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Due to the nonlinearity of the Euler{Poisson equations, it is possible thatthe nonlinear Jeans instability may lead to a faster density growing rate thanthe rate in the standard theory of linearized Jeans instability, whichmotivates us to study the nonlinear Jeans instability. The aim of this articleis to develop a method proving the Jeans instability for slightly nonlinearEuler-Poisson equations in the expanding Newtonian universe. The standardproofs of the Jeans instability rely on the Fourier analysis. However, it isdifficult to generalize Fourier method to a nonlinear setting, and thus thereis no result in the nonlinear analysis of Jeans instability. We firstly developa non-Fourier-based method to reprove the linearized Jeans instability in theexpanding Newtonian universe. Secondly, we generalize this idea to a slightlynonlinear case. This method relies on the Cauchy problem of the Fuchsian systemdue to the recent developments of this system in mathematics. The fullynonlinear Jeans instability for the Euler-Poisson and Einstein-Euler equationsare in progress.

Rigorous proof of slightly nonlinear Jeans instability in the expanding Newtonian universe