Well-posedness for the Prandtl system without analyticity or monotonicity
[Caractère bien posé de l'équation de Prandtl sans monotonie ou analycité]
Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 48 (2015) no. 6, pp. 1273-1325.

Il a longtemps été supposé que l'équation de Prandtl n'est bien posée que sous l'hypothèse de monotonie d'Oleinik, ou pour des données analytiques. Nous montrons qu'elle est en fait localement bien posée pour des données appartenant à la classe Gevrey 7/4 en la variable x. Nous améliorons ainsi le résultat classique d'existence locale de solutions analytiques en la variable x (classe Gevrey 1). La preuve repose sur de nouvelles estimations, faisant appel à des fonctionnelles d'énergie non-quadratiques.

It has been thought for a while that the Prandtl system is only well-posed under the Oleinik monotonicity assumption or under an analyticity assumption. We show that the Prandtl system is actually locally well-posed for data that belong to the Gevrey class 7/4 in the horizontal variable x. Our result improves the classical local well-posedness result for data that are analytic in x (that is Gevrey class 1). The proof uses new estimates, based on non-quadratic energy functionals.

Publié le :
DOI : 10.24033/asens.2270
Classification : 35Q30, 35Q31, 35Q35.
Keywords: Boundary layer, Prandtl equation, Navier-Stokes equation, Gevrey spaces.
Mot clés : Couche limite, équation de Prandtl, équation de Navier-Stokes, espaces de Gevrey. 
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Gérard-Varet, David; Masmoudi, Nader. Well-posedness for the Prandtl system   without analyticity or monotonicity. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 48 (2015) no. 6, pp. 1273-1325. doi : 10.24033/asens.2270. https://www.numdam.org/articles/10.24033/asens.2270/

Alexandre, R.; Wang, Y.-G.; Xu, C.-J.; Yang, T. Well-posedness of the Prandtl equation in Sobolev spaces, J. Amer. Math. Soc., Volume 28 (2015), pp. 745-784 (ISSN: 0894-0347) | DOI | MR

Bedrossian, J.; Masmoudi, N. Asymptotic stability for the Couette flow in the 2D Euler equations, Appl. Math. Res. Express. AMRX, Volume 1 (2014), pp. 157-175 (ISSN: 1687-1200) | MR | Zbl

Brenier, Y. Homogeneous hydrostatic flows with convex velocity profiles, Nonlinearity, Volume 12 (1999), pp. 495-512 (ISSN: 0951-7715) | DOI | MR | Zbl

Cowley, S. J.; Hocking, L. M.; Tutty, O. R. The stability of solutions of the classical unsteady boundary-layer equation, Phys. Fluids, Volume 28 (1985), pp. 441-443 (ISSN: 0031-9171) | DOI | MR | Zbl

Foias, C.; Temam, R. Gevrey class regularity for the solutions of the Navier-Stokes equations, J. Funct. Anal., Volume 87 (1989), pp. 359-369 (ISSN: 0022-1236) | DOI | MR | Zbl

Ferrari, A. B.; Titi, E. S. Gevrey regularity for nonlinear analytic parabolic equations, Comm. Partial Differential Equations, Volume 23 (1998), pp. 1-16 (ISSN: 0360-5302) | DOI | MR | Zbl

Guyon, É.; Hulin, J.-P.; Petit, L., Savoirs actuels, 142, CNRS Éditions - EDP Sciences, 2001

Grenier, E. On the stability of boundary layers of incompressible Euler equations, J. Differential Equations, Volume 164 (2000), pp. 180-222 (ISSN: 0022-0396) | DOI | MR | Zbl

Grenier, E. On the derivation of homogeneous hydrostatic equations, M2AN Math. Model. Numer. Anal., Volume 33 (1999), pp. 965-970 (ISSN: 0764-583X) | DOI | Numdam | MR | Zbl

Gérard-Varet, D.; Dormy, E. On the ill-posedness of the Prandtl equation, J. Amer. Math. Soc., Volume 23 (2010), pp. 591-609 (ISSN: 0894-0347) | DOI | MR | Zbl

Gérard-Varet, D.; Nguyen, T. Remarks on the ill-posedness of the Prandtl equation, Asymptot. Anal., Volume 77 (2012), pp. 71-88 (ISSN: 0921-7134) | MR | Zbl

