Asymptotic expansions with exponential, power, and logarithmic functions for non-autonomous nonlinear differential equat
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Journal of Evolution Equations
Asymptotic expansions with exponential, power, and logarithmic functions for non-autonomous nonlinear differential equations Dat Cao and Luan Hoang
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 timedecaying 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.
Contents 1. 2. 3. 4. 5.
Introduction Preliminaries Basic existence result and large-time estimates Expansions with one secondary base function Expansions with multiple secondary base functions 5.1. Iterated logarithmic expansions 5.2. Mixed power and iterated logarithmic expansions A Appendix REFERENCES
1. Introduction This work is motivated by a deep result by Foias and Saut [12], which is on the longtime behavior of solutions of the Navier–Stokes equations, and its later developments Mathematics Subject Classification: 34E05, 34E10, 41A60 Keywords: Asymptotic expansions, Long-time dynamics, Non-autonomous systems, Dissipative dynamical systems, Perturbations.
J. Evol. Equ.
D. Cao and L. Hoang
in [3,4,13,16–19,21]. In the original work [12], Foias and Saut studied the Navier– Stokes equations written in the functional form (on an appropriate infinite-dimensional space) as (1.1) u t + Au + B(u, u) = 0, where A is a linear operator, and B is a bilinear form. They established the following asymptotic expansion, as t → ∞, u(t) ∼
∞
qk (t)e−μk t ,
(1.2)
k=1
where qk (t)’s are polynomials in t, and μk increases to infinity. Roughly speaking, expansion (1.2) means that for each N , the solution u(t) can be approximated by the finite sum N qk (t)e−μk t , s N (t) := k=1
in the sense that the remainder u(t)−s N (t) decays exponentially faster than the fastest decaying mode e−μ N t in s N (t), see Definition 1.4 for the precise meaning. Expansion (1.2) is studied in more detail in [6–10,13,16] regarding its convergence, approximation in Gevrey spaces, associated invariant nonlinear manifolds and normal form, and connection to the theory of Poincaré–Dulac normal form, applications to statistical solutions and turbulence theory, etc. A similar expansion to (1.2) is also established in [18] for the Navier–Stokes equations of rotating fluids. Besides the Navier–Stokes equations, expans
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