a gentle introduction to well posedness issues in fluid dynamics ( reference request) I am looking for an introduction, either books, survey papers, to well posedness issues in fluid dynamic systems like the Navier Stokes, Euler equations or Stokes' equation and / or even for other model PDE's.
Thank you. 
 A: For Navier-Stokes/Stokes equation, I think two computation-oriented books are pretty good for introduction:


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*Finite element methods for Navier-Stokes equations: theory and algorithms by Girault and Raviart.

*Navier-Stokes Equations Theory and Numerical Analysis by Temam.
Both books have a long dedicated chapter introducing the theory about the function spaces involved ($H(\mathrm{div})$ and $H(\mathbf{curl})$), existence and uniqueness for the solution of the variational problem, the classical velocity-pressure formulation for the incompressible Stokesian flow. Luc Tartar's book has more discussion about the function spaces $H(\mathrm{div})$ and $H(\mathbf{curl})$, difficult material though.
Speaking of Euler equation, this is a totally different beast, and you are entering the realm of hyperbolic conservation laws. We are talking about the propagation of singularities (shockwave), entropy solution, Riemann problem, Cauchy problem, etc. The early researches in mathematical theory in hyperbolic conservation laws are done by R. Courant, P. Lax, and K. Friedrichs.


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*The bible is Supersonic flow and shock waves by Courant and Friedrichs.

*Landau's book Fluid Mechanics, chapter 1 is amazingly written too.
Both books are more physics-oriented, not too much about well-posedness or relevant mathematical theories, they discuss more about how the solutions match the physics, which is more likely to be covered in a fluid dynamics course in physics or engineering department. I am not an expert on Euler equation but I believe there are extensive research in well-posedness of incompressible Euler equations.