Hong, L.; Hunter, J. K. Singularity formation and instability in the unsteady inviscid and viscous Prandtl equations, Commun. Math. Sci., Volume 1 (2003), pp. 293-316 http://projecteuclid.org/euclid.cms/1118152072 (ISSN: 1539-6746) | DOI | MR | Zbl

Kufner, A.; Maligranda, L.; Persson, L.-E., Vydavatelský Servis, Plzeň, 2007, 162 pages (About its history and some related results) (ISBN: 978-80-86843-15-5) | MR | Zbl

Kukavica, I.; Masmoudi, N.; Vicol, V.; Wong, T. K. On the local well-posedness of the Prandtl and the hydrostatic Euler equations with multiple monotonicity regions, SIAM J. Math. Anal., Volume 6 (2014), pp. 3856-3890 | MR

Kukavica, I.; Vicol, V. On the local existence of analytic solutions to the Prandtl boundary layer equations, Commun. Math. Sci., Volume 11 (2013), pp. 269-292 (ISSN: 1539-6746) | DOI | MR | Zbl

Lombardo, M. C.; Cannone, M.; Sammartino, M. Well-posedness of the boundary layer equations, SIAM J. Math. Anal., Volume 35 (2003), pp. 987-1004 (ISSN: 0036-1410) | DOI | MR | Zbl

Levermore, C. D.; Oliver, M. Analyticity of solutions for a generalized Euler equation, J. Differential Equations, Volume 133 (1997), pp. 321-339 (ISSN: 0022-0396) | DOI | MR | Zbl

Masmoudi, N., Handbook of differential equations: evolutionary equations. Vol. III, Elsevier/North-Holland, Amsterdam, 2007, pp. 195-275 | DOI | MR | Zbl

Mouhot, C.; Villani, C. On Landau damping, Acta Math., Volume 207 (2011), pp. 29-201 (ISSN: 0001-5962) | DOI | MR | Zbl

Masmoudi, N.; Wong, T. K. Local-in-time existence and uniqueness of solutions to the Prandtl equations by energy methods (preprint arXiv:1206.3629 ) | MR

Masmoudi, N.; Wong, T. K. On the Hs theory of hydrostatic Euler equations, Arch. Ration. Mech. Anal., Volume 204 (2012), pp. 231-271 (ISSN: 0003-9527) | DOI | MR | Zbl

Oleinik, O. A.; Samokhin, V. N., Applied Mathematics and Mathematical Computation, 15, Chapman & Hall/CRC, Boca Raton, FL, 1999, 516 pages (ISBN: 1-58488-015-5) | MR | Zbl

Oliver, M.; Titi, E. S. Analyticity of the attractor and the number of determining nodes for a weakly damped driven nonlinear Schrödinger equation, Indiana Univ. Math. J., Volume 47 (1998), pp. 49-73 (ISSN: 0022-2518) | DOI | MR | Zbl

Renardy, M. Ill-posedness of the hydrostatic Euler and Navier-Stokes equations, Arch. Ration. Mech. Anal., Volume 194 (2009), pp. 877-886 (ISSN: 0003-9527) | DOI | MR | Zbl

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., Volume 192 (1998), pp. 433-461 (ISSN: 0010-3616) | DOI | MR | Zbl

Xin, Z.; Zhang, L. On the global existence of solutions to the Prandtl's system, Adv. Math., Volume 181 (2004), pp. 88-133 (ISSN: 0001-8708) | DOI | MR | Zbl

