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I have recently been interested in the incompressible Euler equation, but since I am new to the topic, I would like to inquire what are the standard sources/references (for self-study) regarding the existing theory. For instance, are there some good survey papers which also include the classical theory for weak solutions?

In particular, I am interested in a couple of specific questions, mostly in 2D for starters. So, consider the incompressible Euler equation $$ \begin{cases} \partial_tu + u \cdot \nabla u+\nabla p &=\ \ \ 0, \\ \qquad \qquad \ \ \ \nabla \cdot u &= \ \ \ 0. \end{cases} $$ In two dimensions the counterexamples by Scheffer and Schnirelman show the nonuniqueness of $L^2$ weak solutions. In my understanding this only means that the concept of such irregular weak solutions is not the correct one. Now my question is threefold:

  1. What is known regarding the well-posedness of the problem in 2D with the "correct" definition of solutions (whatever it may be)? Are there some classical existence theorems (e.g. for suitably defined weak solutions) in 3D where the uniqueness/regularity is open?

  2. If one defines the weak solutions e.g. in $C(0;T; L^2(\Omega)) \cap L^2(0;T; W^{1,2}(\Omega))$ instead of merely $L^2$ can one hope for better behavior (e.g. uniqueness, regularity etc) than for merely $L^2$ weak solutions? Is something known about existence of weak solutions in such spaces (also for higher dimensions than 2D)?

  3. For Navier-Stokes equations in two dimensions the Serrin result (as later improved by Struwe) shows e.g. that if $u \in L^4(\Omega \times (0, T))$, then $u$ is regular. Is there something similar for the Euler equation? That is, is there a "Serrin result" for the Euler equation? In particular, if the weak solution for the Euler equation is bounded, does it imply further regularity as in the case of Navier-Stokes equations?

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    $\begingroup$ a starting point might be: Majda-Bertozzi: Vorticity and incompressible flow $\endgroup$ – marcoromito Apr 9 '14 at 12:28
  • $\begingroup$ Thanks. I have to see whether I can find the book somewhere. I was also able to find some survey papers by googling, but they did not quite answer all of my questions. $\endgroup$ – Juhana Siljander Apr 11 '14 at 14:05
  • $\begingroup$ Actually it seems that the book is available online. I quickly browsed the book and it seems that most of the existence and uniqueness results are global in the sense that one assumes something from the initial velocity/vorticity. So I am wondering are there any purely local regularity results or is this completely hopeless? $\endgroup$ – Juhana Siljander Apr 11 '14 at 14:22
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    $\begingroup$ Something on the line of a ``Serrin result'' is the Beale-Kato-Majda criterion, that holds for Navier-Stokes as well as Euler (not sure if it is on the book, but look for the paper). For local regularity of Euler, well, I do not know. Euler, especially in 3d might be a very bad guy (look at the examples of Scheffer, Shnirelman, and the recent works of De Lellis etc.) $\endgroup$ – marcoromito Apr 12 '14 at 13:55
  • $\begingroup$ So if I understood correctly, the Beale-Kato-Majda criterion gives regularity under appropriate assumptions on the initial vorticity $\omega_0$ and if $\omega$ is $L^1$ in time and $L^\infty$ in space. But unlike the Serrin result, this is not purely local in the sense that the initial data plays a role in the result. Concerning weak solutions the best result I could find was the one by De Lellis and Székelyhidi that local boundedness does not imply uniqueness. So altogether bounded velocity is not enough, but bounded vorticity is (at least in some sense). $\endgroup$ – Juhana Siljander Apr 13 '14 at 12:56
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Classical existence and regularity results can be found in the survey http://arxiv.org/abs/math/0703406 (Euler Equations of Incompressible Ideal Fluids, by C. Bardos and E. S. Titi -- http://www.mccme.ru/~ansobol/otarie/slides/Russ-Math-Surveys-Euler-Bardos.pdf is the published version).

P.S. This paper http://arxiv.org/abs/1402.4957 (Cauchy's almost forgotten Lagrangian formulation of the Euler equation for 3D incompressible flow, by Uriel Frisch and Barbara Villone) gives some very interesting historical background about 3D Euler equation.

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