In this paper we study the hydrostatic limit of the Navier-Stokes-alpha model in a very thin strip domain. We derive some Prandtl-type limit equations for this model and we prove the global well-posedness of the limit system for small initial conditions in an appropriate analytic function space.
Léo Glangetas
,
Van-Sang Ngo
,
El Mehdi Said
. HYDROSTATIC LIMIT OF THE NAVIER-STOKES-ALPHA MODEL*[J]. Acta mathematica scientia, Series B, 2023
, 43(5)
: 1945
-1980
.
DOI: 10.1007/s10473-023-0502-z
[1] Alexandre R, Wang Y, Xu C J, Yang T. Well-posedness of the Prandtl equation in Sobolev spaces. J Amer Math Soc, 2015, 28:745-784
[2] Bahouri H, Chemin J Y, Danchin R.Fourier Analysis and Nonlinear Partial Differential Equations. Grundlehren der Mathematischen Wissenschaften 343. Heidelberg: Springer, 2011
[3] Bourgeois A J, Beale J T. Validity of the quasigeostrophic model for large-scale flow in the atmosphere and ocean. SIAM J Math Anal, 1994, 25: 1023-1068
[4] Busuioc A V. Sur les équations $\alpha$ Navier-Stokes dans un ouvert borné. C R Acad Sci Paris Série I,2002, 334: 823-826
[5] Camassa R, Holm D D. An integrable shallow water equation with peaked solitons. Phys Rev Lett, 1993, 71(11): 1661-1664
[6] Chemin J Y.Le système de Navier-Stokes incompressible soixante dix ans après Jean Leray//Actes des Journées Mathéatique à la Mémoire de Jean Leray. Séminaire & Congrés, 9. Soc Math France, Paris, 2004: 99-123
[7] Chemin J Y, Gallagher I, Paicu M. Global regularity for some classes of large solutions to the Navier-Stokes equations. Annals of Mathematics, 2011, 173: 983-1012
[8] Chemin J Y, Lerner N. Flot de champs de vecteurs non Lipschitziens et équations de Navier-Stokes. Journal of Differential Equations,1992, 121: 314-328
[9] Chen S, Foias C, Holm D D, Olson E, Titi E S, Wynne S. The Camassa-Holm equations as a closure model for turbulent channel and pipe flow. Phys Rev Lett, 1998, 81: 5338-5341
[10] Cheskidov A. Boundary layer for the Navier-Stokes-alpha model of fluid turbulence. Arch Ration Mech Anal, 2004, 172(3): 333-362
[11] Dietert H, Gérard-Varet D. Well-posedness of the Prandtl equation without any structural assumption. Ann PDE,2019, 5(1): Art 8
[12] Foias C, Holm D D, Titi E S. The three-dimensional viscous Camassa-Holm equations and their relation to the Navier-Stokes equation and turbulence theory. J Dynamics Differential Equations, 2002, 14(1): 1-35
[13] Gérard-Varet D, Masmoudi N, Vicol V. Well-posedness of the hydrostatic Navier-Stokes equations. Anal PDE,2020, 13(5): 1417-1455
[14] Ghosal S. Mathematical and physical constraints on large-eddy simulation of turbulence. AIAA J, 1999, 37(4): 425-433
[15] Holm D D, Marsden J E, Ratiu T. Euler-Poincaré models of ideal fluids with nonlinear dispersion. Phys Rev Lett,1998, 80(19): 4173--4177
[16] Kouranbaeva S. The Camassa-Holm equation as a geodesic flow on the diffeomorphism group. J Math Phys, 1999, 40(2): 857-868
[17] Kukavica I, Temam R, Vicol V, Ziane M. Local existence and uniqueness for the hydrostatic Euler equations on a bounded domain. J Differential Equations, 2011, 250(3): 1719-1746
[18] Li W X, Masmoudi N, Yang T. Well-posedness in Gevrey function space for 3D Prandtl equations without structural assumption. Commun Pure Appl Math, 2022, 75(8): 1755-1797
[19] Lions J L, Temam R, Wang S. On the equations of the large-scale ocean. Nonlinearity, 1992, 5: 1007-1053
[20] Marsden J E, Shkoller S. Global well-posedness for the LANS-$\alpha$ equations on bounded domains. Philos Trans Roy Soc London Ser A, 2001, 359: 1449-1468
[21] Masmoudi N, Wong T K. Local-in-time existence and uniqueness of solutions to the Prandtl equations by energy methods. Comm Pure Appl Math, 2015, 68: 1683-1741
[22] Misiolek G. A shallow water equation as a geodesic flow on the Bott-Virasoro group. J Geom Phys, 1998, 24(3): 203-208
[23] Paicu M, Zhang P, Zhang Z. On the hydrostatic approximation of the Navier-Stokes equations in a thin strip. Adv Math, 2020, 372: 107293
[24] Pedlosky J. Geophysical Fluid Dynamics.New York: Springer-Verlag, 1987
[25] Renardy M. Ill-posedness of the hydrostatic Euler and Navier-Stokes equations. Arch Ration Mech Anal, 2009, 194(3): 877-886