  • Wang, Fei; Zhu, Yichun The growth mechanism of boundary layers for the 2D Navier-Stokes equations, Journal of Differential Equations, Volume 416 (2025), p. 973 | DOI:10.1016/j.jde.2024.10.012
  • Jiang, Song; Zhou, Chunhui Stability and related zero viscosity limit of steady plane Poiseuille-Couette flows with no-slip boundary condition, Journal of Differential Equations, Volume 420 (2025), p. 52 | DOI:10.1016/j.jde.2024.11.047
  • Chen, Yufeng; Ruan, Lizhi; Yang, Anita The local well-posedness of analytic solution to the boundary layer system for compressible flow in three dimensions, Journal of Differential Equations, Volume 429 (2025), p. 716 | DOI:10.1016/j.jde.2025.02.056
  • De Anna, Francesco; Kortum, Joshua; Zarnescu, Arghir Boundary layers for the upper-convected Beris–Edwards model of nematic liquid crystals, Nonlinearity, Volume 38 (2025) no. 4, p. 045012 | DOI:10.1088/1361-6544/adbc3c
  • Wang, Chao; Wang, Yuxi; Zhang, Ping On the global small solution of 2-D Prandtl system with initial data in the optimal Gevrey class, Advances in Mathematics, Volume 440 (2024), p. 109517 | DOI:10.1016/j.aim.2024.109517
  • Gérard-Varet, David; Maekawa, Yasunori; Masmoudi, Nader Optimal Prandtl expansion around a concave boundary layer, Analysis PDE, Volume 17 (2024) no. 9, p. 3125 | DOI:10.2140/apde.2024.17.3125
  • Pan, Xinghong; Xu, Chao-Jiang Global Gevrey-2 solutions of the 3D axially symmetric Prandtl equations, Analysis and Applications, Volume 22 (2024) no. 07, p. 1195 | DOI:10.1142/s0219530524500167
  • Paicu, Marius; Yu, Tianyuan; Zhu, Ning On the hydrostatic Navier–Stokes equations with Gevrey class 2 data, Calculus of Variations and Partial Differential Equations, Volume 63 (2024) no. 3 | DOI:10.1007/s00526-024-02677-w
  • Lin, Xue-yun; Liu, Cheng-jie; Zhang, Ting Magneto-micropolar boundary layers theory in Sobolev spaces without monotonicity: well-posedness and convergence theory, Calculus of Variations and Partial Differential Equations, Volume 63 (2024) no. 3 | DOI:10.1007/s00526-024-02672-1
  • Pan, Xinghong; Xu, Chaojiang Global Tangentially Analytical Solutions of the 3D Axially Symmetric Prandtl Equations, Chinese Annals of Mathematics, Series B, Volume 45 (2024) no. 4, p. 573 | DOI:10.1007/s11401-024-0029-1
  • Grenier, Emmanuel; Nguyen, Toan T. On nonlinear instability of Prandtl's boundary layers: The case of Rayleigh's stable shear flows, Journal de Mathématiques Pures et Appliquées, Volume 184 (2024), p. 71 | DOI:10.1016/j.matpur.2024.02.001
  • Peng, Lei; Li, Jingyu; Mei, Ming; Zhang, Kaijun Characteristic boundary layers in the vanishing viscosity limit for the Hunter-Saxton equation, Journal of Differential Equations, Volume 386 (2024), p. 164 | DOI:10.1016/j.jde.2023.12.020
  • Gao, Jincheng; Li, Minling; Yao, Zheng-an Higher regularity and asymptotic behavior of 2D magnetic Prandtl model in the Prandtl-Hartmann regime, Journal of Differential Equations, Volume 386 (2024), p. 294 | DOI:10.1016/j.jde.2023.12.030
  • Chen, Yuhui; Huang, Jingchi; Li, Minling Global existence and decay estimates of solutions for the compressible Prandtl type equations with small analytic data, Journal of Functional Analysis, Volume 286 (2024) no. 7, p. 110322 | DOI:10.1016/j.jfa.2024.110322
  • Argenziano, Andrea; Cannone, Marco; Sammartino, Marco Navier–Stokes Equations in the Half Space with Non Compatible Data, Journal of Mathematical Fluid Mechanics, Volume 26 (2024) no. 2 | DOI:10.1007/s00021-024-00863-6
