×

Global existence of weak solutions to the three-dimensional Prandtl equations with a special structure. (English) Zbl 1353.35215

Summary: 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 Z. Xin and L. Zhang [Adv. Math. 181, No. 1, 88–133 (2004; Zbl 1052.35135)] on the two-dimensional Prandtl equations to the three-dimensional setting.

MSC:

35M13 Initial-boundary value problems for PDEs of mixed type
35Q35 PDEs in connection with fluid mechanics
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
76D10 Boundary-layer theory, separation and reattachment, higher-order effects
76N20 Boundary-layer theory for compressible fluids and gas dynamics

Citations:

Zbl 1052.35135
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] R. Alexandre, Well-posedness of the Prandtl equation in Sobolev spaces,, J. Amer. Math. Soc., 28, 745 (2015) · Zbl 1317.35186
[2] J. W. Barrett, Reflections on Dubinskiĭs nonlinear compact embedding theorem,, Publ. Inst. Math., 91, 95 (2012) · Zbl 1265.46037
[3] R. E. Caflisch, Existence and singularities for the Prandtl boundary layer equations,, Z. Angew. Math. Mech., 80, 733 (2000) · Zbl 0951.76582
[4] W. E., Boundary layer theory and the zero-viscosity limit of the Navier-Stokes equation,, Acta Math. Sin. (Engl. Ser.), 16, 207 (2000) · Zbl 0961.35101
[5] C.-J. Liu, A well-posedness Theory for the Prandtl equations in three space variables,, <a href= (2014)
[6] C.-J. Liu, On the ill-posedness of the Prandtl equations in three space dimensions,, Arch. Rational Mech. Anal., 220, 83 (2016) · Zbl 1341.35120
[7] N. Masmoudi, Local-in-time existence and uniqueness of solutions to the Prandtl equations by energy methods,, Comm. Pure Appl. Math., 68, 1683 (2015) · Zbl 1326.35279
[8] F. K. Moore, Three-dimensional boundary layer theory,, Adv. Appl. Mech., 4, 159 (1956)
[9] O. A. Oleinik, On the properties of solutions of some elliptic boundary value problems,, Matem. Sb., 30, 695 (1952)
[10] O. A. Oleinik, <em>Mathematical Models in Boundary Layer Theory</em>,, Chapman and Hall/CRC (1999) · Zbl 0928.76002
[11] L. Prandtl, Über flüssigkeitsbewegungen bei sehr kleiner Reibung,, in Verh. Int. Math. Kongr., 484 (1904)
[12] M. Sammartino, 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, 433 (1998) · Zbl 0913.35103
[13] Z. P. Xin, Viscous boundary layers and their stability (I),, J. Partial Differential Equations, 11, 97 (1998) · Zbl 0906.35057
[14] Z. P. Xin, On the global existence of solutions to the Prandtl’s system,, Adv. in Math., 181, 88 (2004) · Zbl 1052.35135
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.