# American Institute of Mathematical Sciences

December  2016, 9(6): 2011-2029. doi: 10.3934/dcdss.2016082

## Global existence of weak solutions to the three-dimensional Prandtl equations with a special structure

 1 Department of Mathematics, City University of Hong Kong, Hong Kong, China 2 School of Mathematical Sciences, MOE-LSC and SHL-MAC, Shanghai Jiao Tong University, Shanghai 200240, China 3 Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong

Received  April 2015 Revised  September 2016 Published  November 2016

The global existence of weak solutions to the three space dimensional Prandtl equations is studied under some constraint on its structure. This is a continuation of our recent study on the local existence of classical solutions with the same structure condition. It reveals the sufficiency of the monotonicity condition on one component of the tangential velocity field and the favorable condition on pressure in the same direction that leads to global existence of weak solutions. This generalizes the result obtained by Xin-Zhang [14] on the two-dimensional Prandtl equations to the three-dimensional setting.
Citation: Cheng-Jie Liu, Ya-Guang Wang, Tong Yang. Global existence of weak solutions to the three-dimensional Prandtl equations with a special structure. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 2011-2029. doi: 10.3934/dcdss.2016082
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##### References:
 [1] Jingrui Wang, Keyan Wang. Almost sure existence of global weak solutions to the 3D incompressible Navier-Stokes equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 5003-5019. doi: 10.3934/dcds.2017215 [2] Jiahong Wu. Regularity results for weak solutions of the 3D MHD equations. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 543-556. doi: 10.3934/dcds.2004.10.543 [3] Ning Ju. The global attractor for the solutions to the 3D viscous primitive equations. Discrete & Continuous Dynamical Systems - A, 2007, 17 (1) : 159-179. doi: 10.3934/dcds.2007.17.159 [4] Yong Yang, Bingsheng Zhang. On the Kolmogorov entropy of the weak global attractor of 3D Navier-Stokes equations:Ⅰ. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2339-2350. doi: 10.3934/dcdsb.2017101 [5] Boyan Jonov, Thomas C. Sideris. Global and almost global existence of small solutions to a dissipative wave equation in 3D with nearly null nonlinear terms. Communications on Pure & Applied Analysis, 2015, 14 (4) : 1407-1442. doi: 10.3934/cpaa.2015.14.1407 [6] Claude Bardos, E. S. Titi. Loss of smoothness and energy conserving rough weak solutions for the $3d$ Euler equations. Discrete & Continuous Dynamical Systems - S, 2010, 3 (2) : 185-197. doi: 10.3934/dcdss.2010.3.185 [7] Jishan Fan, Fucai Li, Gen Nakamura. Global existence and low Mach number limit to the 3D compressible magnetohydrodynamic equations in a bounded domain. Conference Publications, 2015, 2015 (special) : 387-394. doi: 10.3934/proc.2015.0387 [8] Hantaek Bae, Rafael Granero-Belinchón, Omar Lazar. On the local and global existence of solutions to 1d transport equations with nonlocal velocity. Networks & Heterogeneous Media, 2019, 14 (3) : 471-487. doi: 10.3934/nhm.2019019 [9] Boling Guo, Guoli Zhou. Finite dimensionality of global attractor for the solutions to 3D viscous primitive equations of large-scale moist atmosphere. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4305-4327. doi: 10.3934/dcdsb.2018160 [10] Michael Röckner, Rongchan Zhu, Xiangchan Zhu. A remark on global solutions to random 3D vorticity equations for small initial data. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 4021-4030. doi: 10.3934/dcdsb.2019048 [11] Zhuan Ye. Remark on exponential decay-in-time of global strong solutions to 3D inhomogeneous incompressible micropolar equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (12) : 6725-6743. doi: 10.3934/dcdsb.2019164 [12] Junxiong Jia, Jigen Peng, Kexue Li. On the decay and stability of global solutions to the 3D inhomogeneous MHD system. Communications on Pure & Applied Analysis, 2017, 16 (3) : 745-780. doi: 10.3934/cpaa.2017036 [13] Yuncheng You, Caidi Zhao, Shengfan Zhou. The existence of uniform attractors for 3D Brinkman-Forchheimer equations. Discrete & Continuous Dynamical Systems - A, 2012, 32 (10) : 3787-3800. doi: 10.3934/dcds.2012.32.3787 [14] Ning Ju. The finite dimensional global attractor for the 3D viscous Primitive Equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 7001-7020. doi: 10.3934/dcds.2016104 [15] Joachim Naumann, Jörg Wolf. On Prandtl's turbulence model: Existence of weak solutions to the equations of stationary turbulent pipe-flow. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1371-1390. doi: 10.3934/dcdss.2013.6.1371 [16] Huicheng Yin, Lin Zhang. The global existence and large time behavior of smooth compressible fluid in an infinitely expanding ball, Ⅱ: 3D Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1063-1102. doi: 10.3934/dcds.2018045 [17] Mei Wang, Zilai Li, Zhenhua Guo. Global weak solution to 3D compressible flows with density-dependent viscosity and free boundary. Communications on Pure & Applied Analysis, 2017, 16 (1) : 1-24. doi: 10.3934/cpaa.2017001 [18] Feng Cheng, Chao-Jiang Xu. On the Gevrey regularity of solutions to the 3D ideal MHD equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (11) : 6485-6506. doi: 10.3934/dcds.2019281 [19] Aníbal Rodríguez-Bernal, Alejandro Vidal-López. A note on the existence of global solutions for reaction-diffusion equations with almost-monotonic nonlinearities. Communications on Pure & Applied Analysis, 2014, 13 (2) : 635-644. doi: 10.3934/cpaa.2014.13.635 [20] Kunio Hidano, Dongbing Zha. Remarks on a system of quasi-linear wave equations in 3D satisfying the weak null condition. Communications on Pure & Applied Analysis, 2019, 18 (4) : 1735-1767. doi: 10.3934/cpaa.2019082

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