It is well known that the 2d incompressible Navier-Stokes equations under periodic boundary conditions always have global smooth solutions, given smooth initial conditions.
I tried searching for a proof that the 2d incompressible Euler equations under periodic boundary conditions always have global smooth solutions, given smooth initial conditions, but couldn't find one. Is this still an open problem?
A famous result of Beale, Kato and Majda shows that the maximum norm of the vorticity has to become infinite if smooth solutions do not exist globally. This is true even in three dimensions. In two dimensions, the $L^\infty$ norm of the vorticity is conserved, so global existence follows immediately.
I was a bit loathe of writing an answer since I am not a PDE guy and in fact have only a somewhat sketchy grasp of the kind of boundaries people in this business are interested in. However, since nobody came forward, I can supply at least a partial answer. Bear with me for a moment through the geometry setup I am outlining below:
Lets assume that your boundary is nice (e.g. piecewise smooth). For arguments sake I will assume that the domain we are working on is a square $S$ in $\mathbb{R}^2$ so that we have a concrete example. Then by the usual construction we identify the square with a torus $\mathbb{T}$ which is a compact (smooth) manifold without boundary. The identification translates the Riemannian metric on $S$ (i.e. the flat euclidean metric) to a Riemannian metric on $\mathbb{T}$ (where we get the flat Riemannian metric on the torus). The point here is that the incompressible 2D Euler equation under periodic boundary conditions gets translated by this technique to the incompressible Euler equation on the torus $\mathbb{T}$. Important note here: The Euler equation depends on metric data, in the vector space case this is the Euclidean metric. However, since we transported the metric to obtain a Riemannian metric on $\mathbb{T}$ we have now a preferred choice of Riemannian metric (which is necessary to write an Euler equation on a compact manifold). The Euler equation with respect to this Riemannian metric is now exactly the PDE you obtain from your original problem by subjecting the original PDE to the identification. Since we can go back and forth between the two perspectives, clearly we only have to prove that the incompressible Euler equation on a compact manifold depends smoothly on the initial data.
Let's stop here for a moment. You asked for a proof for periodic boundary incompressible Euler having smooth solutions if the initial data is smooth. I have now substituted your problem by a manifold valued one. This seems to be a horrible idea! Well, yes and no. Yes, if you hoped for elementary PDE methods to prove the dependence (since I am no expert in this I can not help with that...). However, here comes the no part: If you are happy to use (infinite-dimensional) differential geometry, we can now cite a seminal paper in the area and are done (I know PDE people do not like to do this, but this is the best I got, sorry!)
So after the above disclaimer, we have to see that incompressible Euler on a compact manifold has the desired property. This is well known an da classical result due to:
D.G. Ebin and J. Marsden: Groups of Diffeomorphisms and the Motion of an Incompressible Fluid, Annals of Mathematics Second Series, Vol. 92, No. 1 (1970), pp. 102-163 (free version here: 1
Before I give you the complete reference let me explain where the infinite-dimensional comes from: You start out with the Euler equation on your finite dimensional manifold and then due to a trick discovered by Arnold you can rewrite it as an ODE (!!!) but on an infinite-dimensional manifold (here the manifold of Sobolev class diffeomorphisms). The Ebin-Marsden article then proves via this method first local well-posedness for the Euler equation. Then the kind of parameter-dependence you are after is established in Theorem 15.2. Citing Statement (iii) (Regularity of solutions): If $u_0$ (the initial condition, i.e. a vector field) is $C^\infty$, then the solution $u_t$ of the Euler problem is $C^\infty$.
Now we got what you asked for, albeit with heavier method then you probably expected. Note however, that results for incompressible Euler (local well posedness, and parameter dependence) given by Ebin-Marsden's paper appear, historically speaking, for the first time in this generality in the cited paper. I hear that PDE people can do it now without infinite-dimensional methods (A very irate PDE professor claimed this when he shouted me down while I was giving a talk on a related topic while I was still a PhD student) however, I do not know them. So I hope that you found something useful in my answer. One last note: Beyond the results mentioned, 1 contains also a wealth of other information on Euler and how solutions depend on initial conditions etc. (though the paper tends to be quite heavy if you do not speak differential geometry)
