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Well-posedness of the Prandtl equation in Sobolev spaces

Authors: R. Alexandre, Y.-G. Wang, C.-J. Xu and T. Yang
Journal: J. Amer. Math. Soc. 28 (2015), 745-784
MSC (2010): Primary 35M13, 35Q35, 76D10, 76D03, 76N20
Published electronically: June 6, 2014
MathSciNet review: 3327535
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Abstract | References | Similar Articles | Additional Information

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.

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  • [1] 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 (2007m:35001)
  • [2] Russel. E. Caflisch and Marco Sammartino, Existence and singularities for the Prandtl boundary layer equations, 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 (2002b:76047),$ \langle $733::AID-ZAMM733$ \rangle $3.0.CO;2-L
  • [3] Jacques Chazarain and Alain Piriou, Introduction to the theory of linear partial differential equations, Studies in Mathematics and its Applications, vol. 14, North-Holland Publishing Co., Amsterdam-New York, 1982. Translated from the French. MR 678605 (83j:35001)
  • [4] Weinan E, Boundary layer theory and the zero-viscosity limit of the Navier-Stokes equation, Acta Math. Sin. (Engl. Ser.) 16 (2000), no. 2, 207-218. MR 1778702 (2001d:76037),
  • [5] Weinan E and Bjorn Engquist, Blowup of solutions of the unsteady Prandtl's equation, Comm. Pure Appl. Math. 50 (1997), no. 12, 1287-1293. MR 1476316 (99c:35196),$ \langle $1287::AID-CPA4$ \rangle $3.0.CO;2-4
  • [6] David Gérard-Varet and Emmanuel Dormy, On the ill-posedness of the Prandtl equation, J. Amer. Math. Soc. 23 (2010), no. 2, 591-609. MR 2601044 (2011f:35259),
  • [7] David Gérard-Varet and Toan Trong Nguyen, Remarks on the ill-posedness of the Prandtl equation, Asymptot. Anal. 77 (2012), no. 1-2, 71-88. MR 2952715
  • [8] Emmanuel Grenier, On the nonlinear instability of Euler and Prandtl equations, Comm. Pure Appl. Math. 53 (2000), no. 9, 1067-1091. MR 1761409 (2001i:76056),$ \langle $1067::AID-CPA1$ \rangle $3.3.CO;2-H
  • [9] Yan Guo and Toan Nguyen, A note on Prandtl boundary layers, Comm. Pure Appl. Math. 64 (2011), no. 10, 1416-1438. MR 2849481 (2012m:35239),
  • [10] 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 (83j:58014),
  • [11] 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 (2005k:76035)
  • [12] Lars Hörmander, The analysis of linear partial differential operators, Vol. I-IV, Springer Verlag, Berlin, 1985.
  • [13] Lars Hörmander, The boundary problems of physical geodesy, Arch. Ration. Mech. Anal. 62 (1976), no. 1, 1-52. MR 0602181 (58 #29202a)
  • [14] 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 (electronic). MR 2049030 (2005a:76137),
  • [15] 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 (2007b:35004)
  • [16] Jürgen Moser, A new technique for the construction of solutions of nonlinear differential equations, Proc. Natl. Acad. Sci. USA 47 (1961), 1824-1831. MR 0132859 (24 #A2695)
  • [17] John Nash, The imbedding problem for Riemannian manifolds, Ann. of Math. (2) 63 (1956), 20-63. MR 0075639 (17,782b)
  • [18] Olga A. Oleinik, The Prandtl system of equations in boundary layer theory, Soviet Math Dokl. 4 (1963), 583-586.
  • [19] Olga A. Oleinik and VN Samokhin, Mathematical models in boundary layer theory, Applied Mathematics and Mathematical Computation, vol. 15, Chapman & Hall/CRC, Boca Raton, FL, 1999. MR 1697762 (2000c:76021)
  • [20] Ludwig Prandtl, Über Flüssigkeitsbewegungen bei sehr kleiner Reibung. In ``Verh. Int. Math. Kongr., Heidelberg 1904,'', Teubner, 1905.
  • [21] 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 (99d:35129b),
  • [22] 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 (2005f:35219),

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Additional Information

R. Alexandre
Affiliation: Department of Mathematics, Shanghai Jiao Tong University, Shanghai, 200240, P. R. China and Arts et Métiers ParisTech, Paris 75013, France

Y.-G. Wang
Affiliation: Department of Mathematics, and MOE-LSC, Shanghai Jiao Tong University, Shanghai, 200240, P. R. China

C.-J. Xu
Affiliation: School of Mathematics, Wuhan University 430072, Wuhan, P. R. China, and Université de Rouen, UMR 6085-CNRS, Mathématiques, Avenue de l’Université, BP.12, 76801 Saint Etienne du Rouvray, France

T. Yang
Affiliation: Department of Mathematics, City University of Hong Kong, Hong Kong, P. R. China

Keywords: Prandtl equation, well-posedness theory, Sobolev spaces, energy method, monotonic velocity field, Nash-Moser-H\"ormander iteration.
Received by editor(s): March 7, 2012
Received by editor(s) in revised form: April 10, 2014
Published electronically: June 6, 2014
Article copyright: © Copyright 2014 American Mathematical Society

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