The Scientific World Journal
Volume 2016 (2016), Article ID 3781760, 6 pages
http://dx.doi.org/10.1155/2016/3781760

Research Article

Blowup Phenomenon of Solutions for the IBVP of the Compressible Euler Equations in Spherical Symmetry

Ka Luen Cheung and Sen Wong

Department of Mathematics and Information Technology, The Hong Kong Institute of Education, 10 Lo Ping Road, Tai Po, New Territories, Hong Kong

Received 3 December 2015; Accepted 19 January 2016

Academic Editor: Martin Bohner

Copyright © 2016 Ka Luen Cheung and Sen Wong. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The blowup phenomenon of solutions is investigated for the initial-boundary value problem (IBVP) of the -dimensional Euler equations with spherical symmetry. We first show that there are only trivial solutions when the velocity is of the form for any value of or any positive integer . Then, we show that blowup phenomenon occurs when and . As a corollary, the blowup properties of solutions with velocity of the form are obtained. Our analysis includes both the isentropic case and the isothermal case .

1. Introduction and Main Results

In this paper, we consider the -dimensional Euler equations for compressible fluid:

with boundary condition

where , , and represent the density, velocity, and pressure of the fluid, respectively. is the unit normal vector on the unit sphere. The -law for is given by (1)₃. The fluid is called isentropic if and is called isothermal if .

Euler equation (1) is one of the most important fundamental equations in inviscid fluid dynamics. Many interesting fluid dynamic phenomena can be described by system (1) [1, 2]. The Euler equations 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 long-standing. Thus, the singularity formation in fluid mechanics has been attracting the attention of a number of researchers [3–11].

In particular, in [3, 4], the authors obtain blowup results for the IBVP of the Euler equations, namely, system (1) with boundary condition (2). By making use of the finite propagation speed property [5, 6], they are able to apply the integration method to derive differential inequalities and show that if the initial weighted functionals of velocity or momentum are large enough, then blowup occurs.

In [10], the authors consider the solutions of (1) with velocity of the form

and show that, by using the standard argument of phase diagram, the solutions will be expanding if and satisfy some inequalities. It is natural to consider the more general velocity form:

for the IBVP of system (1) in spherical symmetry, where is a time-dependent drifting function.

For solutions in spherical symmetry, namely, and , system (1) together with (2) is transformed to

where is the length of the spatial variable .

Our main contributions in this paper are stated as follows.

Theorem 1. There are only trivial solutions to the -dimensional Euler system (5) of the form for any real and integer in the isentropic and isothermal cases except the case . For , one has the following two cases.

(1)If , then, for any ,, and , as .(2)If , then, for any ,. Moreover, in the region where , and as .

As a corollary, we also obtain the following.

Corollary 2. Let be a solution for (5) with , , and . Then satisfies

for some constant . Furthermore, one has the following five cases.

(1)If , then the solution as for some finite .(2)If , then is bounded above and the solution as .(3)If and , is bounded above and the solution as .(4)If and , then the solution is trivial.(5)If and , then the solution as .

2. Lemmas

It is well-known that is always positive if the initial datum is set to be positive. Thus, we suppose in the following to avoid the trivial solutions .

Lemma 3. For , one has

Proof. From (5)₁, one has

Multiply both sides by . Then, the result follows.

Lemma 4. For , one has

Proof. From (5)₂, one has

and the results follow.

Similarly, we have the following two lemmas for .

Lemma 5. For , one has

Lemma 6. For , one has

Lastly, one has the following lemma that will be used to prove that there are only trivial solutions when and .

Lemma 7. Consider the following dynamical system

with or . If , , and

then (17) is equivalent to

where

Otherwise, the solution to (17) is trivial.

Proof. If , then it is clear that is the only solution. So we suppose . One has from (17) that

If and , then is the only solution.
If and , then is the only solution.
So, we suppose both and are not zero.
From (21), one has

for some constant .
From (21) and (22), one has

Thus, from (17)₁, one has

If , then is the only solution. So we suppose it is zero; that is, (18) holds. Then, if , is the only solution. So we suppose . Thus, we have

Conversely, if one has (18) and (19), then system (17) is satisfied. The proof is complete.

3. Proofs of Main Results

Proposition 8. Assume . Then there are only trivial solutions to the -dimensional Euler system (5) of the form with .

