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I am interested in studying fluid dynamics and am searching for a good introductory textbook. I know just the very basics of fluids on the physics side. For mathematical prerequisites, I have completed a course on integration theory, and have a basic understanding of functional analysis and PDE, though by no means am I an expert.

I have done some searching online and most of the books I have come across lean more towards physics, numerical methods, and/or mathematical modeling. What I am instead looking for is something that is more mathematical and analysis/PDE oriented (and ideally one that does not assume too much of this material and introduces it as necessary), targeted towards early graduate students in mathematics. A good example of what I would like are Professor Tao's lecture notes.

Are there any textbooks in this direction?

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  • $\begingroup$ The 2d-case has already many established mathematical results and often serve as a toy/test case. A nice book is amazon.com/… "Mathematics of Two-Dimensional Turbulence" by S.Kuksin and A.Shirikyan $\endgroup$
    – Thomas Kojar
    Commented May 5, 2023 at 19:42

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A few possibilities:

  • JC Robinson, JL Rodrigo, & W Sadowski (2016) Classical theory of the three-dimensional Navier-Stokes equations.

  • OA Ladyzhenskaya (1963) The mathematical theory of viscous incompressible flow.

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  • $\begingroup$ That first book seems to be exactly what I need, thank you! If you have happened to have read through it before, how is it? Is it well suited for self-study and with only a minimal background in PDE? $\endgroup$
    – CBBAM
    Commented May 3, 2023 at 1:27
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    $\begingroup$ I haven't read all of it but the bits I read it seemed well suited to self-study and didn't require too much knowledge of PDE theory. $\endgroup$
    – Hollis Williams
    Commented May 3, 2023 at 5:06
  • $\begingroup$ Thank you, I'll give it a try then. $\endgroup$
    – CBBAM
    Commented May 3, 2023 at 5:47
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    $\begingroup$ No problem, would you mind accepting my answer if you believe this answers your question? $\endgroup$
    – Hollis Williams
    Commented May 3, 2023 at 7:16
  • $\begingroup$ Of course, sorry about not doing so earlier. $\endgroup$
    – CBBAM
    Commented May 3, 2023 at 8:17
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An older, classic text is Mathematical Theory of Compressible Fluid Flow by Richard von Mises.

More recent text books include

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  • $\begingroup$ Thank you for your suggestions. These all seem like wonderful books, but I am looking for something more on the PDE side. For example, existence, uniqueness, use of Sobolev spaces, weak solutions...etc. $\endgroup$
    – CBBAM
    Commented May 2, 2023 at 7:07
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Constantin, Foias, Navier-Stokes Equations (1988).

This classic is not a textbook exactly, but gives a lot of detail and should be pretty readable.

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You can get the free ~900 pages book

or this one on compressible flow:

Good luck

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