The aim of this article is to extend previous works about the asymptotics of an ill-prepared fast rotating, highly stratified incompressible Navier–Stokes system. Thanks to improved Strichartz and a priori estimates, we are able not only to cover a case which was unanswered in our previous work (allowing bigger ill-prepared initial data) but also to improve the convergence rates and reduce some assumptions on the initial data. In passing we also widen the range of some parameters and compare two methods to obtain dispersive estimates.
A.Babin, A.Mahalov and B.Nicolaenko, Strongly stratified limit of 3D primitive equations in an infinite layer, Contemporary Mathematics283 (2001).
2.
H.Bahouri, J.-Y.Chemin and R.Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der mathematischen Wissenschaften, Vol. 343, Springer Verlag, 2011.
3.
T.Beale and A.Bourgeois, Validity of the quasi-geostrophic model for large scale flow in the atmosphere and ocean, SIAM Journal of Mathematical Analysis25 (1994), 1023–1068. doi:10.1137/S0036141092234980.
4.
P.Bougeault and R.Sadourny, Dynamique de l’atmosphère et de l’océan, Editions de l’Ecole polytechnique, 2001.
5.
M.Cannone, C.Miao and L.Xue, Global regularity for the supercritical dissipative quasi-geostrophic equation with large dispersive forcing, Proc. Lond. Math. Soc.106(3) (2013), 650–674. doi:10.1112/plms/pds046.
6.
F.Charve, Global well-posedness and asymptotics for a geophysical fluid system, Communications in Partial Differential Equations29(11 & 12) (2004), 1919–1940.
7.
F.Charve, Convergence of weak solutions for the primitive system of the quasi-geostrophic equations, Asymptotic Analysis42 (2005), 173–209.
8.
F.Charve, Asymptotics and vortex patches for the quasi-geostrophic approximation, Journal de mathématiques pures et appliquées85 (2006), 493–539. doi:10.1016/j.matpur.2005.10.002.
9.
F.Charve, Global well-posedness for the primitive equations with less regular initial data, Annales de la faculté des Sciences de Toulouse17(2) (2008), 221–238.
10.
F.Charve, A priori estimates for the 3D quasi-geostrophic system, J. Math. Anal. Appl444(2) (2016), 911–946. doi:10.1016/j.jmaa.2016.06.056.
11.
F.Charve, Asymptotics and lower bound for the lifespan of solutions to the primitive equations, Acta Applicandae Mathematicae158(1) (2018), 11–47. doi:10.1007/s10440-018-0172-3.
12.
F.Charve, Global well-posedness and asymptotics for a penalized Boussinesq-type system without dispersion, Communications in Mathematical Sciences16(3) (2018), 791–807. doi:10.4310/CMS.2018.v16.n3.a9.
13.
F.Charve, Enhanced convergence rates and asymptotics for a dispersive Boussinesq-type system with large ill-prepared data, Pure and Appl. Anal.2(2) (2020), 477–517.
14.
F.Charve and V.-S.Ngo, Asymptotics for the primitive equations with small anisotropic viscosity, Revista Matemática Iberoamericana27(1) (2011), 1–38.
15.
J.-Y.Chemin, A propos d’un problème de pénalisation de type antisymétrique, Journal de Mathématiques pures et appliquées76 (1997), 739–755. doi:10.1016/S0021-7824(97)89967-9.
16.
J.-Y.Chemin, B.Desjardins, I.Gallagher and E.Grenier, Fluids with anisotropic viscosity, Special issue for R. Temam’s 60th birthday, M2AN. Mathematical Modelling and Numerical Analysis34(2) (2000), 315–335. doi:10.1051/m2an:2000143.
17.
J.-Y.Chemin, B.Desjardins, I.Gallagher and E.Grenier, Anisotropy and dispersion in rotating fluids, in: Nonlinear Partial Differential Equations and Their Application, Collège de France Seminar, Studies in Mathematics and Its Applications, Vol. 31, 2002, pp. 171–191.
18.
J.-Y.Chemin, B.Desjardins, I.Gallagher and E.Grenier, Mathematical Geophysics: An Introduction to Rotating Fluids and to the Navier–Stokes Equations, Oxford University Press, 2006.
19.
A.Córdoba and D.Córdoba, A maximum principle applied to quasi-geostrophic equations, Comm. Math. Phys.249 (2004), 511–528. doi:10.1007/s00220-004-1055-1.
20.
B.Cushman-Roisin, Introduction to Geophysical Fluid Dynamics, Prentice-Hall, 1994.
21.
B.Desjardins and E.Grenier, Derivation of the quasigeostrophic potential vorticity equations, Advances in Differential Equations3(5) (1998), 715–752.
