Acta mathematica scientia,Series A ›› 2026, Vol. 46 ›› Issue (2): 616-627.
Previous Articles Next Articles
Weixi Li*(
), Jiaxi Wang(
), Zhan Xu(
)
Received:2025-12-22
Revised:2026-01-27
Online:2026-04-26
Published:2026-04-27
Contact:
Weixi Li
E-mail:wei-xi.li@whu.edu.cn;wangjiaxi@whu.edu.cn;xuzhan@whu.edu.cn
Supported by:CLC Number:
Weixi Li, Jiaxi Wang, Zhan Xu. Long-Time Well-Posedness of the Hyperbolic Prandtl Equations in Gevrey Spaces[J].Acta mathematica scientia,Series A, 2026, 46(2): 616-627.
Add to citation manager EndNote|Reference Manager|ProCite|BibTeX|RefWorks
| [1] | Abdelhedi B. Global existence of solutions for hyperbolic Navier-Stokes equations in three space dimensions. Asymptot Anal, 2019, 112(3/4): 213-225 |
| [2] | Alexandre R, Wang Y G, Xu C J, Yang T. Well-posedness of the Prandtl equation in Sobolev spaces. J Amer Math Soc, 2015, 28(3): 745-784 |
| [3] | Brenier Y, Natalini R, Puel M. On a relaxation approximation of the incompressible Navier-Stokes equations. Proc Amer Math Soc, 2004, 132(4): 1021-1028 |
| [4] | Cattaneo C. Sulla conduzione del calore. Atti Sem Mat Fis Univ Modena, 1949, 3: 83-101 |
| [5] | Chen D, Wang Y, Zhang Z. Well-posedness of the linearized Prandtl equation around a non-monotonic shear flow. Ann Inst H Poincaré Anal Non Linéaire, 2018, 35(4): 1119-1142 |
| [6] | Collot C, Ghoul T E, Masmoudi N. Singularities and unsteady separation for the inviscid two-dimensional Prandtl system. Arch Ration Mech Anal, 2021, 240(3): 1349-1430 |
| [7] | Coulaud O, Hachicha I, Raugel G. Hyperbolic quasilinear Navier-Stokes equations in $\mathbb{R}^2$. J Dynam Differential Equations, 2022, 34(4): 2749-2785 |
| [8] | Dalibard A L, Masmoudi N. Separation for the stationary Prandtl equation. Publ Math Inst Hautes études Sci, 2019, 130: 187-297 |
| [9] | Dietert H, Gérard-Varet D. Well-posedness of the Prandtl equations without any structural assumption. Ann PDE, 2019, 5(1): Art 8 |
| [10] | Gérard-Varet D, Dormy E. On the ill-posedness of the Prandtl equation. J Amer Math Soc, 2010, 23(2): 591-609 |
| [11] | Gérard-Varet D, Masmoudi N. Well-posedness for the Prandtl system without analyticity or monotonicity. Ann Sci éc Norm Supér, 2015, 48(6): 1273-1325 |
| [12] | Grenier E. On the nonlinear instability of Euler and Prandtl equations. Comm Pure Appl Math, 2000, 53(9): 1067-1091 |
| [13] | Grenier E, Guo Y, Nguyen T T. Spectral instability of characteristic boundary layer flows. Duke Math J, 2016, 165(16): 3085-3146 |
| [14] | Grenier E, Guo Y, Nguyen T T. Spectral instability of general symmetric shear flows in a two-dimensional channel. Adv Math, 2016, 292: 52-110 |
| [15] | Guo Y, Nguyen T. A note on Prandtl boundary layers. Comm Pure Appl Math, 2011, 64(10): 1416-1438 |
| [16] | Ignatova M, Vicol V. Almost global existence for the Prandtl boundary layer equations. Arch Ration Mech Anal, 2016, 220(2): 809-848 |
| [17] | Kukavica I, Masmoudi N, Vicol V, Wong T K. On the local well-posedness of the Prandtl and hydrostatic Euler equations with multiple monotonicity regions. SIAM J Math Anal, 2014, 46(6): 3865-3890 |
| [18] | Kukavica I, Temam R, Vicol V C, Ziane M. Local existence and uniqueness for the hydrostatic Euler equations on a bounded domain. J Differential Equations, 2011, 250(3): 1719-1746 |
| [19] | Li W X, Masmoudi N, Yang T. Well-posedness in Gevrey function space for 3D Prandtl equations without structural assumption. Comm Pure Appl Math, 2022, 75(8): 1755-1797 |
| [20] | Li W X, Ngo V S, Xu C J. Boundary layer analysis for the fast horizontal rotating fluids. Commun Math Sci, 2019, 17(2): 299-338 |
| [21] | Li W X, Wu D, Xu C J. Gevrey class smoothing effect for the Prandtl equation. SIAM J Math Anal, 2016, 48(3): 1672-1726 |
| [22] | Li W X, Xu R. Gevrey well-posedness of the hyperbolic Prandtl equations. Commun Math Res, 2022, 38(4): 605-624 |
| [23] | Li W X, Yang T. Well-posedness in Gevrey function spaces for the Prandtl equations with non-degenerate critical points. J Eur Math Soc, 2020, 22(3): 717-775 |
| [24] | Liu C J, Wang Y G, Yang T. On the ill-posedness of the Prandtl equations in three-dimensional space. Arch Ration Mech Anal, 2016, 220(1): 83-108 |
| [25] | Liu C J, Wang Y G, Yang T. A well-posedness theory for the Prandtl equations in three space variables. Adv Math, 2017, 308: 1074-1126 |
