Long time well-posdness of Prandtl system with small and analytic initial data

In this paper, we investigate the long time existence and uniqueness of small solution to $d,$ for $d=2,3,$ dimensional Prandtl system with small initial data which is analytic in the horizontal variables. In particular, we prove that $d$ dimensional Prandtl system has a unique solution with the life-span of which is greater than $\e^{-\f43}$ if both the initial data and the value on the boundary of the tangential velocity of the outflow are of size $\e.$ We mention that the tool developed in \cite{Ch04, CGP} to make the analytical type estimates and the special structure of the nonlinear terms to this system play an essential role in the proof of this result.

[1]  Zhouping Xin,et al.  On the global existence of solutions to the Prandtl's system , 2004 .

[2]  J. Bony,et al.  Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires , 1980 .

[3]  Radjesvarane Alexandre,et al.  Well-posedness of the Prandtl equation in Sobolev spaces , 2012, 1203.5991.

[4]  M. Paicu.  Équation anisotrope de Navier-Stokes dans des espaces critiques , 2005 .

[5]  Nader Masmoudi,et al.  Well-posedness for the Prandtl system without analyticity or monotonicity , 2013 .

[6]  Marco Cannone,et al.  Well-Posedness of the Boundary Layer Equations , 2003, SIAM J. Math. Anal..

[7]  Russel E. Caflisch,et al.  Zero Viscosity Limit for Analytic Solutions, of the Navier-Stokes Equation on a Half-Space.¶I. Existence for Euler and Prandtl Equations , 1998 .

[8]  Isabelle Gallagher,et al.  Global regularity for some classes of large solutions to the Navier-Stokes equations , 2008, 0807.1265.

[9]  Igor Kukavica,et al.  On the Local Well-posedness of the Prandtl and Hydrostatic Euler Equations with Multiple Monotonicity Regions , 2014, SIAM J. Math. Anal..

[10]  Yan Guo,et al.  A note on Prandtl boundary layers , 2010, 1011.0130.

[11]  Toàn Nguyên,et al.  Remarks on the ill-posedness of the Prandtl equation , 2009, Asymptot. Anal..

[12]  E Weinan,et al.  Boundary Layer Theory and the Zero-Viscosity Limit of the Navier-Stokes Equation , 2000 .

[13]  Zhifei Zhang,et al.  Global regularity for the Navier–Stokes equations with some classes of large initial data , 2011 .

[14]  Emmanuel Dormy,et al.  On the ill-posedness of the Prandtl equation , 2009, 0904.0434.

[15]  E Weinan,et al.  BLOWUP OF SOLUTIONS OF THE UNSTEADY PRANDTL'S EQUATION , 1997 .

[16]  Zhifei Zhang,et al.  Global well-posedness for 3D Navier–Stokes equations with ill-prepared initial data , 2013, Journal of the Institute of Mathematics of Jussieu.

[17]  Zero Viscosity Limit for Analytic Solutions of the Primitive Equations , 2016 .

[18]  W. Tollmien,et al.  Über Flüssigkeitsbewegung bei sehr kleiner Reibung , 1961 .

[19]  V. N. Samokhin,et al.  Mathematical Models in Boundary Layer Theory , 1999 .

[20]  R. Danchin,et al.  Fourier Analysis and Nonlinear Partial Differential Equations , 2011 .

[21]  Ping Zhang,et al.  Global Solutions to the 3-D Incompressible Anisotropic Navier-Stokes System in the Critical Spaces , 2011 .

[22]  Ping Zhang,et al.  On the Global Wellposedness to the 3-D Incompressible Anisotropic Navier-Stokes Equations , 2007 .