and 1 more sections

Key Result

Proposition 1.5

Let $I=(\theta-\delta,\theta+\delta)$ with $\delta>\pi/2k$ and $\widehat{I}=(\theta-\epsilon_*, \theta+\epsilon_*)$, $\epsilon_*:=\delta-\pi/2k>0$. Let $\phi(t,Y)\in {\mathscr O}_{\{1/k\}}(S_{0}(I)\times \Omega)$ with ${\phi}(0,Y)=0$. Then $({\mathscr B}_{k,\theta}{\phi})(\xi,Y)$ is holomorphically holds in $(S(\widehat{I}_\epsilon)\cup \{0<|\xi|<r\})\times \Omega$. $\phi(t,Y)$ is represented by

Theorems & Definitions (42)

  • Definition 1.1
  • Definition 1.2
  • Remark 1.3
  • Definition 1.4
  • Proposition 1.5
  • Lemma 1.6
  • Lemma 1.7
  • proof
  • Definition 1.8
  • Remark 1.9
  • ...and 32 more