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Sen Wong, Manwai Yuen, "Blowup Phenomena for the Compressible Euler and Euler-Poisson Equations with Initial Functional Conditions", The Scientific World Journal, vol. 2014, Article ID 580871, 5 pages, 2014. https://doi.org/10.1155/2014/580871
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 . For time , we can define a functional associated with the solution of the equations and some testing function . When the pressure function of the governing equations is of the form , where is the density function, is a constant, and , we can show that the nontrivial solutions with nonslip boundary condition will blow up in finite time if satisfies some initial functional conditions defined by the integrals of . Examples of the testing functions include , , , , and . The corresponding blowup result for the 1-dimensional nonradial symmetric case is also given.
The compressible isentropic Euler or Euler-Poisson equations for fluids can be written as where is a constant related to the unit ball in . As usual, and 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 . If , it is a system with pressure. If , it is a pressureless system.
When , the system is self-attractive. The system (1) is the Newtonian description of gaseous stars (cf. [1, 2]). When , the system comprises the Euler-Poisson equations with repulsive forces and can be applied as a semiconductor model . When , 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 Poisson equation (1) becomes
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 , Makino et al.  first investigated the blowup of “tame solutions.” In 1990, Makino and Perthame further analyzed the corresponding solutions for the equations with gravitational forces . Subsequently, Perthame  studied the blowup results for the -dimensional pressureless system with repulsive forces . Additional results of the Euler system can be found in [9–12].
In this paper, we introduce the nonslip boundary condition , which is expressed by for all and with the constant .
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 by . Consider the Euler or Euler-Poisson equations (1) in . For pressureless fluids or , and the nontrivial classical solutions with radial symmetry and the first boundary condition (6), we have the following results.
(a) For the attractive forces , if satisfies the following initial functional condition: with a total mass of the fluid and an arbitrary nonnegative and nonzero testing function satisfying the following properties:(1) exists,(2) is increasing,then the solutions blow up in finite time.
(b) For the nonattractive forces or , if satisfies the following initial functional condition: then the solutions blow up on or before the finite time .
2. 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.
Proof. The density function conserves its nonnegative nature.
The mass equation (1) with the material derivative could be integrated as for .
For the nontrivial density initial condition in radial symmetry, , we have (Here we multiplied the function on both sides.)
Subsequently, we take integration with respect to from to for or : (a) For , we have with the total mass Then we apply the integration by parts to deduce Inequality (18) with the first boundary condition (6) becomes with and or .
Note that by property and is increasing by property .
Now, we define the assistant functional: We then use the Cauchy-Schwarz inequality to obtain for , In view of (23) and (19), we get as by property .
It is well known that, with the initial condition the Riccati inequality (25) will blow up on or before the finite time .
(b) For or , 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 .
The proof is completed.
Remark 2. For the physical explanation of the functional , readers may refer to Sideris’ paper .
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 for all and for ,(ii)the radius of convergence is not less than .Actually, power series (or real analytic functions) with the above properties constitute a large class of examples for . Concrete examples include and . Moreover, there are examples with some : , , and , where the constant can be arbitrary.
3. 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 . By considering and instead, one may suppose . Let be a nonnegative and nonzero testing function, such that is increasing for and the functional is given by (a) For or , if the initial functional satisfies then the solutions blow up in finite time.(b) For , if , then the solutions blow up on or before the finite time .
Proof. For the 1-dimensional case, (1) becomes
For , 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
As , for all , we get
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 :
and the following estimate:
we get the following.
(a) For or ,
(b) For , Thus, the result immediately follows.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
The work of this paper was partially funded by Dean’s Research Grant FLASS/ECR-9 from the Hong Kong Institute of Education.
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