Difference between Galerkin approximations and Yosida approximation in Open Problem Navier Stokes Equations To approximate a solutions of Navier-Stokes equation, we use sometime Galerkin approximation and some time we use Yosida approximations. Why we use two different approximations for approximate the Navier - Stokes equation?
In fact there a several many approximation in the literature but we are only concentrate about those two approximation.
My Idea : I think to get more regular solution we use yosida approximations. i.e what we solution get from Yosida approximation is more regular than solutions get from Galerkin approximation.
Can anyone figure out this problem?
Thank You.
 A: As you mentioned, there are many approximations used to approximate a PDE by ODE in order to use the classical theory of ordinary differential equations like  Picard Lindelof. But what makes one appropriate than other is the problematic here. In general Galerkine approximation scheme is a discretisation method. It is compatible and appropriate when your domain is periodic such as three dimensional torus $\mathbb{T}^3$ in this case the Stokes operator coincides with Laplacian $\Delta$ and the  eigenvectors of Stokes operator are used to truncate up to the $n^{th}$ eigenvector to pass from an infinite dimensional space Hilbert spaces to a finite dimensional space hence a finite number of equations. Galerkine method in such situation is suitable because in that periodic case you can decompose your solution as Fourier serie and truncate by Galerkine method the Fourier serie of the solution which is  not the case when you are working in the whole space since your solution is not periodic. Now, when you are dealing with an unbounded space you can use other methods such as mollifiers and in general Fourier analysis since it is suitable in unbounded domains.
