Well-posedness of the Prandtl equation in Sobolev spaces. (English) Zbl 1317.35186
Summary: 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.
MSC:
| 35Q35 | PDEs in connection with fluid mechanics |
| 35M13 | Initial-boundary value problems for PDEs of mixed type |
| 76D10 | Boundary-layer theory, separation and reattachment, higher-order effects |
| 76D03 | Existence, uniqueness, and regularity theory for incompressible viscous fluids |
| 76N20 | Boundary-layer theory for compressible fluids and gas dynamics |
Keywords:
Prandtl equation; well-posedness theory; Sobolev spaces; energy method; monotonic velocity field; Nash-Moser-Hörmander iterationReferences:
| [1] | Alinhac, Serge; G{\'e}rard, Patrick, Pseudo-differential operators and the Nash-Moser theorem, Graduate Studies in Mathematics 82, viii+168 pp. (2007), American Mathematical Society, Providence, RI · Zbl 1121.47033 |
| [2] | Caflisch, Russel. E.; Sammartino, Marco, Existence and singularities for the Prandtl boundary layer equations, Z. Angew. Math. Mech., 80, 11-12, 733\textendash 744 pp. (2000) · Zbl 1050.76016 · doi:10.1002/1521-4001(200011)80:\(11/12\langle |
| [3] | Chazarain, Jacques; Piriou, Alain, Introduction to the theory of linear partial differential equations, Studies in Mathematics and its Applications 14, xiv+559 pp. (1982), North-Holland Publishing Co., Amsterdam\textendash New York · Zbl 0487.35002 |
| [4] | E, Weinan, Boundary layer theory and the zero-viscosity limit of the Navier-Stokes equation, Acta Math. Sin. (Engl. Ser.), 16, 2, 207\textendash 218 pp. (2000) · Zbl 0961.35101 · doi:10.1007/s101140000034 |
| [5] | E, Weinan; Engquist, Bjorn, Blowup of solutions of the unsteady Prandtl’s equation, Comm. Pure Appl. Math., 50, 12, 1287\textendash 1293 pp. (1997) · Zbl 0908.35099 · doi:10.1002/(SICI)1097-0312(199712)50:\(12\langle |
| [6] | G{\'e}rard-Varet, David; Dormy, Emmanuel, On the ill-posedness of the Prandtl equation, J. Amer. Math. Soc., 23, 2, 591\textendash 609 pp. (2010) · Zbl 1197.35204 · doi:10.1090/S0894-0347-09-00652-3 |
| [7] | G{\'e}rard-Varet, David; Nguyen, Toan Trong, Remarks on the ill-posedness of the Prandtl equation, Asymptot. Anal., 77, 1-2, 71\textendash 88 pp. (2012) · Zbl 1238.35178 |
| [8] | Grenier, Emmanuel, On the nonlinear instability of Euler and Prandtl equations, Comm. Pure Appl. Math., 53, 9, 1067\textendash 1091 pp. (2000) · Zbl 1048.35081 · doi:10.1002/1097-0312(200009)53:\(9\langle |
| [9] | Guo, Yan; Nguyen, Toan, A note on Prandtl boundary layers, Comm. Pure Appl. Math., 64, 10, 1416\textendash 1438 pp. (2011) · Zbl 1232.35126 · doi:10.1002/cpa.20377 |
| [10] | Hamilton, Richard S., The inverse function theorem of Nash and Moser, Bull. Amer. Math. Soc. (N.S.), 7, 1, 65\textendash 222 pp. (1982) · Zbl 0499.58003 · doi:10.1090/S0273-0979-1982-15004-2 |
| [11] | Hong, Lan; Hunter, John K., Singularity formation and instability in the unsteady inviscid and viscous Prandtl equations, Commun. Math. Sci., 1, 2, 293\textendash 316 pp. (2003) · Zbl 1084.76020 |
| [12] | H\"{o}rmander, Lars, \it The analysis of linear partial differential operators, I\textendash IV (1985), Springer Verlag, Berlin · Zbl 0601.35001 |
| [13] | H{\"o}rmander, Lars, The boundary problems of physical geodesy, Arch. Ration. Mech. Anal., 62, 1, 1\textendash 52 pp. (1976) · Zbl 0331.35020 |
| [14] | Lombardo, Maria Carmela; Cannone, Marco; Sammartino, Marco, Well-posedness of the boundary layer equations, SIAM J. Math. Anal., 35, 4, 987\textendash 1004 (electronic) pp. (2003) · Zbl 1053.76013 · doi:10.1137/S0036141002412057 |
| [15] | M{\'e}tivier, Guy, Small viscosity and boundary layer methods, Modeling and Simulation in Science, Engineering and Technology, xxii+194 pp. (2004), Birkh\"auser Boston, Inc., Boston, MA · Zbl 1133.35001 · doi:10.1007/978-0-8176-8214-9 |
| [16] | Moser, J{\"u}rgen, A new technique for the construction of solutions of nonlinear differential equations, Proc. Natl. Acad. Sci. USA, 47, 1824\textendash 1831 pp. (1961) · Zbl 0104.30503 |
| [17] | Nash, John, The imbedding problem for Riemannian manifolds, Ann. of Math. (2), 63, 20\textendash 63 pp. (1956) · Zbl 0070.38603 |
| [18] | Oleinik, Olga A., The Prandtl system of equations in boundary layer theory, Soviet Math Dokl., 4, 583\textendash 586 pp. (1963) · Zbl 0134.45004 |
| [19] | Oleinik, Olga A.; Samokhin, VN, Mathematical models in boundary layer theory, Applied Mathematics and Mathematical Computation 15, x+516 pp. (1999), Chapman & Hall/CRC, Boca Raton, FL · Zbl 0928.76002 |
| [20] | Prandtl, Ludwig, \`“{U}ber Fl\'”ussigkeitsbewegungen bei sehr kleiner Reibung. In “{Verh. Int. Math. Kongr., Heidelberg 1904,”}, 484\textendash 494 pp. (1905), Teubner |
| [21] | Sammartino, Marco; Caflisch, Russel E., 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, 2, 463\textendash 491 pp. (1998) · Zbl 0913.35103 · doi:10.1007/s002200050305 |
| [22] | Xin, Zhouping; Zhang, Liqun, On the global existence of solutions to the Prandtl’s system, Adv. Math., 181, 1, 88\textendash 133 pp. (2004) · Zbl 1052.35135 · doi:10.1016/S0001-8708(03)00046-X |
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