Perturbational blowup solutions to the compressible Euler equations with damping
 Ka Luen Cheung^{1}Email authorView ORCID ID profile
Received: 3 January 2016
Accepted: 12 February 2016
Published: 27 February 2016
Abstract
Background
The Ndimensional isentropic compressible Euler system with a damping term is one of the most fundamental equations in fluid dynamics. Since it does not have a general solution in a closed form for arbitrary wellposed initial value problems. Constructing exact solutions to the system is a useful way to obtain important information on the properties of its solutions.
Method
In this article, we construct two families of exact solutions for the onedimensional isentropic compressible Euler equations with damping by the perturbational method. The two families of exact solutions found include the cases \(\gamma >1\) and \(\gamma =1\), where \(\gamma\) is the adiabatic constant.
Results
With analysis of the key ordinary differential equation, we show that the classes of solutions include both blowup type and global existence type when the parameters are suitably chosen. Moreover, in the blowup cases, we show that the singularities are of essential type in the sense that they cannot be smoothed by redefining values at the odd points.
Conclusion
The two families of exact solutions obtained in this paper can be useful to study of related numerical methods and algorithms such as the finite difference method, the finite element method and the finite volume method that are applied by scientists to simulate the fluids for applications.
Keywords
Mathematics Subject Classfication
Background and Main results
System (1) is one of the most fundamental equations in fluid dynamics. Many interesting fluid dynamic phenomena can be described by system (1) (Lions , 1998a; Lions , 1998b). The Euler equations \((\alpha =0)\) are also the special case of the noted Navier–Stokes equations, whose problem of whether there is a formation of singularity is still open and longstanding. Thus, the singularity formation in fluid mechanics has been attracting the attention of a number of researchers (Sideris 1985; Xin 1998; Suzuki 2013; Lei et al. 2013; Li and Wang 2006; Li et al. 2013).
Among others, we mention that in 2003, Sideris–Thomases–Wang (Sideris et al. 2003) obtained results for the three dimensional compressible Euler equations with a linear damping term with assumption \(\gamma >1\), that is, system (1) with \(N=3\) and \(\gamma >1\). They discovered that damping prevents the formation of singularities in small amplitude flows, but large solutions may still break down. They formulated the Euler system as a symmetric hyperbolic system, established the finite speed of propagation of the solution, and some energy estimates to obtain local existence as well as global existence of the solution. For larger solution, they showed that the solution will blow up in a finite time by establishing certain differential inequalities.
Theorem 1
Remark 2
The ordinary differential equation (O.D.E.) (6) will be analyzed in section 2 and it is wellknown by the theory of ordinary differential equations that the solutions of system (7) exist and is \(C^2\) as long as f and g, which are functions of \(\ddot{a}\), \(\dot{a}\) and a, are continuous.
Theorem 3
 (i) :

If \(\xi >0\) and \(a_1\ge 0\), then the solution (4) is a global solution.
 (ii) :

If \(\xi >0, a_1<0\) and \(a_0>a_1/\alpha\), then the solution (4) is a global solution.
 (iii) :

If \(\xi <0\), then the solution (4) blows up on a finite time.
 (iv) :

If \(\xi =0\) and \(a_1>0\), then the solution (4) blows up on the finite time \(T=\frac{1}{\alpha }\ln \frac{a_1}{a_1+a_0\alpha }>0\).
 (v) :

If \(\xi =0\), and \(a_1<0\) and \(a_0<a_1/\alpha\), then the solution (4) blows up on the finite time \(T=\frac{1}{\alpha }\ln \frac{a_1}{a_1+a_0\alpha }>0\).
Moreover, we show that the singularity formations in the cases iii), iv) and v) above are of essential type in the sense that the singularities cannot be smoothed by redefining values at the odd points. This is an improvement of the corresponding results in Yuen (2011).
For \(\gamma =1\), we obtain the following theorem.
Theorem 4
Theorem 5
 (i) :

If \(\xi >0\) and \(a_1\ge 0\), then the solution (8) is a global solution.
 (ii) :

