Stabilities for Nonisentropic Euler-Poisson Equations

We establish the stabilities and blowup results for the nonisentropic Euler-Poisson equations by the energy method. By analysing the second inertia, we show that the classical solutions of the system with attractive forces blow up in finite time in some special dimensions when the energy is negative. Moreover, we obtain the stabilities results for the system in the cases of attractive and repulsive forces.


Introduction
The compressible nonisentropic Euler ( = 0) or Euler-Poisson ( = ±1) system for fluids can be written as where ≥ 0 is the frictional damping constant and ( ) is a constant related to the unit ball in R . (1) = 2, (2) = 2 and where ( ) is the volume of the unit ball in R . As usual, = ( , ) ≥ 0, = ( , ) ∈ R , and ( , ) are the density, the velocity, and the entropy, respectively. is the pressure function, for which the constants ≥ 0 and ≥ 1.
When = 1, the system is self-attractive. The system (1) is the Newtonian description of gaseous stars [1]. When = −1, the system comprises the Euler-Poisson equations with repulsive forces and can be used as a semiconductor model [2,3]. When = 0, the system comprises the compressible Euler equations and can be applied as a classical model in fluid mechanics [3]. For more classical and recent results in these systems, readers can refer to [1,[4][5][6][7][8][9][10].
It is well known that the solution for the Poisson equation (1) 4 can be written as where is the Green's function for the Poisson equation in the -dimensional spaces defined by Notation. In the following discussion, classical solutions ( , , ) are 1 solutions with compact support Ω = Ω( ) for each fixed time . We also denote the total mass by , where where 0 = 0 ( ) := (0, ). Lastly, we will denote 2 The Scientific World Journal

Lemmas
In this section, we establish some lemmas for the proof of the main results. The following lemma will be used to derive the energy functional for > 1; namely, is conserved in time if the system (1) is not damped.
Proof. We have Note that, by Divergence Theorem, Thus, Next, the results of the following two lemmas will be used in the derivations of the energy functionals for both > 1 and = 1. It will be shown that in Section 3 the energy functional for = 1 is which is conversed in time if the system (1) is not damped.
Proof. We have One can check a detail proof of the following equality in the Appendix: Thus, by (16) and (17). Thus, the proof is complete.
Thus, the first equality in (19) holds. Next, Thus, the second equality in (19) holds.
The Scientific World Journal 3 The lemma below is crucial to obtaining the energy functional for = 1. Comparing the left hand sides of (8) and (22), we note that the left hand side of (22) (given in the next lemma), which contains the term ln − 1, is nontrivial to be found.
Proof. Note that Thus, The proof is complete.

Main Results
In this section, we find out the energy functionals for the system (1) in the case of > 1 (Proposition 5) and = 1 (Proposition 6). Moreover, we establish the stabilities results (Proposition 8) and a blowup result (Proposition 9) for system (1).

Proposition 5. For the classical solution ( , , ) of system (1) with > 1, let
Then,̇( where( ) is the devertive of ( ) with respect to . Thus, ( ) is a decreasing function and is conserved if the system is not damped.
Proof. By Lemma 1, By Lemma 2, By Lemma 3, Thus, the proof is complete.

Proposition 6.
For the classical solution ( , , ) of system (1) with = 1, let Then,̇( where( ) is the derivative of ( ) with respect to . Thus, ( ) is a decreasing function and is conserved if the system is not damped.
Proof. By Lemma 2, By Lemma 3, By Lemma 4, Thus, the proof is complete.

4
The Scientific World Journal We havë is a classical solution of system (1), and is defined by (5). Proof.
We split the last term of the above equality into three parts. Firstly, Secondly, Thirdly, where ∇ is the gradient operator with respect to the spatial variable . Note that For = 2, Thus, The result for = 2 is established. For ≥ 3, Thus, The results for ≥ 3 are also established. Now, we are ready to present the stability results.