  • Wang, Chao; Wang, Yuxi OPTIMAL GEVREY STABILITY OF HYDROSTATIC APPROXIMATION FOR THE NAVIER-STOKES EQUATIONS IN A THIN DOMAIN, Journal of the Institute of Mathematics of Jussieu, Volume 23 (2024) no. 4, p. 1521 | DOI:10.1017/s1474748023000282
  • Aarach, Nacer Global well-posedness of 2D Hyperbolic perturbation of the Navier–Stokes system in a thin strip, Nonlinear Analysis: Real World Applications, Volume 76 (2024), p. 104014 | DOI:10.1016/j.nonrwa.2023.104014
  • Qin, Yuming; Dong, Xiaolei Local well-posedness of solutions to 2D mixed Prandtl equations in Sobolev space without monotonicity and lower bound, Nonlinear Analysis: Real World Applications, Volume 80 (2024), p. 104140 | DOI:10.1016/j.nonrwa.2024.104140
  • Paicu, Marius; Yu, Tianyuan; Zhu, Ning Hydrostatic approximation and optimal convergence rate for the Second-Grade fluid system, Nonlinearity, Volume 37 (2024) no. 7, p. 075008 | DOI:10.1088/1361-6544/ad48f2
  • Wang, Peixin; Li, Qian Zero dissipation limit of the anisotropic Boussinesq equations with Navier-slip and Neumann boundary conditions, Physica D: Nonlinear Phenomena, Volume 468 (2024), p. 134301 | DOI:10.1016/j.physd.2024.134301
  • Qin, Yuming; Dong, Xiaolei; Wang, Xiuqing Survey on the Prandtl Equations and Related Boundary Layer Equations, Prandtl Equations and Related Boundary Layer Equations (2024), p. 1 | DOI:10.1007/978-981-97-4565-4_1
  • Qin, Yuming; Dong, Xiaolei; Wang, Xiuqing Global Well-Posedness of Solutions to the 2D Prandtl-Hartmann Equations in an Analytic Framework, Prandtl Equations and Related Boundary Layer Equations (2024), p. 205 | DOI:10.1007/978-981-97-4565-4_2
  • Qin, Yuming; Dong, Xiaolei; Wang, Xiuqing Local Well-Posedness of Solutions to 2D Magnetic Prandtl Model in the Prandtl-Hartmann Regime, Prandtl Equations and Related Boundary Layer Equations (2024), p. 271 | DOI:10.1007/978-981-97-4565-4_5
  • Lin, Xueyun; Zou, Lin Well-Posedness in Gevrey Function Space for the 3D Axially Symmetric MHD Boundary Layer Equations Without Structural Assumption, Results in Mathematics, Volume 79 (2024) no. 2 | DOI:10.1007/s00025-023-02112-0
  • De Anna, Francesco; Kortum, Joshua; Scrobogna, Stefano Quantitative aspects on the ill-posedness of the Prandtl and hyperbolic Prandtl equations, Zeitschrift für angewandte Mathematik und Physik, Volume 75 (2024) no. 2 | DOI:10.1007/s00033-023-02179-3
  • Tan, Zhong; Wu, Zhonger, 2023 | DOI:10.2139/ssrn.4349314
  • Guo, Yan; Wang, Yue; Zhang, Zhifei Dynamic Stability for Steady Prandtl Solutions, Annals of PDE, Volume 9 (2023) no. 2 | DOI:10.1007/s40818-023-00160-x
  • Guo, Yan; Iyer, Sameer Validity of steady Prandtl layer expansions, Communications on Pure and Applied Mathematics, Volume 76 (2023) no. 11, p. 3150 | DOI:10.1002/cpa.22109
  • Tan, Zhong; Wu, Zhonger Global small solutions of MHD boundary layer equations in Gevrey function space*, Journal of Differential Equations, Volume 366 (2023), p. 444 | DOI:10.1016/j.jde.2023.04.019
  • Gerard-Varet, David; Iyer, Sameer; Maekawa, Yasunori Improved Well-Posedness for the Triple-Deck and Related Models via Concavity, Journal of Mathematical Fluid Mechanics, Volume 25 (2023) no. 3 | DOI:10.1007/s00021-023-00809-4
  • De Anna, Francesco; Kortum, Joshua; Scrobogna, Stefano Gevrey-Class-3 Regularity of the Linearised Hyperbolic Prandtl System on a Strip, Journal of Mathematical Fluid Mechanics, Volume 25 (2023) no. 4 | DOI:10.1007/s00021-023-00821-8
