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 and 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.

[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.

[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.

[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.

[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.

[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.

[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.

[8]

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

[9]

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

[10]

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

[11]

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

[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.

[13]

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

[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.

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.

[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.

[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.

[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.

[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.

[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.

[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.

[8]

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

[9]

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

[10]

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

[11]

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

[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.

[13]

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

[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.

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