Notes on INS: spectral decompostion in the time variable ?

The method of Galerkin, used by Hopf and other people to study solutions of INS, is based on a spectral decomposition in space variables (say Fourier decomposition, or spectral decomposition with respect to the Stokes operator).

I wonder if there’s any work out there which is based on a  spectral decomposition in the time variable ?!

In such a decomposition, the initial condition itself will be decomposed into a series of space functions, whose terms are the coefficients (constant with respect to the time variable) of the series (in the time variable) of the solution.

Can one prove local existence (in time), or existence for small initial values, with this time-decomposition method ?!

Remark: In general, the solution is NOT analytic in the time variable (?), though it may depend analytically on the initial value (?), so can’t use power series in time.

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