Quasi-periodic solutions for the incompressible Navier-Stokes equations with nonlocal diffusion

Shuguan Ji; Yanshuo Li; Shuguan Ji; Yanshuo Li

doi:10.3934/era.2023363

Electronic Research Archive

2023, Volume 31, Issue 12: 7182-7194. doi: 10.3934/era.2023363

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Quasi-periodic solutions for the incompressible Navier-Stokes equations with nonlocal diffusion

Shuguan Ji ^,,
Yanshuo Li

School of Mathematics and Statistics and Center for Mathematics and Interdisciplinary Sciences, Northeast Normal University, Changchun 130024, China

Received: 31 July 2023 Revised: 05 November 2023 Accepted: 06 November 2023 Published: 10 November 2023

This paper studied the incompressible Navier-Stokes (NS) equations with nonlocal diffusion on $ \mathbb{T}^d (d \ge 2) $. Driven by a time quasi-periodic force, the existence of time quasi-periodic solutions in the Sobolev space was established. The proof was based on the decomposition of the unknowns into the spatial average part and spatial oscillating one. The former were sought under the Diophantine non-resonance assumption, and the latter by the contraction mapping principle. Moreover, by constructing suitable time weighted function space and using the Banach fixed point theorem, the asymptotic stability of quasi-periodic solutions and the exponential decay of perturbation were proved.
- Navier-Stokes (NS) equations,
- nonlocal diffusion,
- quasi-periodic solutions,
- asymptotic stability,
- exponential decay
Citation: Shuguan Ji, Yanshuo Li. Quasi-periodic solutions for the incompressible Navier-Stokes equations with nonlocal diffusion[J]. Electronic Research Archive, 2023, 31(12): 7182-7194. doi: 10.3934/era.2023363

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Abstract

This paper studied the incompressible Navier-Stokes (NS) equations with nonlocal diffusion on $ \mathbb{T}^d (d \ge 2) $. Driven by a time quasi-periodic force, the existence of time quasi-periodic solutions in the Sobolev space was established. The proof was based on the decomposition of the unknowns into the spatial average part and spatial oscillating one. The former were sought under the Diophantine non-resonance assumption, and the latter by the contraction mapping principle. Moreover, by constructing suitable time weighted function space and using the Banach fixed point theorem, the asymptotic stability of quasi-periodic solutions and the exponential decay of perturbation were proved.

