×

Global hydrostatic approximation of the hyperbolic Navier-Stokes system with small Gevrey class 2 data. (English) Zbl 1491.35317

Summary: We investigate the hydrostatic approximation of a hyperbolic version of Navier-Stokes equations, which is obtained by using the Cattaneo type law instead of the Fourier law, evolving in a thin strip \(\mathbb{R} \times (0, \epsilon)\). The formal limit of these equations is a hyperbolic Prandtl type equation. We first prove the global existence of solutions to these equations under a uniform smallness assumption on the data in the Gevrey class 2. Then we justify the limit globally-in-time from the anisotropic hyperbolic Navier-Stokes system to the hyperbolic Prandtl system with such Gevrey class 2 data. Compared with [M. Paicu et al., Adv. Math. 372, Article ID 107293, 41 p. (2020; Zbl 1446.35105)] for the hydrostatic approximation of the 2-D classical Navier-Stokes system with analytic data, here the initial data belongs to the Gevrey class 2, which is very sophisticated even for the well-posedness of the classical Prandtl system (see [H. Dietert and D. Gérard-Varet, Ann. PDE 5, No. 1, Paper No. 8, 51 p. (2019; Zbl 1428.35355)] and [C. Wang, Y. Wang, and P. Zhang, “On the global small solution of 2-D Prandtl system with initial data in the optimal Gevrey class”, Preprint, arXiv:2103.00681]); furthermore, the estimate of the pressure term in the hyperbolic Prandtl system give rise to additional difficulties.

MSC:

