Infinite Series Asymptotic Expansions for Decaying Solutions of Dissipative Differential Equations with Non-smooth Nonlinearity

Cao, Dat; Hoang, Luan; Kieu, Thinh

doi:10.1007/s12346-021-00502-9

Cited by 6 publications

(19 citation statements)

References 23 publications

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“…The proof in [15] follows that of Foias and Saut in [13]. For systems of ODE, this result is re-established in [12] using a different method.…”

Section: Introductionmentioning

confidence: 63%

“…However, these asymptotic expansions can be obtained independently from the first approximation in (1.2). The interested reader can find other results on the asymptotic expansions for different ODE and PDE without forcing functions in [12,18,20,23,24], and with forcing functions in [9-11, 16, 17, 19]. In particular, [12] obtains the asymptotic expansions of nontrivial, decaying solutions of (1.3) in the case F (u) is not smooth in any neighborhood of the origin.…”

Section: Introductionmentioning

confidence: 99%

“…The interested reader can find other results on the asymptotic expansions for different ODE and PDE without forcing functions in [12,18,20,23,24], and with forcing functions in [9-11, 16, 17, 19]. In particular, [12] obtains the asymptotic expansions of nontrivial, decaying solutions of (1.3) in the case F (u) is not smooth in any neighborhood of the origin. Roughly speaking, F (u) can be approximated, near the origin, by a finite sum or a series of positively homogeneous terms of possibly nonintegral degrees, see Definition 3.2 below.…”

Section: Introductionmentioning

confidence: 99%

“…Roughly speaking, F (u) can be approximated, near the origin, by a finite sum or a series of positively homogeneous terms of possibly nonintegral degrees, see Definition 3.2 below. A crucial step in [12] is the very first asymptotic approximation (1.2).…”

Section: Introductionmentioning

confidence: 99%

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Long-time behavior of solutions of superlinear systems of differential equations

Hoang¹

2022

Preprint

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This paper establishes the precise asymptotic behavior, as time t tends to infinity, for nontrivial, decaying solutions of genuinely nonlinear dissipative systems of ordinary differential equations. The lowest order term in these systems, when the spatial variables are small, is not linear, but rather positively homogeneous of a degree greater than one. We prove that the solution behaves like ξt −p , as t → ∞, for a nonzero vector ξ and an explicit number p > 0.

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“…The proof in [15] follows that of Foias and Saut in [13]. For systems of ODE, this result is re-established in [12] using a different method.…”

Section: Introductionmentioning

confidence: 63%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Long-time behavior of solutions of superlinear systems of differential equations

Hoang¹

2022

Preprint

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“…It is developed further for general nonlinear ordinary differential equations (ODEs). Specifically, it is investigated first in [32] for autonomous analytic systems, and recently in [9] for systems with non-smooth nonlinearity, and in [8,26] for nonautonomous systems. In particular, the forcing functions in [26] are much more general and complex than (1.4) and (1.5) above.…”

Section: Introductionmentioning

confidence: 99%

The Navier-Stokes equations with body forces decaying coherently in time

Hoang¹

2022

Preprint

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The long-time behavior of solutions of the three-dimensional Navier-Stokes equations in a periodic domain is studied. The time-dependent body force decays, as time t tends to infinity, in a coherent manner. In fact, it is assumed to have a general and complicated asymptotic expansion which involves complex powers of e t , t, ln t, or other iterated logarithmic functions of t. We prove that all Leray-Hopf weak solutions admit an asymptotic expansion which is independent of the solutions and is uniquely determined by the asymptotic expansion of the body force. The proof makes use of the complexifications of the Gevrey-Sobolev spaces together with those of the Stokes operator and the bilinear form of the Navier-Stokes equations.

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Asymptotic Expansions for the Lagrangian Trajectories from Solutions of the Navier–Stokes Equations

Hoang

2020

Commun. Math. Phys.

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We study the three-dimensional Navier-Stokes equations in a periodic domain with the force decaying in time. Although the force has a certain coherent decay, as time tends to infinity, it can be too complicated for the previous theory of asymptotic expansions to be applicable. To deal with this issue, we systematically develop a new theory of asymptotic expansions containing the so-called subordinate variables which can be defined recursively. We apply it to obtain an asymptotic expansion for any Leray-Hopf weak solutions. The expansion, in fact, is constructed explicitly and the impact of the subordinate variables can be clearly specified. The complexifications of the Gevrey-Sobolev spaces, and of the Stokes and bilinear operators of the Navier-Stokes equations are utilized to facilitate such a construction.

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Infinite Series Asymptotic Expansions for Decaying Solutions of Dissipative Differential Equations with Non-smooth Nonlinearity

Cited by 6 publications

References 23 publications

Long-time behavior of solutions of superlinear systems of differential equations

Long-time behavior of solutions of superlinear systems of differential equations

The Navier-Stokes equations with body forces decaying coherently in time

Asymptotic Expansions for the Lagrangian Trajectories from Solutions of the Navier–Stokes Equations

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