Fourier would be acceptable if and only if there were only linear solution to the problem. No?

My reasoning is the following : NS also describes non linear behaviour (turbulent flows that are sensitive to the initial condition thus that may result in solutions that looks like white noise).

Linear algebrae can't be used to solve non linear phenomena and thus if a solution is looking like temporal noise, I guess the fourier of the time serie would also would like a solution with a density of probability that would be constant on all spectrum of frequencies that are compatible with the physics (no energy travel faster then the speed of light, hence, there is an upper limit to the frequency).

Am I stupid to expect non linear algebrae involved like Liouville theorem ?