Analytical Solutions to the Navier-Stokes-Poisson Equations with Density-dependent Viscosity and with Pressure

We study some particular solutions to the Navier-Stokes-Poisson equations with density-dependent viscosity and with pressure, in radial symmetry. With extension of the previous known blowup solutions for the Euler-Poisson equations / pressureless Navier-Stokes-Poisson with density-dependent viscosity, we constructed the corresponding analytical blowup solutions for the Navier-Stokes-Poisson Equations with density-dependent viscosity and with pressure.


Introduction
The evolution of a self-gravitating fluid can be formulated by the isentropic Euler-Poisson equations of the following form: ρ[ u t + ( u · ∇) u)] + ∇P = −δρ∇Φ + vis(ρ, u), where α(N ) is a constant related to the unit ball in R N : α(1) = 2; α(2) = 2π and for N ≥ 3, where V ol(N ) is the volume of the unit ball in R N and Γ is a Gamma function. And as usual, ρ = ρ(t, x) and u = u(t, x) ∈ R N are the density and the velocity respectively. P = P (ρ) is the pressure. And Λ is the background constant.
When δ = 1, the system can model fluids that are self-gravitating , such as gaseous stars. In addition, the evolution of the simple cosmology can be modelled by the dust distribution without pressure term. This describes the stellar systems of collisionless and gravitational n-body systems [8]. And the pressureless Euler-Poisson equations can be derived from the Vlasov-Poisson-Boltzmann model with the zero mean free path [10]. For N = 3, the equations (1) are the classical (non-relativistic) descriptions of a galaxy in astrophysics. See [2] and [3], for details about the systems.
When δ = −1, the system is the compressible Euler-Poisson equations with repulsive forces. The equation (1) 3 is the Poisson equation through which the potential with repulsive forces is determined by the density distribution of the electrons. In this case, the system can be viewed as a semiconductor model. See [5] and [6] for detailed analysis of the system.
When δ = 0, the potential forces are ignored. The system is called Euler / Navier-Stokes equations.
See [4], [16] and [17] for detailed analysis of the system In the above system, the self-gravitational potential field Φ = Φ(t, x) is determined by the density ρ through the Poisson equation.
And vis(ρ, u) is the viscosity function: In this article, we seek the radial solutions . By the standard computation, the Euler-Poisson equations in radial symmetry can be written in the following form: Under a common assumption for, the viscosity function can be defined: where κ and θ ≥ 0 are the constants. For the study of the above system, the readers may refer [13], [16], [18] and [21]. In particular, when θ = 0, that returns the expression for the only V -dependent viscosity function: Here, if the vis(ρ, u) = 0, the system is called the Euler-Poisson equations. In this case, the equations can be viewed as a prefect gas model. For N = 3, (1) is a classical (nonrelativistic) description of a galaxy, in astrophysics. See [3], [15] for details about the system. P = P (ρ) is the pressure. The γ-law can be applied on the pressure P (ρ), i.e.
which is a common hypothesis. The constant γ = c P /c v ≥ 1, where c P , c v are the specific heats per unit mass under constant pressure and constant volume respectively, is the ratio of the specific heats, that is, the adiabatic exponent in (8). With K = 0, we call that the system is pressureless.
For the local existence of the Euler-Poisson equations, the interested reader may refer the results of Makino [14], Gambin [9] and Bezard [1].
In the following, we always seek solutions in radial symmetry. Thus, the Poisson equation (1) 3 is transformed to In this paper, we concern about blowup solutions for the N -dimensional pressureless Navier-Stokes-Poisson equations with the density-dependent viscosity. And our aim is to construct a family of blowup solutions.
Historically in astrophysics, Goldreich and Weber [11] constructed the analytical blowup (collapsing) solutions of the 3-dimensional Euler-Poisson equations for γ = 4/3 and for the non-rotating gas spheres. After that, Makino [15] obtained the rigorously mathematical proof of the existence of such kind of blowup solutions. And in [7], Deng, Xiang and Yang extended the above blowup Recently, Yuen obtained the blowup solutions in R 2 with γ = 1 in [21].
For the construction of analytical solutions to the Euler, Navier-Stokes and pressureless Navier-Stokes-Poisson equations, readers may refer to the recent results in [20], [19], [23] and [24].
In this article, it is natural to extend the more general results to cover the full system (5) --Navier-Stokes-Poisson equations with density-dependent viscosity and with pressure. In short, we successfully deduce the system (5) into the much simpler ordinary differential equations.

Theorem 1 For the Navier-Stokes-Poisson equations in
exists a family of solutions, those are: where f (z) and a(t) are the following functions and the finite Z µ is the first zero of f (z): (1) with Λ = 0, and where m, n > 0 and α are constant; In particular, for m < 0, the solutions (14) and (15) blow up at the finite time T = −m/n.

Separable Solutions
Before we present the proof of Theorem 1, Lemmas 3 and 12 of [22] are quoted here.

Lemma 2 (Lemma 3 of [22]) For the equation of conservation of mass in radial symmetry:
there exist solutions, with the form f ≥ 0 ∈ C 1 and a(t) > 0 ∈ C 1 .
It is clear to check our solutions to satisfy the Navier-Stokes-Poisson equations: Proof. As we use the functional structure of the above lemma, our solutions (13) fit well to the mass equation (5) 1 .
Then, we have: In the theorem, we require the ordinary differential equation for y(z) is Therefore, our solutions satisfy the momentum equation.
Then, we have: and with the new variable z := r/a(t): In the theorem, we require the ordinary differential equation for f (z): Therefore, our solutions satisfy the momentum equation.

Remark 3
If the complex number solutions (ρ, u) ∈ C N +1 is considered, we have the corresponding solutions for δΛ < 0: where i is the complex constant.