Global C ∞ regularity of the steady Prandtl equation with favorable pressure gradient
Wang, Yue1 ; Zhang, Zhifei2
1 a School of Mathematical Sciences, Capital Normal University, Beijing 100048, China
2 b School of Mathematical Sciences, Peking University, 100871, Beijing, China
Annales de l'I.H.P. Analyse non linéaire, novembre – décembre 2021, Tome 38 (2021) no. 6, pp. 1989-2004.
Le texte intégral des articles récents est réservé aux abonnés de la revue. Consultez l'article sur le site de la revue.

In the case of favorable pressure gradient, Oleinik obtained the global-in-x solutions to the steady Prandtl equations with low regularity (see Oleinik and Samokhin [9], P.21, Theorem 2.1.1). Due to the degeneracy of the equation near the boundary, the question of higher regularity of Oleinik's solutions remains open. See the local-in-x higher regularity established by Guo and Iyer [5]. In this paper, we prove that Oleinik's solutions are smooth up to the boundary y=0 for any x>0, using further maximum principle techniques. Moreover, since Oleinik only assumed low regularity on the data prescribed at x=0, our result implies instant smoothness (in the steady case, x=0 is often considered as initial time).

Reçu le :
Révisé le :
Accepté le :
DOI : 10.1016/j.anihpc.2021.02.007
Mots-clés : Global C∞ regularity, Steady Prandtl equations, Favorable pressure gradient
Wang, Yue 1 ; Zhang, Zhifei 2

1 a School of Mathematical Sciences, Capital Normal University, Beijing 100048, China
2 b School of Mathematical Sciences, Peking University, 100871, Beijing, China 
@article{AIHPC_2021__38_6_1989_0,
     author = {Wang, Yue and Zhang, Zhifei},
     title = {Global {\protect\emph{C}
}               \protect\textsuperscript{\ensuremath{\infty}} regularity of the steady {Prandtl} equation with favorable pressure gradient},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {1989--2004},
     publisher = {Elsevier},
     volume = {38},
     number = {6},
     year = {2021},
     doi = {10.1016/j.anihpc.2021.02.007},
     mrnumber = {4327905},
     zbl = {1475.35099},
     language = {en},
     url = {https://www.numdam.org/articles/10.1016/j.anihpc.2021.02.007/}
}
TY  - JOUR
AU  - Wang, Yue
AU  - Zhang, Zhifei
TI  - Global C
               ∞ regularity of the steady Prandtl equation with favorable pressure gradient
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2021
SP  - 1989
EP  - 2004
VL  - 38
IS  - 6
PB  - Elsevier
UR  - https://www.numdam.org/articles/10.1016/j.anihpc.2021.02.007/
DO  - 10.1016/j.anihpc.2021.02.007
LA  - en
ID  - AIHPC_2021__38_6_1989_0
ER  - 
%0 Journal Article
%A Wang, Yue
%A Zhang, Zhifei
%T Global C
               ∞ regularity of the steady Prandtl equation with favorable pressure gradient
%J Annales de l'I.H.P. Analyse non linéaire
%D 2021
%P 1989-2004
%V 38
%N 6
%I Elsevier
%U https://www.numdam.org/articles/10.1016/j.anihpc.2021.02.007/
%R 10.1016/j.anihpc.2021.02.007
%G en
%F AIHPC_2021__38_6_1989_0
Wang, Yue; Zhang, Zhifei. Global C
               ∞ regularity of the steady Prandtl equation with favorable pressure gradient. Annales de l'I.H.P. Analyse non linéaire, novembre – décembre 2021, Tome 38 (2021) no. 6, pp. 1989-2004. doi : 10.1016/j.anihpc.2021.02.007. https://www.numdam.org/articles/10.1016/j.anihpc.2021.02.007/

[1] Dalibard, A.; Masmoudi, N. Separation for the stationary Prandtl equation, Publ. Math. Inst. Hautes Études Sci., Volume 130 (2019), pp. 187-297 | DOI | MR | Zbl

[2] E, W. Boundary layer theory and the zero-viscosity limit of the Navier-Stokes equation, Acta Math. Sin., Volume 16 (2000), pp. 207-218 | DOI | MR | Zbl

[3] Gerard-Varet, D.; Maekawa, Y. Sobolev stability of Prandtl expansions for the steady Navier-Stokes equations, Arch. Ration. Mech. Anal., Volume 233 (2019), pp. 1319-1382 | DOI | MR | Zbl

[4] Guo, Y.; Iyer, S. Validity of steady Prandtl layer expansions | arXiv | DOI | Zbl

[5] Guo, Y.; Iyer, S. Regularity and expansion for steady Prandtl equations | arXiv | DOI | Zbl

[6] Iyer, S. On global-in-x stability of Blasius profiles, Arch. Ration. Mech. Anal., Volume 237 (2020), pp. 951-998 | DOI | MR | Zbl

[7] Iyer, S.; Masmoudi, N. Global-in-x stability of steady Prandtl expansions for 2D Navier-Stokes flows | arXiv

[8] Krylov, N.V. Lectures on Elliptic and Parabolic Equations in Hölder Spaces, Graduate Studies in Mathematics, vol. 12, American Mathematical Society, Providence, RI, 1996 | MR | Zbl

[9] Oleinik, O.A.; Samokhin, V.N. Mathematical Models in Boundary Layer Theory, Applied Mathematics and Mathematical Computation, vol. 15, Chapman & Hall/CRC, Boca Raton, FL, 1999 | MR | Zbl

[10] Prandtl, L., Heidelberg (1904), pp. 484-491

[11] Serrin, J. Asymptotic behaviour of velocity profiles in the Prandtl boundary layer theory, Proc. R. Soc. Lond. A, Volume 299 (1967), pp. 491-507 | DOI | MR | Zbl

[12] Shen, W.; Wang, Y.; Zhang, Z. Boundary layer separation and local behavior for the steady Prandtl equation | arXiv | DOI | Zbl

[13] Xin, Z.; Zhang, L. On the global existence of solutions to the Prandtl's system, Adv. Math., Volume 181 (2004), pp. 88-133 | DOI | MR | Zbl

Cité par Sources :