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Proceedings of the Royal Society of Edinburgh Section A: Mathematics Proceedings of the Royal Society of Edinburgh Section A: Mathematics

Article contents

Long-time asymptotic expansions for Navier-Stokes equations with power-decaying forces

Part of: Equations of mathematical physics and other areas of application Infinite-dimensional dissipative dynamical systems Partial differential equations Incompressible viscous fluids Representations of solutions

Published online by Cambridge University Press:  22 January 2019

Dat Cao  and
Luan Hoang
Dat Cao
Affiliation:
Department of Mathematics and Statistics, Texas Tech University, Box 41042, Lubbock, TX79409-1042, USA (dat.cao@ttu.edu, luan.hoang@ttu.edu)
Luan Hoang
Affiliation:
Department of Mathematics and Statistics, Texas Tech University, Box 41042, Lubbock, TX79409-1042, USA (dat.cao@ttu.edu, luan.hoang@ttu.edu)
Corresponding
E-mail address:
dat.cao@ttu.edu
luan.hoang@ttu.edu
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Abstract

The Navier-Stokes equations for viscous, incompressible fluids are studied in the three-dimensional periodic domains, with the body force having an asymptotic expansion, when time goes to infinity, in terms of power-decaying functions in a Sobolev-Gevrey space. Any Leray-Hopf weak solution is proved to have an asymptotic expansion of the same type in the same space, which is uniquely determined by the force, and independent of the individual solutions. In case the expansion is convergent, we show that the next asymptotic approximation for the solution must be an exponential decay. Furthermore, the convergence of the expansion and the range of its coefficients, as the force varies are investigated.

Keywords

Navier-Stokes asymptotic expansion power-decaying

MSC classification

Primary: 35C20: Asymptotic expansions
Secondary: 35B40: Asymptotic behavior of solutions 35Q30: Navier-Stokes equations 37L99: None of the above, but in this section 76D05: Navier-Stokes equations
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 2 , April 2020 , pp. 569 - 606
Copyright
Copyright © Royal Society of Edinburgh 2019