If \(\xi >0, a_1<0\) and \(a_0>a_1/\alpha\), then the solution (8) is a global solution.
 (iii) :

If \(\xi <0\), then the solution (8) blows up on a finite time.
 (iv) :

If \(\xi =0\) and \(a_1>0\), then the solution (8) blows up on the finite time \(T=\frac{1}{\alpha }\ln \frac{a_1}{a_1+a_0\alpha }>0\).
 (v) :

If \(\xi =0\), and \(a_1<0\) and \(a_0<a_1/\alpha\), then the solution (8) blows up on the finite time \(T=\frac{1}{\alpha }\ln \frac{a_1}{a_1+a_0\alpha }>0\).
Analysis of an O.D.E.
Lemma 6
Proof
Lemma 7

Case 1. If \(\xi >0 \quad \text {and} \quad a_1\ge 0, \quad \text{then} \quad T^*=+\infty\) .

Case 2. If \(\xi >0, a_1<0 \quad \text {and} \quad a_0>a_1/\alpha , \quad \text {then} \quad T^*=+\infty\) .

Case 3. If \(\xi <0, \quad \text {then} \quad T^*<+\infty\).
Proof
Remark 8
The case for \(\xi =0\) will be analyzed in the proof of Theorem 3.
Proofs of the Theorems
Proof of Theorem 1
We divide the proof into steps.
Step 1. In the first step, we show a lemma.
Lemma 9
Proof of Lemma 9
Next, we prove Theorem 3 as follows.
Proof of Theorem 3
For \(\xi >0\), case i) and case ii) of Theorem 3 follow from Case 1. and Case 2. of Lemma 7.
Proof of Theorems 4 and 5
Note that (10) is a special case of (12) and the arguments in the proof of Theorem 3 hold for \(\gamma =1\). Thus, the results for Theorem 5 follows. \(\square\)
Conclusion
The complicated Euler equations with a damping term (1) do not have a general solution in a closed form for arbitrary wellposed initial value problems. Thus, numerical methods and algorithms such as the finite difference method, the finite element method and the finite volume method are applied by scientists to simulate the fluids for applications in real world. Thus, our exact solutions in this article provide concrete examples for researchers to test their numerical methods and algorithms.
Declarations
Acknowledgements
This research paper is partially supported by the Grant: MIT/SRG02/1516 from the Department of Mathematics and Information Technology of the Hong Kong Institute of Education.
Competing interests
The authors declare that they have no competing interests.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
 Lei Z, Du Y, Zhang QT (2013) Singularities of solutions to compressible Euler equations with vacuum. Math Res Lett 20:41–50View ArticleGoogle Scholar
 Li D, Miao CX, Zhang XY (2013) On the isentropic compressible Euler equation with adiabatic index \(\gamma =1\). Pac J Math 262:109–128View ArticleGoogle Scholar
 Li T, Wang D (2006) Blowup phenomena of solutions to the Euler equations for compressible fluid flow. J Diff Equ 221:91–101View ArticleGoogle Scholar
 Lions PL (1998) Mathematical topics in fluid mechanics, vol 1. Clarendon Press, OxfordGoogle Scholar
 Lions PL (1998) Mathematical topics in fluid mechanics, vol 2. Clarendon Press, OxfordGoogle Scholar
 Sideris TC (1985) Formation of singularities in threedimensional compressible fluids. Commun Math Phys 101:475–485View ArticleGoogle Scholar
 Sideris TC, Thomases B, Wang D (2003) Long time behavior of solutions to the 3D compressible Euler equations with damping. Comm Partial Differ Equ 28:795–816View ArticleGoogle Scholar
 Suzuki T (2013) Irrotational blowup of the solution to compressible Euler equation. J Math Fluid Mech 15:617–633View ArticleGoogle Scholar
 Xin ZP (1998) Blowup of smooth solutions to the compressible Navier–Stokes equations with compact density. Commun Pure Appl Math LI:0229–0240View ArticleGoogle Scholar
 Yuen MW (2011) Perturbational blowup solutions to the compressible 1dimensional Euler equations. Phys Lett A 375:3821–3825View ArticleGoogle Scholar