22.
A.Dutrifoy, Slow convergence to vortex patches in quasi-geostrophic balance, Arch. Ration. Mech. Anal.171(3) (2004), 417–449. doi:10.1007/s00205-003-0295-0.
23.
A.Dutrifoy, Examples of dispersive effects in non-viscous rotating fluids, Journal de Mathématiques Pures et Appliquées84(9) (2005), 331–356. doi:10.1016/j.matpur.2004.09.007.
24.
Y.Giga, K.Inui, A.Mahalov and J.Saal, Uniform global solvability of the rotating Navier–Stokes equations for nondecaying initial data, Ind. U. Math. J.57(6) (2008), 2775–2791. doi:10.1512/iumj.2008.57.3795.
25.
G.H.Hardy and J.E.Littlewood, On certain inequalities connected with the calculus of variations, J. London Math. Soc.5 (1930), 34–39. doi:10.1112/jlms/s1-5.1.34.
26.
M.Hieber and Y.Shibata, The Fujita-Kato approach to the Navier–Stokes equations in the rotational framework, Math. Zeit.265 (2010), 481–491. doi:10.1007/s00209-009-0525-8.
27.
T.Hmidi, Régularité hölderienne des poches de tourbillon visqueuses, Journal de Mathématiques pures et appliquées84(11) (2005), 1455–1495. doi:10.1016/j.matpur.2005.01.004.
28.
D.Iftimie, The approximation of the quasi-geostrophic system with the primitive systems, Asymptotic Analysis21(2) (1999), 89–97.
29.
T.Iwabuchi, A.Mahalov and R.Takada, Global solutions for the incompressible rotating stably stratified fluids, Math. Nachr290(4) (2017), 613–631. doi:10.1002/mana.201500385.
30.
T.Iwabuchi and R.Takada, Global solutions for the Navier–Stokes equations in the rotational framework, Math. Ann.357(2) (2013), 727–741. doi:10.1007/s00208-013-0923-4.
31.
T.Iwabuchi and R.Takada, Dispersive effect of the Coriolis force and the local well-posedness for the Navier–Stokes equations in the rotational framework, Funkcial. Ekvac.58(3) (2015), 365–385. doi:10.1619/fesi.58.365.
32.
Y.Koh, S.Lee and R.Takada, Dispersive estimates for the Navier–Stokes equations in the rotational framework, Advances in differential equations19(9–10) (2014), 857–878.
33.
Y.Koh, S.Lee and R.Takada, Strichartz estimates for the Euler equations in the rotational framework, J. Differential Equations265 (2014), 707–744. doi:10.1016/j.jde.2013.09.017.
34.
S.Lee and R.Takada, Dispersive estimates for the stably stratified Boussinesq equations, Indiana University Mathematics Journal66(6) (2017), 2037–2070. doi:10.1512/iumj.2017.66.6179.
35.
E.H.Lieb, Sharp constants in the Hardy–Littlewood–Sobolev and related inequalities, Ann. of Math.118 (1983), 349–374. doi:10.2307/2007032.
36.
W.Littman, Fourier transforms of surface-carried measures and differentiability of surface averages, Bull. Amer. Math. Soc.69 (1963), 766–770. doi:10.1090/S0002-9904-1963-11025-3.
37.
V.-S.Ngo, Rotating fluids with small viscosity, International Mathematics Research Notices10 (2009), 1860–1890.
38.
V.-S.Ngo and S.Scrobogna, Dispersive effects of weakly compressible and fast rotating inviscid fluids, Discrete Contin. Dyn. Syst.38(2) (2018), 749–789. doi:10.3934/dcds.2018033.
S.Scrobogna, Derivation of limit equations for a singular perturbation of a 3D periodic Boussinesq system, Discrete Contin. Dyn. Syst.37(12) (2017), 5979–6034. doi:10.3934/dcds.2017259.
41.
S.Scrobogna, Highly rotating fluids with vertical stratification for periodic data and anisotropic diffusion, Rev. Mat. Iberoam.34(1) (2018), 1–58. doi:10.4171/RMI/980.
42.
S.Scrobogna, Global existence and convergence of nondimensionalized incompressible Navier–Stokes equations in low-Froude number regime, Discrete Contin. Dyn. Syst.40(9) (2020), 5471–5511. doi:10.3934/dcds.2020235.
43.
S.L.Sobolev, On a theorem of functional analysis, Mat. Sbornik5 (1938), 471–479.
44.
E.Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, 1970.
45.
E.Stein, Harmonic Analysis, Real-Variable Methods, Orthogonality and Oscillatory Integrals, Princeton Mathematical Series, Vol. 43, Princeton University Press, 1993.
