We study the Euler-Poisson equations of describing the evolution of the gaseous star in astrophysics. Firstly, we construct a family of analytical blowup solutions for the isothermal case in R 2 . Furthermore the blowup rate of the above solutions is also studied and some remarks about the applicability of such solutions to the Navier-Stokes-Poisson equations and the drift-diffusion model in semiconductors are included. Finally, for the isothermal case (γ = 1), the result of Makino and Perthame for the tame solutions is extended to show that the life span of such solutions must be finite if the initial data is with compact support.

In this paper, we investigate some new interesting solution structures of the (2+1)-dimensional bidirectional Sawada–Kotera (bSK) equation. We obtain soliton molecules by introducing velocity resonance. On the basis of soliton molecules, asymmetric solitons are obtained by changing the distance between two solitons of molecules. Based on the N-soliton solutions, several novel types of mixed solutions are generated, which include the mixed breather-soliton molecule solution by the module resonance of the wave number and partial velocity resonance, the mixed lump-soliton molecule solution obtained by partial velocity resonance and partial long wave limits, and the mixed solutions composed of soliton molecules (asymmetric solitons), lump waves, and breather waves. Some plots are presented to clearly illustrate the dynamic features of these solutions.

In this paper, we study the blowup of the N -dim Euler or Euler-Poisson equations with repulsive forces, in radial symmetry. We provide a novel integration method to show that the non-trivial classical solutions (ρ, V ), with compact support in [0, R], where R > 0 is a positive constant and in the sense which ρ(t, r) = 0 and V (t, r) = 0 for r ≥ R, under the initial conditionblow up on or before the finite time T = R 3 /(2H0) for pressureless fluids or γ > 1.

By the extension of the 3-dimensional analytical solutions of Goldreich and Weber [6] with adiabatic exponent γ = 4/3, to the (classical) Euler-Poisson equations without cosmological constant, the self-similar (almost re-collapsing) time-periodic solutions with negative cosmological constant (Λ < 0) are constructed. The solutions with time-periodicity are novel. On basing these solutions, the time-periodic and almost re-collapsing model is conjectured, for some gaseous stars.

We study the construction of analytical non-radially solutions for the 1-dimensional compressible adiabatic Euler equations in this article. We could design the perturbational method to construct a new class of analytical solutions. In details, we perturb the linear velocity: