In this paper, we prove the well-posedness of the linearized Prandtl equation around a non-monotonic shear flow in Gevrey class for any . This result is almost optimal by the ill-posedness result proved by Gérard-Varet and Dormy, who construct a class of solution with the growth like for the linearized Prandtl equation around a non-monotonic shear flow.
@article{AIHPC_2018__35_4_1119_0,
author = {Chen, Dongxiang and Wang, Yuxi and Zhang, Zhifei},
title = {Well-posedness of the linearized {Prandtl} equation around a non-monotonic shear flow},
journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
pages = {1119--1142},
publisher = {Elsevier},
volume = {35},
number = {4},
year = {2018},
doi = {10.1016/j.anihpc.2017.11.001},
mrnumber = {3795028},
zbl = {1392.35220},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.anihpc.2017.11.001/}
}
TY - JOUR AU - Chen, Dongxiang AU - Wang, Yuxi AU - Zhang, Zhifei TI - Well-posedness of the linearized Prandtl equation around a non-monotonic shear flow JO - Annales de l'I.H.P. Analyse non linéaire PY - 2018 SP - 1119 EP - 1142 VL - 35 IS - 4 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.anihpc.2017.11.001/ DO - 10.1016/j.anihpc.2017.11.001 LA - en ID - AIHPC_2018__35_4_1119_0 ER -
%0 Journal Article %A Chen, Dongxiang %A Wang, Yuxi %A Zhang, Zhifei %T Well-posedness of the linearized Prandtl equation around a non-monotonic shear flow %J Annales de l'I.H.P. Analyse non linéaire %D 2018 %P 1119-1142 %V 35 %N 4 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.anihpc.2017.11.001/ %R 10.1016/j.anihpc.2017.11.001 %G en %F AIHPC_2018__35_4_1119_0
Chen, Dongxiang; Wang, Yuxi; Zhang, Zhifei. Well-posedness of the linearized Prandtl equation around a non-monotonic shear flow. Annales de l'I.H.P. Analyse non linéaire, Tome 35 (2018) no. 4, pp. 1119-1142. doi : 10.1016/j.anihpc.2017.11.001. http://www.numdam.org/articles/10.1016/j.anihpc.2017.11.001/
[1] Well-posedness of the Prandtl equation in Sobolev spaces, J. Am. Math. Soc., Volume 28 (2015), pp. 745–784 | MR | Zbl
[2] D. Chen, Y. Wang, Z. Zhang, Well-posedness of the Prandtl equation with monotonicity in Sobolev Spaces, submitted for publication. | MR
[3] On the ill-posedness of the Prandtl equation, J. Am. Math. Soc., Volume 23 (2010), pp. 591–609 | MR | Zbl
[4] Well-posedness for the Prandtl system without analyticity or monotonicity, Ann. Sci. Éc. Norm. Supér., Volume 48 (2015), pp. 1273–1325 | MR | Zbl
[5] Remarks on the ill-posedness of the Prandtl equation, Asymptot. Anal., Volume 77 (2012), pp. 71–88 | MR | Zbl
[6] A note on Prandtl boundary layers, Commun. Pure Appl. Math., Volume 64 (2011), pp. 1416–1438 | MR | Zbl
[7] On the local well-posedness of the Prandtl and the hydrostatic Euler equations with multiple monotonicity regions, SIAM J. Math. Anal., Volume 46 (2014), pp. 3865–3890 | DOI | MR | Zbl
[8] Well-posedness in Gevrey space for the Prandtl equations with non-degenerate critical points | arXiv | DOI | Zbl
[9] Well-posedness of the boundary layer equations, SIAM J. Math. Anal., Volume 35 (2003), pp. 987–1004 | DOI | MR | Zbl
[10] Local-in-time existence and uniqueness of solutions to the Prandtl equations by energy methods, Commun. Pure Appl. Math., Volume 68 (2015), pp. 1683–1741 | DOI | MR
[11] Mathematical Models in Boundary Layer Theory, Applied Mathematics and Mathematical Computation, vol. 15, Chapman & Hall/CRC, Boca Raton, FL, 1999 | MR | Zbl
[12] Über Flüssigkeitsbewegung bei sehr kleiner Reibung, Verhandlung des III Intern. Math.-Kongresses, Heidelberg, 1904, pp. 484–491 | JFM
[13] Zero viscosity limit for analytic solutions, of the Navier–Stokes equation on a half-space. I. Existence for Euler and Prandtl equations, Commun. Math. Phys., Volume 192 (1998), pp. 433–461 | MR | Zbl
[14] On the global existence of solutions to the Prandtl's system, Adv. Math., Volume 181 (2004), pp. 88–133 | MR | Zbl
[15] Long time well-posedness of Prandtl system with small and analytic initial data, J. Funct. Anal., Volume 270 (2016), pp. 2591–2615 | DOI | MR | Zbl
Cité par Sources :
