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:
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?
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)?
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?