  • Chen, Dongxiang; Li, Xiaoli Long time well‐posedness of two dimensional Magnetohydrodynamic boundary layer equation without resistivity, Mathematical Methods in the Applied Sciences, Volume 46 (2023) no. 9, p. 10186 | DOI:10.1002/mma.9110
  • Bachmann, L.; De Anna, F.; Schlömerkemper, A.; Şengül, Y. Existence of solutions for stress-rate type strain-limiting viscoelasticity in Gevrey spaces, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, Volume 381 (2023) no. 2263 | DOI:10.1098/rsta.2022.0374
  • Guo, Yan; Iyer, Sameer Steady Prandtl layer expansions with external forcing, Quarterly of Applied Mathematics, Volume 81 (2023) no. 2, p. 375 | DOI:10.1090/qam/1655
  • Ning, Liu; Ping, Zhang Global stability of monotone shear flows for the 2-D Prandtl system in Sobolev spaces, SCIENTIA SINICA Mathematica (2023) | DOI:10.1360/ssm-2023-0011
  • Gao, Chen; Zhang, Liqun On the steady Prandtl boundary layer expansions, Science China Mathematics, Volume 66 (2023) no. 9, p. 1993 | DOI:10.1007/s11425-022-2025-5
  • Li, Wei-Xi; Xu, Rui; Yang, Tong Global well-posedness of a Prandtl model from MHD in Gevrey function spaces, Acta Mathematica Scientia, Volume 42 (2022) no. 6, p. 2343 | DOI:10.1007/s10473-022-0609-7
  • Qin, Yuming; Dong, Xiaolei Local existence of solutions to 2D Prandtl equations in a weighted Sobolev space, Analysis and Mathematical Physics, Volume 12 (2022) no. 1 | DOI:10.1007/s13324-021-00615-z
  • Jeong, In-Jee; Oh, Sung-Jin On the Cauchy Problem for the Hall and Electron Magnetohydrodynamic Equations Without Resistivity I: Illposedness Near Degenerate Stationary Solutions, Annals of PDE, Volume 8 (2022) no. 2 | DOI:10.1007/s40818-022-00134-5
  • Li, Wei‐Xi; Masmoudi, Nader; Yang, Tong Well‐Posedness in Gevrey Function Space for 3D Prandtl Equations without Structural Assumption, Communications on Pure and Applied Mathematics, Volume 75 (2022) no. 8, p. 1755 | DOI:10.1002/cpa.21989
  • Aarach, Nacer Hydrostatic approximation of the 2D MHD system in a thin strip with a small analytic data, Journal of Mathematical Analysis and Applications, Volume 509 (2022) no. 2, p. 125949 | DOI:10.1016/j.jmaa.2021.125949
  • Kukavica, Igor; Nguyen, Trinh T.; Vicol, Vlad; Wang, Fei On the Euler+Prandtl Expansion for the Navier-Stokes Equations, Journal of Mathematical Fluid Mechanics, Volume 24 (2022) no. 2 | DOI:10.1007/s00021-021-00645-4
  • Zou, Lin; Lin, Xueyun Magnetic effects on the solvability of 2D incompressible magneto-micropolar boundary layer equations without resistivity in Sobolev spaces, Nonlinear Analysis, Volume 224 (2022), p. 113080 | DOI:10.1016/j.na.2022.113080
  • Wang, Chao; Wang, Yuxi On the Hydrostatic Approximation of the MHD Equations in a Thin Strip, SIAM Journal on Mathematical Analysis, Volume 54 (2022) no. 1, p. 1241 | DOI:10.1137/21m1425360
  • Chen, Qi; Wu, Di; Zhang, Zhifei On the L∞ stability of Prandtl expansions in the Gevrey class, Science China Mathematics, Volume 65 (2022) no. 12, p. 2521 | DOI:10.1007/s11425-021-1896-5
  • Paicu, Marius; Zhang, Ping Global hydrostatic approximation of the hyperbolic Navier-Stokes system with small Gevrey class 2 data, Science China Mathematics, Volume 65 (2022) no. 6, p. 1109 | DOI:10.1007/s11425-021-1956-8
  • Paicu, Marius; Zhang, Ping Global Existence and the Decay of Solutions to the Prandtl System with Small Analytic Data, Archive for Rational Mechanics and Analysis, Volume 241 (2021) no. 1, p. 403 | DOI:10.1007/s00205-021-01654-3
