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The Cauchy problem is studied for very general systems of evolutionequations, where the time derivative of solution is written by Fouriermultipliers in space and analytic nonlinearity, with no other structuralrequirement. We construct a function space for the Fourier transform embeddedin the space of distributions, and establish the global wellposedness with nosize restriction. The major restriction on the initial data is that the Fouriertransform is supported on the half space, decaying at the boundary in the senseof measure. We also require uniform integrability for the orthogonal directionsin the distribution sense, but no other condition. In particular, the initialdata may be much more rough than the tempered distributions, and may growpolynomially at the spatial infinity. A simpler argument is also presented forthe solutions locally integrable in the frequency. When the Fourier support isslightly more restricted to a conical region, the generality of equations isextremely wide, including those that are even locally illposed in the standardfunction spaces, such as the backward heat equations, as well as those withinfinite derivatives and beyond the natural boundary of the analyticnonlinearity. As more classical examples, our results may be applied to theincompressible and compressible Navier-Stokes and Euler equations, thenonlinear diffusion and wave equations, and so on. The major drawback of theFourier support restriction is that the solutions cannot be real valued.

Global wellposedness of general nonlinear evolution equations for distributions on the Fourier half space