Blowup Phenomena for the Compressible Euler and Euler-Poisson Equations with Initial Functional Conditions

We study, in the radial symmetric case, the finite time life span of the compressible Euler or Euler-Poisson equations in R N. For time t ≥ 0, we can define a functional H(t) associated with the solution of the equations and some testing function f. When the pressure function P of the governing equations is of the form P = Kρ γ, where ρ is the density function, K is a constant, and γ > 1, we can show that the nontrivial C 1 solutions with nonslip boundary condition will blow up in finite time if H(0) satisfies some initial functional conditions defined by the integrals of f. Examples of the testing functions include r N−1ln(r + 1), r N−1 e r, r N−1(r 3 − 3r 2 + 3r + ε), r N−1sin((π/2)(r/R)), and r N−1sinh r. The corresponding blowup result for the 1-dimensional nonradial symmetric case is also given.


Introduction
The compressible isentropic Euler ( = 0) or Euler-Poisson where ( ) is a constant related to the unit ball in . As usual, = ( , ) ≥ 0 and = ( , ) ∈ R are the density and the velocity, respectively. = ( ) is the pressure function. The -law for the pressure term ( ) can be expressed as for which the constant ≥ 1. If > 0, it is a system with pressure. If = 0, it is a pressureless system. When = −1, the system is self-attractive. The system (1) is the Newtonian description of gaseous stars (cf. [1,2]). When = 1, the system comprises the Euler-Poisson equations with repulsive forces and can be applied as a semiconductor model [3]. When = 0, the system comprises the compressible Euler equations and can be applied as a classical model in fluid mechanics [4,5].
The solutions in radial symmetry are expressed by with the radius = ( The equations in radial symmetry can be expressed in the following form: The blowup phenomena have attracted the attention of many mathematicians. Regarding the Euler equations 2 The Scientific World Journal ( = 0), Makino et al. [6] first investigated the blowup of "tame solutions. " In 1990, Makino and Perthame further analyzed the corresponding solutions for the equations with gravitational forces ( = −1) [7]. Subsequently, Perthame [8] studied the blowup results for the 3-dimensional pressureless system with repulsive forces ( = 1). Additional results of the Euler system can be found in [9][10][11][12].
In this paper, we introduce the nonslip boundary condition [13], which is expressed by for all ≥ 0 and with the constant > 0.
In 2011, Yuen used the integration method to show the 1 blowup phenomenon with a "radial dependent" initial functional: for = 1 [14] and > 0 [15]. Following the integration method, we observe that the functional (7) could be generalized to have the following result.

Theorem 1. Define the functional associated with the testing function by
and denote the initial functional (0) by 0 . Consider the Euler or Euler-Poisson equations (1) in . For pressureless fluids ( = 0) or > 1, and the nontrivial classical 1 solutions ( , ) with radial symmetry and the first boundary condition (6), we have the following results.
(a) For the attractive forces ( = −1), if 0 satisfies the following initial functional condition: with a total mass of the fluid and an arbitrary nonnegative and nonzero 1 [0, ] testing function ( ) satisfying the following properties: then the solutions blow up on or before the finite time = (2 ∫ 0 ( ) )/ 0 .

The Generalized Integration Method
The key ideas in obtaining the above results are (i) to design the right form of generalized functional and find the right class of testing functions and (ii) to transform the nonlinear partial differential equations into the Riccati inequality.
For the nontrivial density initial condition in radial symmetry, 0 ( ) ̸ = 0, we have (Here we multiplied the function ( ) on both sides.) Subsequently, we take integration with respect to from 0 to for > 1 or = 0: (a) For = −1, we have The Scientific World Journal 3 with the total mass Then we apply the integration by parts to deduce Inequality (18) with the first boundary condition (6) becomes with ( ) = ( ) and > 1 or = 0. Note that (0) = 0 by property 1 and is increasing by property 2. Now, we define the assistant functional: We then use the Cauchy-Schwarz inequality to obtain In view of (23) and (19), we get as ( ) ≥ (1/ ) ( ) by property 2.
It is well known that, with the initial condition 2 0 the Riccati inequality (25) will blow up on or before the finite time .
(b) For = 0 or 1, by a similar analysis, one can show that Finally, if we set the initial condition Thus, the solutions blow up on or before the finite time = (2 ∫ 0 ( ) )/ 0 . The proof is completed.

Remark 2.
For the physical explanation of the functional ( ), readers may refer to Sideris' paper [16]. For the construction of testing functions with the desired properties as required in Theorem 1, one recalls the class of power series: with the following properties: (i) all ≥ 0 for all and = 0 for < − 1, (ii) the radius of convergence is not less than .

The 1-Dimensional Nonradial Symmetric Case
In the 1-dimensional case, we can apply a similar argument to gain the result for the nonradial symmetric fluids.

Theorem 3. Suppose and have compact support on [ , ]
and vanish at the boundaries: for all ≥ 0. By considering ( , − ) and ( , − ) instead, one may suppose ≥ 0. Let ( ) be a nonnegative and nonzero 1 [ , ] testing function, such that ( )/ is increasing for > and the functional is given by For ̸ = 1, one has Then, we multiply the above equation by ( ) on both sides, taking integration with respect to from to and using integration by parts, to yield Using the properties of ( ) and the Cauchy-Schwarz inequality (as in the proof of Theorem 1), we obtain On the other hand, by using the following explicit form of Φ : Φ ( , ) = 2 (∫ ( , ) − ∫ ( , ) ) (39) and the following estimate: we get the following.
Thus, the result immediately follows.