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I am currently working on a PhD project (1st year) involving the Euler equations. I would like to have a strong background on this, especially from the mathematical theory points of view. (since I found that many books written in this topic focus much on the numerical points of view). Also, I found some very good books on the Navier-Stoke equation, but is there any specific literature focusing only on Euler equation? Briefly, I need a book that:

  • explain the derivation of Euler equation. (from physical points of view)
  • explain the roles of each terms in the equation.
  • explain the mathematical theory developed on it. (analyze the existence and uniqueness of solution, well-posed, convergence of approximation solution (by numerical methods), limitation of today's knowledge on solving it, ...)
  • some parts about numerical experiments (with analysis and explanation). Thank you for all of your recommendation.
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  • $\begingroup$ see also herehttp://www.mccme.ru/~ansobol/otarie/slides/Russ-Math-Surveys-Euler-Bardos.pdf $\endgroup$ – Dr. Sonnhard Graubner Nov 12 '16 at 17:46
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Dont altogether dismiss the numerical books. Modern numerical methods for conservation laws make extensive use of the theory (even if only of the one-dimensional theory, which is in fact most of what there is) but many of the numerical books are a bit superficial (jumping on a bandwagon) My recommendations would be

  1. Godlewski, E., & Raviart, P. A. (2013). Numerical approximation of hyperbolic systems of conservation laws (Vol. 118). Springer Science & Business Media.

  2. Smoller, Joel. Shock waves and reaction—diffusion equations. Vol. 258. Springer Science & Business Media, 2012.

  3. Courant, R., & Friedrichs, K. O. (1999). Supersonic flow and shock waves (Vol. 21). Springer Science & Business Media.(Reprint of a 1948 classic, but you have to know this stuff)

And no, Im not getting any kickback from Springer. Generally, look for books about conservation laws in general, not specifically about the Euler equations.

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