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Nonlinear gravitational instability is a crucial way to comprehend theclustering of matter and the formation of nonlinear structures in both theUniverse and stellar systems. However, with the exception of a few exactparticular solutions for pressureless matter, there are only someapproximations and numerical and phenomenological approaches to study thenonlinear gravitational instability instead of mathematically rigorousanalysis. We construct a family of particular solutions of the Euler-Poissonsystem that exhibits the nonlinear gravitational instability of matter withinhomogeneous pressure and entropy (i.e., the cold center and hot rim) in theexpanding Newtonian universe. Despite the density perturbations beinghomogeneous, the pressure is not, resulting in significant nonlinear effects.By making use of our prior work on nonlinear analysis of a class ofdifferential equations \cite{Liu2022b}, we estimate that the growth rate of thedensity contrast is approximately $\sim \exp(t^{\frac{2}{3}})$, much fasterthan the growth rate anticipated by classical linear Jeans instability ($\simt^{\frac{2}{3}}$). Our main motivation for constructing this family ofsolutions is to provide a family of reference solutions for conducting a fullynonlinear analysis of inhomogeneous perturbations of density contrast. We willpresent the general results in a mathematical article \cite{Liu2023b}separately. Additionally, we emphasize that our model does not feature anyshell-crossing singularities before mass accretion singularities since we arespecifically interested in analyzing the mathematical mechanics of a pure massaccretion model, which poses limitations on the applicability of our model forunderstanding the realistic nonlinear structure formation.

Fully nonlinear gravitational instabilities for expanding Newtonian universes with inhomogeneous pressure and entropy: Beyond the Tolman's solution