  • Iyer, Sameer; Vicol, Vlad Real Analytic Local Well‐Posedness for the Triple Deck, Communications on Pure and Applied Mathematics, Volume 74 (2021) no. 8, p. 1641 | DOI:10.1002/cpa.21894
  • Li, Wei-Xi; Xu, Rui Well-posedness in Sobolev spaces of the two-dimensional MHD boundary layer equations without viscosity, Electronic Research Archive, Volume 29 (2021) no. 6, p. 4243 | DOI:10.3934/era.2021082
  • Ding, Shijin; Ji, Zhijun; Lin, Zhilin Validity of Prandtl layer theory for steady magnetohydrodynamics over a moving plate with nonshear outer ideal MHD flows, Journal of Differential Equations, Volume 278 (2021), p. 220 | DOI:10.1016/j.jde.2020.12.039
  • Liu, Ning; Zhang, Ping Global small analytic solutions of MHD boundary layer equations, Journal of Differential Equations, Volume 281 (2021), p. 199 | DOI:10.1016/j.jde.2021.02.003
  • Li, Shengxin; Xie, Feng Global solvability of 2D MHD boundary layer equations in analytic function spaces, Journal of Differential Equations, Volume 299 (2021), p. 362 | DOI:10.1016/j.jde.2021.07.025
  • Maekawa, Yasunori Gevrey stability of Rayleigh boundary layer in the inviscid limit, Journal of Elliptic and Parabolic Equations, Volume 7 (2021) no. 2, p. 417 | DOI:10.1007/s41808-021-00128-7
  • Guo, Lianhong; Ji, Zhijun Validity of boundary layer theory for the 3D plane‐parallel nonhomogeneous electrically conducting flows, Mathematical Methods in the Applied Sciences, Volume 44 (2021) no. 11, p. 8862 | DOI:10.1002/mma.7314
  • Wang, Chao; Wang, Yuxi; Zhang, Zhifei Gevrey stability of hydrostatic approximate for the Navier–Stokes equations in a thin domain, Nonlinearity, Volume 34 (2021) no. 10, p. 7185 | DOI:10.1088/1361-6544/ac20a6
  • Lin, Xueyun; Zhang, Ting Almost Global Existence for the 3D Prandtl Boundary Layer Equations, Acta Applicandae Mathematicae, Volume 169 (2020) no. 1, p. 383 | DOI:10.1007/s10440-019-00303-y
  • Paicu, Marius; Zhang, Ping; Zhang, Zhifei On the hydrostatic approximation of the Navier-Stokes equations in a thin strip, Advances in Mathematics, Volume 372 (2020), p. 107293 | DOI:10.1016/j.aim.2020.107293
  • Morisse, Baptiste On hyperbolicity and Gevrey well-posedness. Part one: the elliptic case., Annales Henri Lebesgue, Volume 3 (2020), p. 1195 | DOI:10.5802/ahl.59
  • Kukavica, Igor; Vicol, Vlad; Wang, Fei The Inviscid Limit for the Navier–Stokes Equations with Data Analytic Only Near the Boundary, Archive for Rational Mechanics and Analysis, Volume 237 (2020) no. 2, p. 779 | DOI:10.1007/s00205-020-01517-3
  • Chen, Dongxiang; Ren, Siqi; Wang, Yuxi; Zhang, Zhifei Global well-posedness of the 2-D magnetic Prandtl model in the Prandtl–Hartmann regime, Asymptotic Analysis, Volume 120 (2020) no. 3-4, p. 373 | DOI:10.3233/asy-191593
  • Li, Quanrong; Ding, Shijin Symmetrical Prandtl boundary layer expansions of steady Navier-Stokes equations on bounded domain, Journal of Differential Equations, Volume 268 (2020) no. 4, p. 1771 | DOI:10.1016/j.jde.2019.09.030
  • Liu, Cheng-Jie; Wang, Dehua; Xie, Feng; Yang, Tong Magnetic effects on the solvability of 2D MHD boundary layer equations without resistivity in Sobolev spaces, Journal of Functional Analysis, Volume 279 (2020) no. 7, p. 108637 | DOI:10.1016/j.jfa.2020.108637
  • Wang, Chao; Wang, Yuxi Zero-Viscosity Limit of the Navier–Stokes Equations in a Simply-Connected Bounded Domain Under the Analytic Setting, Journal of Mathematical Fluid Mechanics, Volume 22 (2020) no. 1 | DOI:10.1007/s00021-019-0471-0
