It is interesting to know the exact (analytical) solutions of the incompressible Navier-Stokes equations (2-D and 3-D). I am familiar with the solutions which are, e.g., in the articles: Kim&Moin (1985), Ethier et al. (1994), Sheu et al. (1996). May be someone knows and other exact solutions, not considering of course the well-known ones for a pipe (1-D)? What strict theorems are known about the existence and uniqueness for the NS equations (steady state and transient)?

Thank you.

for 2D and 3D one can use the complex space thery. For 2D one have ixi=-1 in 3D,the same,ixi=-1 and jxj=-1;this products veryfing the equation DDy=0 whit DD the laplace's operator.

Thank you, but could you write in more details. What means 'ixi'? And except Laplace operator in the Navier-Stokes there are also other terms: D(uv)/Dx, Dp/Dx, Du/Dt. What to do with them?

ixi means the product of the versor i for his same. The Navier-Stoke's equation in the incompressible flow reduce to Laplace's equation whit the ipotesys of the viscous terms are trascurable.This thinks divide our fisic space into two region:1 near our body where we use the N.S.'s equation, 2 far a body where we use the laplace's equation.

Into the space complex we have for a point 2D P=x+i*y 3D P=x+i*y+j*z for a function 2D f=f1+i*f2 3D f=f1+i*f2+j*f3 The trigonometric rappresentation (you must use this because you have pair terms)give us 2D P=r*(cos t +i*sin t) 3d P=r*(cos t + i*sin t )* (cos f +j * sin f) I think the product i*j is a new axis. If you calculate the derivative (you can to do egual the derivative along the axis increment) you have flxl=fkxk and flxk=-fkxl, whit fl a l component of f,flxl a partial derivative of fl rispect to xl axys (remember if i*i=-1 you have i=-1/i).When you have obtined this you can veryfi DDf=0 for all f into the space complex.

Tat iz weri goot anser.