This work studies the hydrodynamics of self-gravitating compressible isothermal fluids .
We show that the hydrodynamic evolution equations are scale covariant in absence of viscosity .
Then , we study the evolution of the time-dependent fluctuations around singular and regular isothermal spheres .
We linearize the fluid equations around such stationary solutions and develop a method based on the Laplace transform to analyze their dynamical stability .
We find that the system is stable below a critical size ( X \sim 9.0 in dimensionless variables ) and unstable above ; this criterion is the same as the one found for the thermodynamic stability in the canonical ensemble and it is associated to a center-to-border density ratio of 32.1 .
We prove that the value of this critical size is independent of the Reynolds number of the system .
Furthermore , we give a detailed description of the series of successive dynamic instabilities that appear at larger and larger sizes following the geometric progression X _ { n } \sim 10.7 ^ { n } , n = 1 , 2 , \ldots .
Then , we search for exact solutions of the hydrodynamic equations without viscosity : we provide analytic and numerical axisymmetric soliton-type solutions .
The stability of exact solutions corresponding to a collapsing filament is studied by computing linear fluctuations .
Radial fluctuations growing faster than the background are found for all sizes of the system .
However , a critical size ( X \sim 4.5 ) appears , separating a weakly from a strongly unstable regime .