American Institute of Mathematical Sciences

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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

Cheng-Jie Liu 1, , Ya-Guang Wang 2, and  Tong Yang 3,

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.
Keywords: 3D Prandtl equations, weak solutions, favorable pressure., monotonic velocity field, global existence.
Mathematics Subject Classification: 35M13, 35Q35, 76D03, 76D10, 76N2.
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
References:
[1]

R. Alexandre, Y.-G. Wang, C.-J. Xu and T. Yang, Well-posedness of the Prandtl equation in Sobolev spaces, J. Amer. Math. Soc., 28 (2015), 745-784. doi: 10.1090/S0894-0347-2014-00813-4.  Google Scholar

[2]

J. W. Barrett and E. Süli, Reflections on Dubinskiĭs nonlinear compact embedding theorem, Publ. Inst. Math., 91 (2012), 95-110. doi: 10.2298/PIM1205095B.  Google Scholar

[3]

R. E. Caflisch and M. Sammartino, Existence and singularities for the Prandtl boundary layer equations, Z. Angew. Math. Mech., 80 (2000), 733-744. doi: 10.1002/1521-4001(200011)80:11/12<733::AID-ZAMM733>3.0.CO;2-L.  Google Scholar

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W. E, Boundary layer theory and the zero-viscosity limit of the Navier-Stokes equation, Acta Math. Sin. (Engl. Ser.), 16 (2000), 207-218. doi: 10.1007/s101140000034.  Google Scholar

[5]

C.-J. Liu, Y.-G. Wang and T. Yang, A well-posedness Theory for the Prandtl equations in three space variables, arXiv:1405.5308v2, 2014. Google Scholar

[6]

C.-J. Liu, Y.-G. Wang and T. Yang, On the ill-posedness of the Prandtl equations in three space dimensions, Arch. Rational Mech. Anal., 220 (2016), 83-108. doi: 10.1007/s00205-015-0927-1.  Google Scholar

[7]

N. Masmoudi and T. K. Wong, Local-in-time existence and uniqueness of solutions to the Prandtl equations by energy methods, Comm. Pure Appl. Math., 68 (2015), 1683-1741. doi: 10.1002/cpa.21595.  Google Scholar

[8]

F. K. Moore, Three-dimensional boundary layer theory, Adv. Appl. Mech., 4 (1956), 159-228. doi: 10.1016/S0065-2156(08)70373-9.  Google Scholar

[9]

O. A. Oleinik, On the properties of solutions of some elliptic boundary value problems, Matem. Sb., 30 (1952), 695-702.  Google Scholar

[10]

O. A. Oleinik and V. N. Samokhin, Mathematical Models in Boundary Layer Theory, Chapman and Hall/CRC, 1999.  Google Scholar

[11]

L. Prandtl, Über flüssigkeitsbewegungen bei sehr kleiner Reibung, in Verh. Int. Math. Kongr., Heidelberg, Germany 1904, Teubner, Germany 1905, 484-494. Google Scholar

[12]

M. Sammartino and R. E. Caflisch, Zero viscosity limit for analytic solutions of the Navier-Stokes equations on a half-space, I. Existence for Euler and Prandtl equations, Comm. Math. Phys., 192 (1998), 433-461; II. Construction of the Navier-Stokes solution, Comm. Math. Phys., 192 (1998), 463-491. doi: 10.1007/s002200050305.  Google Scholar

[13]

Z. P. Xin, Viscous boundary layers and their stability (I), J. Partial Differential Equations, 11 (1998), 97-124.  Google Scholar

[14]

Z. P. Xin and L. Zhang, On the global existence of solutions to the Prandtl's system, Adv. in Math., 181 (2004), 88-133. doi: 10.1016/S0001-8708(03)00046-X.  Google Scholar

show all references

References:
[1]

R. Alexandre, Y.-G. Wang, C.-J. Xu and T. Yang, Well-posedness of the Prandtl equation in Sobolev spaces, J. Amer. Math. Soc., 28 (2015), 745-784. doi: 10.1090/S0894-0347-2014-00813-4.  Google Scholar

[2]

J. W. Barrett and E. Süli, Reflections on Dubinskiĭs nonlinear compact embedding theorem, Publ. Inst. Math., 91 (2012), 95-110. doi: 10.2298/PIM1205095B.  Google Scholar

[3]

R. E. Caflisch and M. Sammartino, Existence and singularities for the Prandtl boundary layer equations, Z. Angew. Math. Mech., 80 (2000), 733-744. doi: 10.1002/1521-4001(200011)80:11/12<733::AID-ZAMM733>3.0.CO;2-L.  Google Scholar

[4]

W. E, Boundary layer theory and the zero-viscosity limit of the Navier-Stokes equation, Acta Math. Sin. (Engl. Ser.), 16 (2000), 207-218. doi: 10.1007/s101140000034.  Google Scholar

[5]

C.-J. Liu, Y.-G. Wang and T. Yang, A well-posedness Theory for the Prandtl equations in three space variables, arXiv:1405.5308v2, 2014. Google Scholar

[6]

C.-J. Liu, Y.-G. Wang and T. Yang, On the ill-posedness of the Prandtl equations in three space dimensions, Arch. Rational Mech. Anal., 220 (2016), 83-108. doi: 10.1007/s00205-015-0927-1.  Google Scholar

[7]

N. Masmoudi and T. K. Wong, Local-in-time existence and uniqueness of solutions to the Prandtl equations by energy methods, Comm. Pure Appl. Math., 68 (2015), 1683-1741. doi: 10.1002/cpa.21595.  Google Scholar

[8]

F. K. Moore, Three-dimensional boundary layer theory, Adv. Appl. Mech., 4 (1956), 159-228. doi: 10.1016/S0065-2156(08)70373-9.  Google Scholar

[9]

O. A. Oleinik, On the properties of solutions of some elliptic boundary value problems, Matem. Sb., 30 (1952), 695-702.  Google Scholar

[10]

O. A. Oleinik and V. N. Samokhin, Mathematical Models in Boundary Layer Theory, Chapman and Hall/CRC, 1999.  Google Scholar

[11]

L. Prandtl, Über flüssigkeitsbewegungen bei sehr kleiner Reibung, in Verh. Int. Math. Kongr., Heidelberg, Germany 1904, Teubner, Germany 1905, 484-494. Google Scholar

[12]

M. Sammartino and R. E. Caflisch, Zero viscosity limit for analytic solutions of the Navier-Stokes equations on a half-space, I. Existence for Euler and Prandtl equations, Comm. Math. Phys., 192 (1998), 433-461; II. Construction of the Navier-Stokes solution, Comm. Math. Phys., 192 (1998), 463-491. doi: 10.1007/s002200050305.  Google Scholar

[13]

Z. P. Xin, Viscous boundary layers and their stability (I), J. Partial Differential Equations, 11 (1998), 97-124.  Google Scholar

[14]

Z. P. Xin and L. Zhang, On the global existence of solutions to the Prandtl's system, Adv. in Math., 181 (2004), 88-133. doi: 10.1016/S0001-8708(03)00046-X.  Google Scholar

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