References

[1]	J. R. Mercado, E. P. Guido, A. J. Sánchez-Sesma, M. Íñiguez, A. González, Analysis of the Blasius's formula and the Navier-Stokes fractional equation, in Fluid Dynamics in Physics, Engineering and Environmental Applications, Springer, Berlin, Heidelberg, (2013), 475–480. https://doi.org/10.1007/978-3-642-27723-8_44
[2]	X. C. Zhang, Stochastic Lagrangian particle approach to fractal Navier-Stokes equations, Commun. Math. Phys., 311 (2012), 133–155. https://doi.org/10.1007/s00220-012-1414-2 doi: 10.1007/s00220-012-1414-2
[3]	W. A. Woyczyński, Lévy processes in the physical sciences, in Lévy Processes, Birkhäuser, Boston, MA, (2001), 241–266. https://doi.org/10.1007/978-1-4612-0197-7_11
[4]	J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non-Linéaires (French), Dunod, Paris, 1969.
[5]	J. C. Mattingly, Y. G. Sinai, An elementary proof of the existence and uniqueness theorem for the Navier-Stokes equations, Commun. Contemp. Math., 1 (1999), 497–516. https://doi.org/10.1142/S0219199799000183 doi: 10.1142/S0219199799000183
[6]	L. Tang, Y. Yu, Partial regularity of suitable weak solutions to the fractional Navier-Stokes equations, Commun. Math. Phys., 334 (2015), 1455–1482. https://doi.org/10.1007/s00220-014-2149-z doi: 10.1007/s00220-014-2149-z
[7]	B. Lai, C. Miao, X. Zheng, Forward self-similar solutions of the fractional Navier-Stokes equations, Adv. Math., 352 (2019), 981–1043. https://doi.org/10.1016/j.aim.2019.06.021 doi: 10.1016/j.aim.2019.06.021
[8]	M. Colombo, C. De Lellis, L. De Rosa, Ill-posedness of Leray solutions for the hypodissipative Navier-Stokes equations, Commun. Math. Phys., 362 (2018), 659–688. https://doi.org/10.1007/s00220-018-3177-x doi: 10.1007/s00220-018-3177-x
[9]	L. De Rosa, Infinitely many Leray-Hopf solutions for the fractional Navier-Stokes equations, Commun. Partial Differ. Equations, 44 (2019), 335–365. https://doi.org/10.1080/03605302.2018.1547745 doi: 10.1080/03605302.2018.1547745
[10]	J. Wu, Lower bounds for an integral involving fractional Laplacians and the generalized Navier-Stokes equations in Besov spaces, Commun. Math. Phys., 263 (2006), 803–831. https://doi.org/10.1007/s00220-005-1483-6 doi: 10.1007/s00220-005-1483-6
[11]	J. Serrin, A note on the existence of periodic solutions of the Navier-Stokes equations, Arch. Ration. Mech. Anal., 3 (1959), 120–122. https://doi.org/10.1007/BF00284169 doi: 10.1007/BF00284169
[12]	V. I. Yudovich, Periodic motions of a viscous incompressible fluid, Sov. Math., Dokl., 1 (1960), 168–172.
[13]	G. Prodi, Qualche risultato riquardo alle equazioni di Navier-Stokes nel caso bidimensionale, Rend. Semin. Mat. Univ. Padova, 30 (1960), 1–15.
[14]	G. Prouse, Soluzioni periodiche dell' equazione di Navier-Stokes, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat., 35 (1963), 443–447.
[15]	O. Vejvoda, L. Herrmann, V. Lovicar, M. Sova, I. Straškraba, M. Štědrý, Partial Differential Equations: Time-Periodic Solutions, Martinus Nijhoff Publishers, The Hague, 1981.
[16]	H. Kozono, M. Nakao, Periodic solutions of the Navier-Stokes equations in unbounded domains, Tohoku Math. J., 48 (1996), 33–50. https://doi.org/10.2748/tmj/1178225411 doi: 10.2748/tmj/1178225411
[17]	M. Kyed, Time-periodic Solutions to the Navier-Stokes Equations, Habilitation thesis, Technische Universität in Darmstadt, 2012.
[18]	M. Kyed, The existence and regularity of time-periodic solutions to the three dimensional Navier-Stokes equations in the whole space, Nonlinearity, 27 (2014), 2909–2935. https://doi.org/10.1088/0951-7715/27/12/2909 doi: 10.1088/0951-7715/27/12/2909
[19]	G. P. Galdi, Viscous flow past a body translating by time-periodic motion with zero average, Arch. Ration. Mech. Anal., 237 (2020), 1237–1269. https://doi.org/10.1007/s00205-020-01530-6 doi: 10.1007/s00205-020-01530-6
[20]	G. P. Galdi, A. L. Silvestre, Existence of time-periodic solutions to the Navier-Stokes equations around a moving body, Pac. J. Math., 223 (2006), 251–267. https://doi.org/10.2140/pjm.2006.223.251 doi: 10.2140/pjm.2006.223.251
[21]	G. P. Galdi, H. Sohr, Existence and uniqueness of time-periodic physically reasonable Navier-Stokes flow past a body, Arch. Ration. Mech. Anal., 172 (2004), 363–406. https://doi.org/10.1007/s00205-004-0306-9 doi: 10.1007/s00205-004-0306-9
[22]	M. Yamazaki, The Navier-Stokes equations in the weak-$L^n$ space with time-dependent external force, Math. Ann., 317 (2000), 635–675. https://doi.org/10.1007/PL00004418 doi: 10.1007/PL00004418
[23]	R. Montalto, The Navier-Stokes equation with time quasi-periodic external force: existence and stability of quasi-periodic solutions, J. Dyn. Differ. Equations, 33 (2021), 1341–1362. https://doi.org/10.1007/s10884-021-09944-w doi: 10.1007/s10884-021-09944-w
[24]	P. Baldi, R. Montalto, Quasi-periodic incompressible Euler flows in 3D, Adv. Math., 384 (2021), 107730. https://doi.org/10.1016/j.aim.2021.107730 doi: 10.1016/j.aim.2021.107730
[25]	L. Franzoi, R. Montalto, A KAM approach to the inviscid limit for the 2D Navier-Stokes equations, preprint, arXiv: 2207.11008.
[26]	M. Berti, Z. Hassainia, N. Masmoudi, Time quasi-periodic vortex patches of Euler equation in the plane, Invent. Math., 233 (2023), 1279–1391. https://doi.org/10.1007/s00222-023-01195-4 doi: 10.1007/s00222-023-01195-4
[27]	N. Crouseilles, E. Faou, Quasi-periodic solutions of the 2D Euler equation, Asymptotic Anal., 81 (2013), 31–34. https://doi.org/10.3233/ASY-2012-1117 doi: 10.3233/ASY-2012-1117
[28]	A. Enciso, D. Peralta-Salas, F. T. de Lizaur, Quasi-periodic solutions to the incompressible Euler equations in dimensions two and higher, J. Differ. Equations, 354 (2023), 170–182. https://doi.org/10.1016/j.jde.2023.01.013 doi: 10.1016/j.jde.2023.01.013
[29]	L. Grafakos, Classical Fourier Analysis, Springer, New York, 2014. https://doi.org/10.1007/978-1-4939-1194-3

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