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Asymptotic expansions for the Lagrangian trajectories from solutions of the Navier-Stokes equations. (English) Zbl 1466.35286

Summary: Consider any Leray-Hopf weak solution of the three-dimensional Navier-Stokes equations for incompressible, viscous fluid flows. We prove that any Lagrangian trajectory associated with such a velocity field has an asymptotic expansion, as time tends to infinity, which describes its long-time behavior very precisely.

MSC:

35Q30 Navier-Stokes equations
76D05 Navier-Stokes equations for incompressible viscous fluids
35D30 Weak solutions to PDEs
35C20 Asymptotic expansions of solutions to PDEs
35B65 Smoothness and regularity of solutions to PDEs
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[1] Besse, N.; Frisch, U., A constructive approach to regularity of Lagrangian trajectories for incompressible Euler flow in a bounded domain, Commun. Math. Phys., 351, 2, 689-707 (2017) · Zbl 1371.35201
[2] Camliyurt, G.; Kukavica, I., On the Lagrangian and Eulerian analyticity for the Euler equations, Physica D, 376, 377, 121-130 (2018) · Zbl 1398.76021
[3] Cao, D., Hoang, L.: Asymptotic expansions with exponential, power, and logarithmic functions for non-autonomous nonlinear differential equations. J. Evol. Equ. 1-45 (2020) (accepted). doi:10.1007/s00028-020-00622-w
[4] Cao, D.; Hoang, L., Asymptotic expansions in a general system of decaying functions for solutions of the Navier-Stokes equations, Ann. Mat., 3, 199, 1023-1072 (2020) · Zbl 1442.35294
[5] Cao, D.; Hoang, L., Long-time asymptotic expansions for Navier-Stokes equations with power-decaying forces, Proc. R. Soc. Edinb. Sect. A Math., 150, 2, 569-606 (2020) · Zbl 1439.35366
[6] Constantin, P., Foias, C.: Navier-Stokes equations. Chicago Lectures in Mathematics. University of Chicago Press, Chicago (1988) · Zbl 0687.35071
[7] Constantin, P.; Kukavica, I.; Vicol, V., Contrast between Lagrangian and Eulerian analytic regularity properties of Euler equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33, 6, 1569-1588 (2016) · Zbl 1353.35233
[8] Constantin, P.; La, J., Note on Lagrangian-Eulerian methods for uniqueness in hydrodynamic systems, Adv. Math., 345, 27-52 (2019) · Zbl 1410.35111
[9] Constantin, P.; Vicol, V.; Wu, J., Analyticity of Lagrangian trajectories for well posed inviscid incompressible fluid models, Adv. Math., 285, 352-393 (2015) · Zbl 1422.35135
[10] 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) · Zbl 1246.76019
[11] 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. Nonlinéaire, 26, 5, 1635-1673 (2009) · Zbl 1179.35212
[12] 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) · Zbl 1232.35115
[13] Foias, C.; Hoang, L.; Saut, J-C, Navier and Stokes meet Poincaré and Dulac, J. Appl. Anal. Comput., 8, 3, 727-763 (2018) · Zbl 1457.35024
[14] Foias, C., Manley, O., Rosa, R., Temam, R.: Navier-Stokes equations and turbulence. In: Encyclopedia of Mathematics and its Applications, vol. 83. Cambridge University Press, Cambridge (2001) · Zbl 0994.35002
[15] 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) · Zbl 0565.35087
[16] Foias, C.; Saut, J-C, On the smoothness of the nonlinear spectral manifolds associated to the Navier-Stokes equations, Indiana Univ. Math. J., 33, 6, 911-926 (1984) · Zbl 0572.35081
[17] Foias, C.; Saut, J-C, Linearization and normal form of the Navier-Stokes equations with potential forces, Ann. Inst. H. Poincaré Anal. Nonlinéaire, 4, 1, 1-47 (1987) · Zbl 0635.35075
[18] Foias, C.; Saut, J-C, Asymptotic integration of Navier-Stokes equations with potential forces, I. Indiana Univ. Math. J., 40, 1, 305-320 (1991) · Zbl 0739.35066
[19] Hernandez, M., Mechanisms of Lagrangian analyticity in fluids, Arch. Ration. Mech. Anal., 233, 2, 513-598 (2019) · Zbl 1422.76019
[20] Hoang, LT; Martinez, VR, Asymptotic expansion in Gevrey spaces for solutions of Navier-Stokes equations, Asymptot. Anal., 104, 3-4, 167-190 (2017) · Zbl 1375.35317
[21] Hoang, LT; Martinez, VR, Asymptotic expansion for solutions of the Navier-Stokes equations with non-potential body forces, J. Math. Anal. Appl., 462, 1, 84-113 (2018) · Zbl 1394.35130
[22] Hoang, LT; Titi, ES, Asymptotic expansions in time for rotating incompressible viscous fluids, Ann. l’Inst. Henri Poincaré Anal. Nonlinéaire (2020)
[23] Lang, S., Analysis I (1968), London: Addison-Wesley, London · Zbl 0159.34303
[24] Ma, T., Wang, S.: Geometric theory of incompressible flows with applications to fluid dynamics. In: Mathematical Surveys and Monographs, vol. 119. American Mathematical Society, Providence (2005) · Zbl 1099.76002
[25] Minea, G., Investigation of the Foias-Saut normalization in the finite-dimensional case, J. Dyn. Differ. Equ., 10, 1, 189-207 (1998) · Zbl 0970.34045
[26] Shi, Y., A Foias-Saut type of expansion for dissipative wave equations, Commun. Partial Differ. Equ., 25, 11-12, 2287-2331 (2000) · Zbl 0963.35123
[27] Sueur, F., Smoothness of the trajectories of ideal fluid particles with Yudovich vorticities in a planar bounded domain, J. Differ. Equ., 251, 12, 3421-3449 (2011) · Zbl 1248.76011
[28] Temam, R.: Navier-Stokes equations and nonlinear functional analysis, volume 66 of CBMS-NSF Regional Conference Series in Applied Mathematics, 2nd ed. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (1995) · Zbl 0833.35110
[29] Temam, R.: Navier-Stokes equations. AMS Chelsea Publishing, Providence (2001). Theory and Numerical Analysis, Reprint of the 1984 edition · Zbl 0981.35001
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