  • Wang, Ya‐Guang; Zhu, Shi‐Yong Well‐posedness of thermal boundary layer equation in two‐dimensional incompressible heat conducting flow with analytic datum, Mathematical Methods in the Applied Sciences (2020) | DOI:10.1002/mma.6226
  • Wang, Ya-Guang; Zhu, Shi-Yong Back Flow of the Two-dimensional Unsteady Prandtl Boundary Layer Under an Adverse Pressure Gradient, SIAM Journal on Mathematical Analysis, Volume 52 (2020) no. 1, p. 954 | DOI:10.1137/19m1270355
  • Blechta, Jan; Málek, Josef; Rajagopal, K. R. On the Classification of Incompressible Fluids and a Mathematical Analysis of the Equations That Govern Their Motion, SIAM Journal on Mathematical Analysis, Volume 52 (2020) no. 2, p. 1232 | DOI:10.1137/19m1244895
  • Wang, Fei The Three-Dimensional Inviscid Limit Problem with Data Analytic Near the Boundary, SIAM Journal on Mathematical Analysis, Volume 52 (2020) no. 4, p. 3520 | DOI:10.1137/19m1296094
  • Li, Ya Jun; Wang, Wen Dong Local Well-posedness for Linearized Degenerate MHD Boundary Layer Equations in Analytic Setting, Acta Mathematica Sinica, English Series, Volume 35 (2019) no. 8, p. 1402 | DOI:10.1007/s10114-019-8067-4
  • Dietert, Helge; Gérard-Varet, David Well-Posedness of the Prandtl Equations Without Any Structural Assumption, Annals of PDE, Volume 5 (2019) no. 1 | DOI:10.1007/s40818-019-0063-6
  • Grenier, Emmanuel; Nguyen, Toan T. L∞ Instability of Prandtl Layers, Annals of PDE, Volume 5 (2019) no. 2 | DOI:10.1007/s40818-019-0074-3
  • Gerard-Varet, David; Maekawa, Yasunori Sobolev Stability of Prandtl Expansions for the Steady Navier–Stokes Equations, Archive for Rational Mechanics and Analysis, Volume 233 (2019) no. 3, p. 1319 | DOI:10.1007/s00205-019-01380-x
  • Liu, Cheng‐Jie; Xie, Feng; Yang, Tong MHD Boundary Layers Theory in Sobolev Spaces Without Monotonicity I: Well‐Posedness Theory, Communications on Pure and Applied Mathematics, Volume 72 (2019) no. 1, p. 63 | DOI:10.1002/cpa.21763
  • Bardos, Claude Zero Viscosity Boundary Effect Limit and Turbulence, Contributions to Partial Differential Equations and Applications, Volume 47 (2019), p. 77 | DOI:10.1007/978-3-319-78325-3_7
  • Huang, Yongting; Liu, Cheng-Jie; Yang, Tong Local-in-time well-posedness for compressible MHD boundary layer, Journal of Differential Equations, Volume 266 (2019) no. 6, p. 2978 | DOI:10.1016/j.jde.2018.08.052
  • Sun, Yimin A blow-up criterion for classical solutions to the Prandtl equations, Journal of Mathematical Physics, Volume 60 (2019) no. 11 | DOI:10.1063/1.5079672
  • Iyer, Sameer Global Steady Prandtl Expansion over a Moving Boundary I, Peking Mathematical Journal, Volume 2 (2019) no. 2, p. 155 | DOI:10.1007/s42543-019-00011-4
  • Dalibard, Anne-Laure; Masmoudi, Nader Separation for the stationary Prandtl equation, Publications mathématiques de l'IHÉS, Volume 130 (2019) no. 1, p. 187 | DOI:10.1007/s10240-019-00110-z
  • Liu, Cheng-Jie; Xie, Feng; Yang, Tong Justification of Prandtl Ansatz for MHD Boundary Layer, SIAM Journal on Mathematical Analysis, Volume 51 (2019) no. 3, p. 2748 | DOI:10.1137/18m1219618
  • Chen, Dongxiang; Wang, Yuxi; Zhang, Zhifei Well-posedness of the linearized Prandtl equation around a non-monotonic shear flow, Annales de l'Institut Henri Poincaré C, Analyse non linéaire, Volume 35 (2018) no. 4, p. 1119 | DOI:10.1016/j.anihpc.2017.11.001
  • Grenier, Emmanuel; Nguyen, Toan T. Sublayer of Prandtl Boundary Layers, Archive for Rational Mechanics and Analysis, Volume 229 (2018) no. 3, p. 1139 | DOI:10.1007/s00205-018-1235-3
