Abstract
We study the precise asymptotic behavior of a non-trivial solution that converges to zero, as time tends to infinity, of dissipative systems of nonlinear ordinary differential equations. The nonlinear term of the equations may not possess a Taylor series expansion about the origin. This absence technically cripples previous proofs in establishing an asymptotic expansion, as an infinite series, for such a decaying solution. In the current paper, we overcome this limitation and obtain an infinite series asymptotic expansion, as time goes to infinity. This series expansion provides large time approximations for the solution with the errors decaying exponentially at any given rates. The main idea is to shift the center of the Taylor expansions for the nonlinear term to a non-zero point. Such a point turns out to come from the non-trivial asymptotic behavior of the solution, which we prove by a new and simple method. Our result applies to different classes of non-linear equations that have not been dealt with previously.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bibikov, Y.N.: Local Theory of Nonlinear Analytic Ordinary Differential Equations. In: Lecture Notes in Mathematics, vol. 702. Springer-Verlag, Berlin-New York (1979)
Bruno, A.D.: Local Methods in Nonlinear Differential Equations. Springer Series in Soviet Mathematics. Springer-Verlag, Berlin (1989)
Bruno, A.D.: Power Geometry in Algebraic and Differential Equations. North-Holland Mathematical Library, vol. 57. North-Holland Publishing Co., Amsterdam (2000)
Bruno, A.D.: Asymptotic behavior and expansions of solutions of an ordinary differential equation. Uspekhi Mat. Nauk 59(3), 31–80 (2004)
Bruno, A.D.: Power-logarithmic expansions of solutions of a system of ordinary differential equations. Dokl. Akad. Nauk 419(3), 298–302 (2008)
Bruno, A.D.: Power-exponential expansions of solutions of an ordinary differential equation. Dokl. Akad. Nauk 444(2), 137–142 (2012)
Bruno, A.D.: On complicated expansions of solutions to ODES. Comput. Math. Math. Phys. 58(3), 328–347 (2018)
Cao, D., Hoang, L.: Asymptotic expansions in a general system of decaying functions for solutions of the Navier-Stokes equations. Ann. Mat. Pura Appl. 199(3), 1023–1072 (2020)
Cao, D., Hoang, L.: Asymptotic expansions with exponential, power, and logarithmic functions for non-autonomous nonlinear differential equations, J. Evol. Equ. 21(2), 1179–1225 (2021)
Cao, D., Hoang, L.: Long-time asymptotic expansions for Navier-Stokes equations with power-decaying forces. Proc. Roy. Soc. Edinburgh Sect. A 150(2), 569–606 (2020)
Coddington, E.A., Levinson, N.: Theory of Ordinary Differential Equations. McGraw-Hill Book Company Inc, New York (1955)
Cohen, P.J., Lees, M.: Asymptotic decay of solutions of differential inequalities. Pac. J. Math. 11, 1235–1249 (1961)
Foias, C., Hoang, L., Nicolaenko, B.: On the helicity in 3D-periodic Navier-Stokes equations. I. The non-statistical case. Proc. Lond. Math. Soc. 94(1), 53–90 (2007)
Foias, C., Hoang, L., Nicolaenko, B.: On the helicity in 3D-periodic Navier-Stokes equations. II. The statistical case. Comm. Math. Phys. 290(2), 679–717 (2009)
Foias, C., Hoang, L., Olson, E., Ziane, M.: On the solutions to the normal form of the Navier-Stokes equations. Indiana Univ. Math. J. 55(2), 631–686 (2006)
Foias, C., Hoang, L., Olson, E., Ziane, M.: The normal form of the Navier-Stokes equations in suitable normed spaces. Ann. Inst. H. Poincaré Anal. Non Linéaire 26(5), 1635–1673 (2009)
Foias, C., Hoang, L., Saut, J.-C.: Asymptotic integration of Navier-Stokes equations with potential forces. II. an explicit Poincaré-Dulac normal form. J. Funct. Anal. 260(10), 3007–3035 (2011)
Foias, C., Saut, J.-C.: Asymptotic behavior, as \(t\rightarrow +\infty \), of solutions of Navier-Stokes equations and nonlinear spectral manifolds. Indiana Univ. Math. J. 33(3), 459–477 (1984)
Foias, C., Saut, J.-C.: Linearization and normal form of the Navier-Stokes equations with potential forces. Ann. Inst. H. Poincaré Anal. Non Linéair 4(1), 1–47 (1987)
Foias, C., Saut, J.-C.: Asymptotic integration of Navier-Stokes equations with potential forces. I. Indiana Univ. Math. J. 40(1), 305–320 (1991)
Ghidaglia, J.-M.: Long time behaviour of solutions of abstract inequalities: applications to thermohydraulic and magnetohydrodynamic equations. J. Diff. Equ. 61(2), 268–294 (1986)
Ghidaglia, J.-M.: Some backward uniqueness results. Nonlinear Anal. 10(8), 777–790 (1986)
Hoang, L.: Asymptotic expansions for the Lagrangian trajectories from solutions of the Navier-Stokes equations. Comm. Math. Phys. 383(2), 981–995 (2021)
Hoang, L.T., Martinez, V.R.: Asymptotic expansion for solutions of the Navier-Stokes equations with non-potential body forces. J. Math. Anal. Appl. 462(1), 84–113 (2018)
Hoang, L.T., Titi, E.S.: Asymptotic expansions in time for rotating incompressible viscous fluids. Ann. Inst. H. Poincaré Anal. Non Linéaire 38, 109–137 (2021)
Minea, G.: Investigation of the Foias-Saut normalization in the finite-dimensional case. J. Dynam. Diff. Equ. 10(1), 189–207 (1998)
Shi, Y.: A Foias-Saut type of expansion for dissipative wave equations. Comm. Partial Diff. Equ. 25(11–12), 2287–2331 (2000)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Cao, D., Hoang, L. & Kieu, T. Infinite Series Asymptotic Expansions for Decaying Solutions of Dissipative Differential Equations with Non-smooth Nonlinearity. Qual. Theory Dyn. Syst. 20, 62 (2021). https://doi.org/10.1007/s12346-021-00502-9
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s12346-021-00502-9