Asymptotic expansions with exponential, power, and logarithmic functions for non-autonomous nonlinear differential equations

Abstract

This paper develops further and systematically the asymptotic expansion theory that was initiated by Foias and Saut in (Ann Inst H Poincaré Anal Non Linéaire, 4(1):1–47 1987). We study the long-time dynamics of a large class of dissipative systems of nonlinear ordinary differential equations with time-decaying forcing functions. The nonlinear term can be, but not restricted to, any smooth vector field which, together with its first derivative, vanishes at the origin. The forcing function can be approximated, as time tends to infinity, by a series of functions which are coherent combinations of exponential, power and iterated logarithmic functions. We prove that any decaying solution admits an asymptotic expansion, as time tends to infinity, corresponding to the asymptotic structure of the forcing function. Moreover, these expansions can be generated by more than two base functions and go beyond the polynomial formulation imposed in previous work.

A Appendix

We discuss a particular application of our results to numerical approximations of a nonlinear PDE problem using ODE systems. The presentation below is focused on the ideas without showing technical details.

Consider the Navier–Stokes Eq. (1.3) with a given initial data $u(0)=u_0$. For $m\in \mathbb {N}$, let $P_m$ denote the orthogonal projection to the first m eigenspaces (corresponding to the first m distinct eigenvalues) of the Stokes operator A.

The Galerkin approximation problem is

$$\begin{aligned} \frac{\mathrm{d} u_m}{\mathrm{d}t} + Au_m +B_m(u_m,u_m)=P_m f,\quad u_m(0)=P_m u_0, \end{aligned}$$

(A.1)

where $B_m(u,u)=P_mB(u,u)$. For each $m\in \mathbb {N}$, the approximate system (A.1) is an ODE system in a finite-dimensional space, and $B_m(\cdot ,\cdot )$ is a bilinear form. Thus, the results obtained in previous sections apply.

Consider Type 1, 2, 3 problems as in Sect. 4, that is,

$$\begin{aligned} f(t)\sim \sum _{k=1}^\infty p_k(\phi (t))\psi (t)^{-k}, \end{aligned}$$

(A.2)

where the base functions $\phi (t)$ and $\psi (t)$ are given in Definition 4.2.

Then, the solutions u(t) and $u_m(t)$ have the asymptotic expansions

$$\begin{aligned} u(t)\sim \sum _{k=1}^\infty q_k(\phi (t))\psi (t)^{-k}\text { and } u_m(t)\sim \sum _{k=1}^\infty q_k^{(m)}(\phi (t))\psi (t)^{-k}, \text { respectively.} \end{aligned}$$

(A.3)

The question is whether $q_k^{(m)}$ converges to $q_k$ as $m\rightarrow \infty$ in a certain sense.

First, we roughly have

$$\begin{aligned} B_m(u,u)\rightarrow B(u,u),\ P_m u_0\rightarrow u_0\text { and }P_mf\rightarrow f \text { as }m\rightarrow \infty . \end{aligned}$$

(A.4)

(The normed spaces in which the convergences hold depend on the regularity of u, $u_0$ and f.)

For Types 2 and 3, the polynomials $q_k$’s are independent of the solution u(t), depend only on $p_k$ and $B(\cdot ,\cdot )$. Similarly, for each $m\in \mathbb {N}$, the polynomials $q_k^{(m)}$’s are independent of the individual solution $u_m(t)$, depend only on $P_mp_k$ and $P_mB(\cdot ,\cdot )$. With the convergences in (A.4) and explicit formulas (4.57) and (4.58), it is likely that the coefficients of $q_k^{(m)}$ converge to its corresponding coefficients of $q_k(t)$, as $m\rightarrow \infty$.

For Type 1, we consider the case u(t) is a unique, regular solution on $[0,\infty )$. The construction of polynomial $q_k$, respectively $q_k^{(m)}$, depends on the long-time values of u(t), respectively $u_m(t)$. Therefore, determining the convergence of $q_k^{(m)}$ to $q_k$, as $m\rightarrow \infty$, is more subtle than in the case of Types 2 and 3. However, we only consider the convergence for each fixed k, and, in light of many related estimates in previous work such as [8, 9, 11, 13], it may still be possible to prove such a convergence.