  The high-order accuracy of Fourier method makes it the method of choice in
many large scale simulations. We discuss here the stability of Fourier method
for nonlinear evolution problems, focusing on the two prototypical cases of the
inviscid Burgers' equation and the multi-dimensional incompressible Euler
equations. The Fourier method for such problems with quadratic nonlinearities
comes in two main flavors. One is the spectral Fourier method. The other is the
2/3 pseudo-spectral Fourier method, where one removes the highest 1/3 portion
of the spectrum; this is often the method of choice to maintain the balance of
quadratic energy and avoid aliasing errors. Two main themes are discussed in
this paper. First, we prove that as long as the underlying exact solution has a
minimal C^{1+\alpha} spatial regularity, then both the spectral and the 2/3
pseudo-spectral Fourier methods are stable. Consequently, we prove their
spectral convergence for smooth solutions of the inviscid Burgers equation and
the incompressible Euler equations. On the other hand, we prove that after a
critical time at which the underlying solution lacks sufficient smoothness,
then both the spectral and the 2/3 pseudo-spectral Fourier methods exhibit
nonlinear instabilities which are realized through spurious oscillations. In
particular, after shock formation in inviscid Burgers' equation, the total
variation of bounded (pseudo-) spectral Fourier solutions must increase with
the number of increasing modes and we stipulate the analogous situation occurs
with the 3D incompressible Euler equations: the limiting Fourier solution is
shown to enforce L^2-energy conservation, and the contrast with energy
dissipating Onsager solutions is reflected through spurious oscillations.