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References

1Constantin, P. and Foias, C.. Navier-Stokes equations. Chicago Lectures in Mathematics. (Chicago, IL: University of Chicago Press, 1988).Google Scholar
2Dieudonné, J.. Foundations of modern analysis (New York-London: Academic Press, 1969). Enlarged and corrected printing, Pure and Applied Mathematics, Vol. 10-I.Google Scholar
3Dyer, R. H. and Edmunds, D. E.. Lower bounds for solutions of the Navier-Stokes equations. Proc. London Math. Soc. (3) 18 (1968), 169–178.CrossRefGoogle Scholar
4Foias, C., and Saut, J.-C.. Asymptotic behavior, as t → ∞, of solutions of the Navier-Stokes equations. In Nonlinear partial differential equations and their applications. Collège de France Seminar, Vol. IV (Paris, 1981/1982), pp. 74–86, vol. 84 of Res. Notes in Math (Boston, MA: Pitman, 1983).Google Scholar
5Foias, C. and Saut, J.-C.. Asymptotic behavior, as t → +∞, of solutions of Navier-Stokes equations and nonlinear spectral manifolds. Indiana Univ. Math. J. 33 (1984a), 459–477.CrossRefGoogle Scholar
6Foias, C. and Saut, J.-C.. On the smoothness of the nonlinear spectral manifolds associated to the Navier-Stokes equations. Indiana Univ. Math. J. 33 (1984b), 911–926.CrossRefGoogle Scholar
7Foias, C. and Saut, J.-C.. Linearization and normal form of the Navier-Stokes equations with potential forces. Ann. Inst. H. Poincaré Anal. Non Linéaire 4 (1987), 1–47.CrossRefGoogle Scholar
8Foias, C. and Temam, R.. Gevrey class regularity for the solutions of the Navier-Stokes equations. J. Funct. Anal. 87 (1989), 359–369.CrossRefGoogle Scholar
9Foias, C. and Saut, J.-C.. Asymptotic integration of Navier-Stokes equations with potential forces. I. Indiana Univ. Math. J. 40 (1991), 305–320.CrossRefGoogle Scholar
10Foias, C., Manley, O., Rosa, R. and Temam, R.. Navier-Stokes equations and turbulence, vol. 83 of Encyclopedia of Mathematics and its Applications (Cambridge: Cambridge University Press, 2001).CrossRefGoogle Scholar
11Foias, C., Hoang, L., Olson, E. and Ziane, M.. On the solutions to the normal form of the Navier-Stokes equations. Indiana Univ. Math. J. 55 (2006), 631–686.CrossRefGoogle Scholar
12Foias, C., Hoang, L. and Nicolaenko, B.. On the helicity in 3D-periodic Navier-Stokes equations. I. The non-statistical case. Proc. Lond. Math. Soc. (3) 94 (2007), 53–90.CrossRefGoogle Scholar
13Foias, C., Hoang, L. and Nicolaenko, B.. On the helicity in 3D-periodic Navier-Stokes equations. II. The statistical case. Comm. Math. Phys. 290 (2009a), 679–717.CrossRefGoogle Scholar
14Foias, C., Hoang, L., Olson, E. and Ziane, M.. The normal form of the Navier-Stokes equations in suitable normed spaces. Ann. Inst. H. Poincaré Anal. Non Linéaire 26 (2009b), 1635–1673.CrossRefGoogle Scholar
15Foias, C., Rosa, R. and Temam, R.. Topological properties of the weak global attractor of the three-dimensional Navier-Stokes equations. Discrete Contin. Dyn. Syst. 27 (2010), 1611–1631.CrossRefGoogle Scholar
16Foias, C., Hoang, L. and Saut, J.-C.. Asymptotic integration of Navier-Stokes equations with potential forces. II. An explicit Poincaré-Dulac normal form. J. Funct. Anal. 260 (2011), 3007–3035.CrossRefGoogle Scholar
17Foias, C., Hoang, L. and Saut, J.-C.. Navier and Stokes meet Poincaré and Dulac. J. Appl. Anal. Comput. 8 (2018), 727–763.Google Scholar
18Hoang, L. T. and Martinez, V. R.. Asymptotic expansion in Gevrey spaces for solutions of Navier-Stokes equations. Asymptot. Anal. 104 (2017), 167–190.CrossRefGoogle Scholar
19Hoang, L. T. and Martinez, V. R.. Asymptotic expansion for solutions of the Navier-Stokes equations with non-potential body forces. J. Math. Anal. Appl. 462 (2018), 84–113.CrossRefGoogle Scholar
20Kukavica, I. and Reis, E.. Asymptotic expansion for solutions of the Navier-Stokes equations with potential forces. J. Diff. Equ. 250 (2011), 607–622.CrossRefGoogle Scholar
21Ladyzhenskaya, O.A.. The mathematical theory of viscous incompressible flow, 2nd English edn, revised and enlarged. Translated from the Russian by Richard A. Silverman and John Chu. Mathematics and its Applications, Vol. 2 (New York: Gordon and Breach Science Publishers, 1969).Google Scholar
22Shi, Y.. A Foias-Saut type of expansion for dissipative wave equations. Comm. Partial Diff. Equ. 25 (2000), 2287–2331.CrossRefGoogle Scholar
23Temam, R.. Navier-Stokes equations and nonlinear functional analysis, 2nd edn, vol. 66 of CBMS-NSF Regional Conference Series in Applied Mathematics (Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM), 1995).CrossRefGoogle Scholar
24Temam, R.. Infinite-dimensional dynamical systems in mechanics and physics, 2nd edn, vol. 68 of Applied Mathematical Sciences (New York: Springer-Verlag, 1997).CrossRefGoogle Scholar
25Temam, R.. Navier-Stokes equations (Providence, RI: AMS Chelsea Publishing, 2001). Theory and numerical analysis, Reprint of the 1984 edition.CrossRefGoogle Scholar

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