## 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

### Keywords:

Navier-Stokes equations; Leray-Hopf weak solution
Full Text:

### References:

 [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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.