  • Nguyen, Toan T.; Nguyen, Trinh T. The Inviscid Limit of Navier–Stokes Equations for Analytic Data on the Half-Space, Archive for Rational Mechanics and Analysis, Volume 230 (2018) no. 3, p. 1103 | DOI:10.1007/s00205-018-1266-9
  • Gérard-Varet, David; Maekawa, Yasunori; Masmoudi, Nader Gevrey stability of Prandtl expansions for 2-dimensional Navier–Stokes flows, Duke Mathematical Journal, Volume 167 (2018) no. 13 | DOI:10.1215/00127094-2018-0020
  • Maekawa, Yasunori; Mazzucato, Anna The Inviscid Limit and Boundary Layers for Navier-Stokes Flows, Handbook of Mathematical Analysis in Mechanics of Viscous Fluids (2018), p. 781 | DOI:10.1007/978-3-319-13344-7_15
  • Fei, Mingwen; Tao, Tao; Zhang, Zhifei On the zero-viscosity limit of the Navier–Stokes equations in R+3 without analyticity, Journal de Mathématiques Pures et Appliquées, Volume 112 (2018), p. 170 | DOI:10.1016/j.matpur.2017.09.007
  • Nguyen van yen, Natacha; Waidmann, Matthias; Klein, Rupert; Farge, Marie; Schneider, Kai Energy dissipation caused by boundary layer instability at vanishing viscosity, Journal of Fluid Mechanics, Volume 849 (2018), p. 676 | DOI:10.1017/jfm.2018.396
  • Xu, Xin Uniform regularity for the incompressible Navier-Stokes system with variable density and Navier boundary conditions, Quarterly of Applied Mathematics, Volume 77 (2018) no. 3, p. 553 | DOI:10.1090/qam/1515
  • Xie, Feng; Yang, Tong Global-in-Time Stability of 2D MHD Boundary Layer in the Prandtl–Hartmann Regime, SIAM Journal on Mathematical Analysis, Volume 50 (2018) no. 6, p. 5749 | DOI:10.1137/18m1174969
  • Liu, Cheng-Jie; Xie, Feng; Yang, Tong A note on the ill-posedness of shear flow for the MHD boundary layer equations, Science China Mathematics, Volume 61 (2018) no. 11, p. 2065 | DOI:10.1007/s11425-017-9306-0
  • Gie, Gung-Min; Hamouda, Makram; Jung, Chang-Yeol; Temam, Roger M. The Navier-Stokes Equations in a Periodic Channel, Singular Perturbations and Boundary Layers, Volume 200 (2018), p. 251 | DOI:10.1007/978-3-030-00638-9_6
  • Liu, Cheng-Jie; Wang, Ya-Guang; Yang, Tong A well-posedness theory for the Prandtl equations in three space variables, Advances in Mathematics, Volume 308 (2017), p. 1074 | DOI:10.1016/j.aim.2016.12.025
  • Xu, Chao-Jiang; Zhang, Xu Long time well-posedness of Prandtl equations in Sobolev space, Journal of Differential Equations, Volume 263 (2017) no. 12, p. 8749 | DOI:10.1016/j.jde.2017.08.046
  • Cheng, Feng; Li, Wei‐Xi; Xu, Chao‐Jiang Vanishing viscosity limit of Navier–Stokes Equations in Gevrey class, Mathematical Methods in the Applied Sciences, Volume 40 (2017) no. 14, p. 5161 | DOI:10.1002/mma.4378
  • Gérard-Varet, D.; Prestipino, M. Formal derivation and stability analysis of boundary layer models in MHD, Zeitschrift für angewandte Mathematik und Physik, Volume 68 (2017) no. 3 | DOI:10.1007/s00033-017-0820-x
  • Maekawa, Yasunori; Mazzucato, Anna The Inviscid Limit and Boundary Layers for Navier-Stokes Flows, Handbook of Mathematical Analysis in Mechanics of Viscous Fluids (2016), p. 1 | DOI:10.1007/978-3-319-10151-4_15-1
  • Li, Wei-Xi; Wu, Di; Xu, Chao-Jiang Gevrey Class Smoothing Effect for the Prandtl Equation, SIAM Journal on Mathematical Analysis, Volume 48 (2016) no. 3, p. 1672 | DOI:10.1137/15m1020368
  • Gérard-Varet, David; Masmoudi, Nader Well-posedness issues for the Prandtl boundary layer equations, Séminaire Laurent Schwartz — EDP et applications (2014), p. 1 | DOI:10.5802/slsedp.59

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