Well-posedness of the Prandtl equation in Sobolev spaces

Alexandre, R.; Wang, Y.-G.; Xu, C.-J.; Yang, T.

doi:10.1090/S0894-0347-2014-00813-4

Well-posedness of the Prandtl equation in Sobolev spaces
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by R. Alexandre, Y.-G. Wang, C.-J. Xu and T. Yang

J. Amer. Math. Soc. 28 (2015), 745-784

DOI: https://doi.org/10.1090/S0894-0347-2014-00813-4

Published electronically: June 6, 2014

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Abstract:

We develop a new approach to study the well-posedness theory of the Prandtl equation in Sobolev spaces by using a direct energy method under a monotonicity condition on the tangential velocity field instead of using the Crocco transformation. Precisely, we firstly investigate the linearized Prandtl equation in some weighted Sobolev spaces when the tangential velocity of the background state is monotonic in the normal variable. Then to cope with the loss of regularity of the perturbation with respect to the background state due to the degeneracy of the equation, we apply the Nash-Moser-Hörmander iteration to obtain a well-posedness theory of classical solutions to the nonlinear Prandtl equation when the initial data is a small perturbation of a monotonic shear flow.

References

Serge Alinhac and Patrick Gérard, Pseudo-differential operators and the Nash-Moser theorem, Graduate Studies in Mathematics, vol. 82, American Mathematical Society, Providence, RI, 2007. Translated from the 1991 French original by Stephen S. Wilson. MR 2304160, DOI 10.1090/gsm/082
R. E. Caflisch and M. Sammartino, Existence and singularities for the Prandtl boundary layer equations, ZAMM Z. Angew. Math. Mech. 80 (2000), no. 11-12, 733–744. Special issue on the occasion of the 125th anniversary of the birth of Ludwig Prandtl. MR 1801538, DOI 10.1002/1521-4001(200011)80:11/12<733::AID-ZAMM733>3.0.CO;2-L
Jacques Chazarain and Alain Piriou, Introduction to the theory of linear partial differential equations, Studies in Mathematics and its Applications, vol. 14, North-Holland Publishing Co., Amsterdam-New York, 1982. Translated from the French. MR 678605
Weinan E, Boundary layer theory and the zero-viscosity limit of the Navier-Stokes equation, Acta Math. Sin. (Engl. Ser.) 16 (2000), no. 2, 207–218. MR 1778702, DOI 10.1007/s101140000034
Weinan E and Bjorn Engquist, Blowup of solutions of the unsteady Prandtl’s equation, Comm. Pure Appl. Math. 50 (1997), no. 12, 1287–1293. MR 1476316, DOI 10.1002/(SICI)1097-0312(199712)50:12<1287::AID-CPA4>3.0.CO;2-4
David Gérard-Varet and Emmanuel Dormy, On the ill-posedness of the Prandtl equation, J. Amer. Math. Soc. 23 (2010), no. 2, 591–609. MR 2601044, DOI 10.1090/S0894-0347-09-00652-3
D. Gérard-Varet and T. Nguyen, Remarks on the ill-posedness of the Prandtl equation, Asymptot. Anal. 77 (2012), no. 1-2, 71–88. MR 2952715, DOI 10.3233/ASY-2011-1075
Emmanuel Grenier, On the nonlinear instability of Euler and Prandtl equations, Comm. Pure Appl. Math. 53 (2000), no. 9, 1067–1091. MR 1761409, DOI 10.1002/1097-0312(200009)53:9<1067::AID-CPA1>3.3.CO;2-H
Yan Guo and Toan Nguyen, A note on Prandtl boundary layers, Comm. Pure Appl. Math. 64 (2011), no. 10, 1416–1438. MR 2849481, DOI 10.1002/cpa.20377
Richard S. Hamilton, The inverse function theorem of Nash and Moser, Bull. Amer. Math. Soc. (N.S.) 7 (1982), no. 1, 65–222. MR 656198, DOI 10.1090/S0273-0979-1982-15004-2
Lan Hong and John K. Hunter, Singularity formation and instability in the unsteady inviscid and viscous Prandtl equations, Commun. Math. Sci. 1 (2003), no. 2, 293–316. MR 1980477, DOI 10.4310/CMS.2003.v1.n2.a5
Lars Hörmander, The analysis of linear partial differential operators, Vol. I–IV, Springer Verlag, Berlin, 1985.
Lars Hörmander, The boundary problems of physical geodesy, Arch. Rational Mech. Anal. 62 (1976), no. 1, 1–52. MR 602181, DOI 10.1007/BF00251855
Maria Carmela Lombardo, Marco Cannone, and Marco Sammartino, Well-posedness of the boundary layer equations, SIAM J. Math. Anal. 35 (2003), no. 4, 987–1004. MR 2049030, DOI 10.1137/S0036141002412057
Guy Métivier, Small viscosity and boundary layer methods, Modeling and Simulation in Science, Engineering and Technology, Birkhäuser Boston, Inc., Boston, MA, 2004. Theory, stability analysis, and applications. MR 2151414, DOI 10.1007/978-0-8176-8214-9
Jürgen Moser, A new technique for the construction of solutions of nonlinear differential equations, Proc. Nat. Acad. Sci. U.S.A. 47 (1961), 1824–1831. MR 132859, DOI 10.1073/pnas.47.11.1824
John Nash, The imbedding problem for Riemannian manifolds, Ann. of Math. (2) 63 (1956), 20–63. MR 75639, DOI 10.2307/1969989
Olga A. Oleinik, The Prandtl system of equations in boundary layer theory, Soviet Math Dokl. 4 (1963), 583–586.
O. A. Oleinik and V. N. Samokhin, Mathematical models in boundary layer theory, Applied Mathematics and Mathematical Computation, vol. 15, Chapman & Hall/CRC, Boca Raton, FL, 1999. MR 1697762
Ludwig Prandtl, Über Flüssigkeitsbewegungen bei sehr kleiner Reibung. In “Verh. Int. Math. Kongr., Heidelberg 1904,”, Teubner, 1905.
Marco Sammartino and Russel E. Caflisch, Zero viscosity limit for analytic solutions of the Navier-Stokes equation on a half-space. II. Construction of the Navier-Stokes solution, Comm. Math. Phys. 192 (1998), no. 2, 463–491. MR 1617538, DOI 10.1007/s002200050305
Zhouping Xin and Liqun Zhang, On the global existence of solutions to the Prandtl’s system, Adv. Math. 181 (2004), no. 1, 88–133. MR 2020656, DOI 10.1016/S0001-8708(03)00046-X