| [26] | Liu C J, Yang T. Ill-posedness of the Prandtl equations in Sobolev spaces around a shear flow with general decay. J Math Pures Appl, 2017, 108(2): 150-162 |
| [27] | Masmoudi N, Wong T K. Local-in-time existence and uniqueness of solutions to the Prandtl equations by energy methods. Comm Pure Appl Math, 2015, 68(10): 1683-1741 |
| [28] | Oleinik O A, Samokhin V N. Mathematical Models in Boundary Layer Theory, Boca Raton FL: CRC Press, 1999 |
| [29] | Paicu M, Zhang P. Global existence and the decay of solutions to the Prandtl system with small analytic data. Arch Ration Mech Anal, 2021, 241(1): 403-446 |
| [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, 1998, 192(2): 433-461 |
| [31] | Wang C, Wang Y, Zhang P. On the global small solution of 2-D Prandtl system with initial data in the optimal Gevrey class. Adv Math, 2024, 440: Art 109517 |
| [32] | Weinan E, Engquist B. Blowup of solutions of the unsteady Prandtl's equation. Comm Pure Appl Math, 1997, 50(12): 1287-1293 |
| [33] | Xin Z, Zhang L. On the global existence of solutions to the Prandtl's system. Adv Math, 2004, 181(1): 88-133 |
| [34] | Xu C J, Zhang X. Long time well-posedness of Prandtl equations in Sobolev space. J Differential Equations, 2017, 263(12): 8749-8803 |
| [35] | Yang T.Vector fields of Cancellation for the Prandtl Operators. arXiv:2201.10139 |
| [36] | Zhang P, Zhang Z. Long time well-posedness of Prandtl system with small and analytic initial data. J Funct Anal, 2016, 270(7): 2591-2615 |
| [1] | Hui Liu. Some New Progress for the Problems About Closed Characteristics on Compact Star-Shaped Hypersurfaces [J]. Acta mathematica scientia,Series A, 2026, 46(2): 493-502. |
| [2] | Yaping Guo, Jialin Li, Wenbin Lyu. Global Existence for a Class of Chemotaxis Systems with Signal-Dependent Motility and Generalized Logistic Source [J]. Acta mathematica scientia,Series A, 2026, 46(2): 535-551. |
| [3] | Weiwei Ao, Shanshan Lai. Sharp Estimates for Fully Bubbling Solutions of $G_2$ Toda System [J]. Acta mathematica scientia,Series A, 2026, 46(2): 552-583. |
| [4] | Juncheng Wei, Yifu Zhou. Parabolic Gluing Method and Singularity Formation [J]. Acta mathematica scientia,Series A, 2026, 46(2): 646-668. |
| [5] | Xiaohan Pi, Qiuju Xu. Global Existence and Large Time Behavior of Strong Solutions for Three-Dimensional Compressible Liquid Crystals in a Bounded Domain [J]. Acta mathematica scientia,Series A, 2026, 46(1): 200-214. |
| [6] | Zhengwu Yang, Min Xiao, Ying Zhou, Jie Ding, Jing Zhao, Rutkowski Leszek. Spatiotemporal Dynamics Analysis of a Leslie-Gower Predator-Prey System with Multiple Interactions [J]. Acta mathematica scientia,Series A, 2026, 46(1): 143-156. |
| [7] | Changjian Wang, Gaofeng Zheng. Convergence of Solutions to Dirichlet Problem of Parabolic Allen-Cahn Equation [J]. Acta mathematica scientia,Series A, 2025, 45(6): 1768-1790. |
| [8] | Huifang Jia, Xiangyi Ye. Normalized Solutions to the Schrödinger Equation with Forcing Perturbation Term [J]. Acta mathematica scientia,Series A, 2025, 45(6): 1791-1805. |
| [9] | Xiao Luo, Xi Qin. Limiting Properties of Ground States for the Schrödinger Equation with Mixed Repulsive Nonlinear Terms [J]. Acta mathematica scientia,Series A, 2025, 45(6): 1942-1960. |
| [10] | Mengxia Wu, Tao Yang, Hua Zhang. Another Proof on the Existence of Normalized Solution to a Fourth-Order Schrödinger Equation [J]. Acta mathematica scientia,Series A, 2025, 45(6): 1928-1941. |
| [11] | Huan Luo, Haoguang Li. Global Well-Posedness and Optimal Decay for the Lower Regularity Solution of Boltzmann Equation [J]. Acta mathematica scientia,Series A, 2025, 45(6): 1854-1874. |
| [12] | Hongyu Ye. The Existence of Normalized Solutions for the Quasilinear Schrödinger Equations with $L^2$-Subcritical General Nonlinearity [J]. Acta mathematica scientia,Series A, 2025, 45(6): 1907-1927. |
| [13] | Yumiao Cheng, Mingzi Fang, Youjun Wang. A Class of Local and Nonlocal Elliptic Equations with Nirenberg-Brezis Problem [J]. Acta mathematica scientia,Series A, 2025, 45(5): 1519-1534. |
| [14] | Wang Hanyi, Huang Shiyu, Xiang Jianlin. Mountain-pass Solution and Ground State Solution for a Kirchhoff Type Elliptic Equation [J]. Acta mathematica scientia,Series A, 2025, 45(4): 1041-1057. |
| [15] | Zuo Wenwen, Zhou Shouming. Global Solutions for the Parabolic-Parabolic Keller-Segel Equation with a Logistic Source Term in High Dimensions [J]. Acta mathematica scientia,Series A, 2025, 45(4): 1100-1109. |
|
||