35Q30 Navier-Stokes equations
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
76D05 Navier-Stokes equations for incompressible viscous fluids
76D10 Boundary-layer theory, separation and reattachment, higher-order effects
35L10 Second-order hyperbolic equations
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Aarach N. Global well-posedness of 2D hyperbolic perturbation of the Navier-Stokes system in a thin strip. arXiv:2111.13052, 2021
[2] Abdelhedi, B., Global existence of solutions for hyperbolic Navier-Stokes equations in three space dimensions, Asymptot Anal, 112, 213-225 (2019) · Zbl 1416.35178
[3] Bahouri, H.; Chemin, J-Y; Danchin, R., Fourier Analysis and Nonlinear Partial Differential Equations (2011), Berlin-Heidelberg: Springer-Verlag, Berlin-Heidelberg · Zbl 1227.35004 · doi:10.1007/978-3-642-16830-7
[4] Besson, O.; Laydi, M. R., Some estimates for the anisotropic Navier-Stokes equations and for the hydrostatic approximation, ESAIM Math Model Numer Anal, 26, 855-865 (1992) · Zbl 0765.76017 · doi:10.1051/m2an/1992260708551
[5] Bony, J. M., Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires, Ann Sci Éc Norm Supér (4), 14, 209-246 (1981) · Zbl 0495.35024 · doi:10.24033/asens.1404
[6] Brenier, Y.; Natalini, R.; Puel, M., On a relaxation approximation of the incompressible Navier-Stokes equations, Proc Amer Math Soc, 132, 1021-1028 (2004) · Zbl 1080.35064 · doi:10.1090/S0002-9939-03-07230-7
[7] Cattaneo, C., Sulla conduzione del calore, Atti Sem Mat Fis Univ Modena, 3, 83-101 (1949) · Zbl 0035.26203
[8] Cattaneo, C., Sur une forme de l’équation de la chaleur éliminant le paradoxe d’une propagation instantanée, C R Acad Sci Paris Sér I Math, 247, 431-433 (1958) · Zbl 1339.35135
[9] Chemin, J-Y, Le système de Navier-Stokes incompressible soixante dix ans après Jean Leray, Actes des Journées Mathématiques à la Mémoire de Jean Leray, 99-123 (2004), Paris: Soc Math France, Paris · Zbl 1075.35035
[10] Chemin, J-Y; Gallagher, I.; Paicu, M., Global regularity for some classes of large solutions to the Navier-Stokes equations, Ann of Math (2), 173, 983-1012 (2011) · Zbl 1229.35168 · doi:10.4007/annals.2011.173.2.9
[11] Chemin, J-Y; Lerner, N., Flot de champs de vecteurs non Lipschitziens et équations de Navier-Stokes, J Differential Equations, 121, 314-328 (1995) · Zbl 0878.35089 · doi:10.1006/jdeq.1995.1131
[12] Chemin, J-Y; Zhang, P., On the global wellposedness to the 3-D incompressible anisotropic Navier-Stokes equations, Comm Math Phys, 272, 529-566 (2007) · Zbl 1132.35068 · doi:10.1007/s00220-007-0236-0
[13] Coulaud O, Hachicha I, Raugel G. Hyperbolic quasilinear Navier-Stokes equations in ℝ^2. J Dynam Differential Equations, 2022, in press · Zbl 1504.35222
[14] Dietert, H.; Gérard-Varet, D., Well-posedness of the Prandtl equations without any structural assumption, Ann PDE, 5, 8 (2019) · Zbl 1428.35355 · doi:10.1007/s40818-019-0063-6
[15] Gérard-Varet, D.; Dormy, E., On the ill-posedness of the Prandtl equation, J Amer Math Soc, 23, 591-609 (2010) · Zbl 1197.35204 · doi:10.1090/S0894-0347-09-00652-3
[16] Gérard-Varet, D.; Masmoudi, N., Well-posedness for the Prandtl system without analyticity or monotonicity, Ann Sc Ec Norm Supér (4), 48, 1273-1325 (2015) · Zbl 1347.35201 · doi:10.24033/asens.2270
[17] Gérard-Varet, D.; Masmoudi, N.; Vicol, V., Well-posedness of the hydrostatic Navier-Stokes equations, Anal PDE, 13, 1417-1455 (2020) · Zbl 1451.35109 · doi:10.2140/apde.2020.13.1417
[18] Hachicha, I., Approximations hyperboliques des équations de Navier-Stokes (2013), Évry-Courcouronnes: Université d’Évry-Val d’Essonne, Évry-Courcouronnes
[19] Lions, P-L, Mathematical Topics in Fluid Mechanics. Volume 1, Incompressible Models (1996), Oxford: Clarendon Press, Oxford · Zbl 0866.76002
[20] Lombardo, M. C.; Cannone, M.; Sammartino, M., Well-posedness of the boundary layer equations, SIAM J Math Anal, 35, 987-1004 (2003) · Zbl 1053.76013 · doi:10.1137/S0036141002412057
[21] Paicu, M., Équation anisotrope de Navier-Stokes dans des espaces critiques, Rev Mat Iberoam, 21, 179-235 (2005) · Zbl 1110.35060 · doi:10.4171/RMI/420
[22] Paicu, M.; Raugel, G., Une perturbation hyperbolique des équations de Navier-Stokes, ESAIM Proc, 21, 65-87 (2007) · Zbl 1221.35284 · doi:10.1051/proc:072106
[23] Paicu, M.; Zhang, P., Global solutions to the 3-D incompressible anisotropic Navier-Stokes system in the critical spaces, Comm Math Phys, 307, 713-759 (2011) · Zbl 1237.35129 · doi:10.1007/s00220-011-1350-6
[24] Paicu, M.; Zhang, P., Global existence and the decay of solutions to the Prandtl system with small analytic data, Arch Ration Mech Anal, 241, 403-446 (2021) · Zbl 1472.35307 · doi:10.1007/s00205-021-01654-3
[25] Paicu, M.; Zhang, P.; Zhang, Z., On the hydrostatic approximation of the Navier-Stokes equations in a thin strip, Adv Math, 372, 107293 (2020) · Zbl 1446.35105 · doi:10.1016/j.aim.2020.107293
[26] Paicu, M.; Zhang, Z., Global regularity for the Navier-Stokes equations with some classes of large initial data, Anal PDE, 4, 95-113 (2011) · Zbl 1242.35187 · doi:10.2140/apde.2011.4.95
[27] Paicu, M.; Zhang, Z., Global well-posedness for the 3D Navier-Stokes equations with ill-prepared initial data, J Inst Math Jussieu, 13, 395-411 (2014) · Zbl 1291.35191 · doi:10.1017/S1474748013000212
[28] Racke, R.; Saal, J., Hyperbolic Navier-Stokes equations II: Global existence of small solutions, Evol Equ Control Theory, 1, 217-234 (2012) · Zbl 1371.35223 · doi:10.3934/eect.2012.1.217
[29] Renardy, M., Ill-posedness of the hydrostatic Euler and Navier-Stokes equations, Arch Ration Mech Anal, 194, 877-886 (2009) · Zbl 1292.76011 · doi:10.1007/s00205-008-0207-4
[30] Sammartino, M.; Caflisch, R. E., Zero viscosity limit for analytic solutions, of the Navier-Stokes equation on a half-space. I. Existence for Euler and Prandtl equations, Comm Math Phys, 192, 433-461 (1998) · Zbl 0913.35102 · doi:10.1007/s002200050304
[31] Vernotte, P., Some possible complication in the phenomena of thermal conduction, C R Acad Sci Paris Sér I Math, 252, 2190-2191 (1961)
[32] Wang C, Wang Y, Zhang P. On the global small solution of 2-D Prandtl system with initial data in the optimal Gevrey class. arXiv:2103.00681, 2021
[33] Zhang, P.; Zhang, Z., Long time well-posedness of Prandtl system with small and analytic inital data, J Funct Anal, 270, 2591-2615 (2016) · Zbl 1337.35113 · doi:10.1016/j.jfa.2016.01.004
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.