Electronic Journal of Differential Equations,
Vol. 2007(2007), No. 79, pp. 1-8.

Title: Degeneracy in the Blasius problem

Author: Faiz Ahmad (King Abdulaziz Univ., Jeddah, Saudi Arabia)

Abstract:
 The Navier-Stokes equations for the boundary layer are  transformed, 
 by a similarity transformation, into the ordinary Blasius  differential 
 equation which, together with appropriate boundary conditions  
 constitutes the Blasius problem,  
 $$
 f'''(\eta )+\frac{1}{2}f(\eta )f''(\eta)=0,\quad 
 f(0)=0,\;  f'(0)=0,\; f'(\infty )=1.  
 $$
 The well-posedness of the Navier-Stokes equations is an open problem.  We
 solve this problem, in the case of constant flow in a boundary layer,  by
 showing that the Blasius problem is ill-posed.  If the second condition is
 replaced by $f'(0)=-\lambda $,  then degeneracy occurs for 
 $0<\lambda <\lambda _{c}\simeq 0.354$.  We investigate the problem
 analytically to explain this phenomenon.  We derive a simple equation 
 $g(\alpha ,\lambda )=0$, whose roots,  for a fixed $\lambda $, determine the
 solutions of the problem.  It is found that the equation has exactly two
 roots for  $0<\lambda <\lambda _{c}$ and no root beyond this point.  Since
 an arbitrarily small perturbation of the boundary condition  gives rise to
 an additional solution, which can be markedly different from  the
 unperturbed solution, the Blasius problem is ill-posed.
  
Submitted June 27, 2006. Published May 25, 2007.
Math Subject Classifications: 34A12, 34A34, 47J06.
Key Words: Navier-Stokes equations; Blasius problem; degeneracy; 
           Wang equation; well-posed problem; ill-posed problem.