Proof. First, we set

From (5)₄, we have

Then,

For , if , then from (27). It follows from (7) and (9) that is independent of and , respectively. Thus, is a constant.
For and , after substituting (28), (29), and (30) into (9), (11), and (7), respectively, we see that (7) becomes

for all , where are functions of only. More precisely, one has

where

Note that we omitted as it is irrelevant in the proof.
If , then the powers and are different and unique among the powers in (31). In this case, one has

As the two constants and cannot be both zero for , we conclude that . Hence, and is a constant.
For , we have Table 1.
For or , as is unique among other powers, one has

As , we conclude that . Thus, and is a constant.
For , as and are different and unique among other powers, one has

which is reduced to

for . As , we conclude that . Thus, and is a constant.
For , as is unique among other powers, one has

As , we conclude that . Thus, and is a constant.
Next, we consider the case .
For , the corresponding equation of (31) is

for all , where are functions of only. As is not a rational function, one has that all . In particular, one has

As and cannot be both zero for , we conclude that and the solutions are trivial.
For and , one can show that there are only trivial solutions with similar procedures. The proof is complete.

Table 1

1/3
1/2

()
0
1/2
1
5

()
2/3
1
4/3
4

()
−1/3
0
1/3
3

()
4/3
3/2
5/3
3

()
1/3
1/2
2/3
2

()
−2/3
−1/2
−1/3
1

1 ()
1
1
1
1

0 ()
0

0
0

−1 ()
−1
−1
−1
−1

Using similar analysis as that given for the case in Proposition 8, we obtain the following proposition for the case .

Proposition 9. Assume . Then there are only trivial solutions to the -dimensional Euler system (5) of the form with .

Next, the crucial case will be analyzed as follows.

Proposition 10. Assume . Then there are only trivial solutions to the -dimensional Euler system (5) of the form with .

Proof. For and , one has

(41)₁ and (41)₂ are equivalent to

where

Note that is equivalent to and is equivalent to

Thus, we have either or . Moreover, condition (18) is equivalent to

which is impossible for . Thus, we conclude by Lemma 7 that there are only trivial solutions.

Next, we consider the remaining case .

Proposition 11. Assume . Then there are only trivial solutions to the -dimensional Euler system (5) of the form with and . For , one has the following two cases.

(1)If , then, for any , , and , as .(2)If , then, for any , . Moreover, in the region where , and as .

Proof. For , the corresponding system of (41) is

where

It is clear that from (46)₄ and (48) we have only the trivial solutions if . So we suppose . Then (46) is equivalent to

Note that (49)₁ is a special case of equation in [10] when we set the parameter in (7) to be zero. Thus, by Theorem  2.1 in [10], the results and in the proposition follow.

Remark 12. From (49)₂ and (15), the density function is given by

Thus, the total mass is finite if and is infinite if . From and in the proposition, we see that blowup can occur only when the total mass is infinite.

Proof of Theorem 1. Theorem 1 is followed from Propositions 8–11.

Finally, we are ready to present the proof of Corollary 2.

Proof of Corollary 2. Let in (49)₁. One has

It follows that

where . Thus, satisfies (6). Consider (6) with initial conditions

Set

First, note that if is finite, then the one-sided limit must be zero. More precisely, if is finite and , then we can extend the solution by solving (6) with initial condition . This contradicts the definition of .
Next, suppose , and then

where . It follows that

From (55), one has

As the coefficient of is negative, we see that will be negative if is sufficiently large. This implies that is finite.
On the other hand, from (55), one has

Thus,

Thus, and as .
For , one has

(i),(ii),(iii)

From (i), (ii), and (iii) above, we see that is bounded above by

Moreover, we have . This is because if is finite, then is bounded by some positive number . But, from (6), one has

which implies that is unbounded as the coefficient of is positive.
Next, we show that

If is finite, then (64) is clearly held. If is not finite, then

Thus, for , as .
As the cases for can be verified trivially, the proof is complete.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This research paper is partially supported by Grant no. MIT/SRG02/15-16 from the Department of Mathematics and Information Technology of the Hong Kong Institute of Education.

References

P. L. Lions, Mathematical Topics in Fluid Mechanics, vol. 1, Clarendon Press, Oxford, UK, 1998.

P. L. Lions, Mathematical Topics in Fluid Mechanics, vol. 2, Clarendon Press, Oxford, UK, 1998.

X. Zhu and A. Tu, “Blowup of the axis-symmetric solutions for the IBVP of the isentropic Euler equations,” Nonlinear Analysis, vol. 95, pp. 99–106, 2014.

X. Zhu, “Blowup of the solutions for the IBVP of the isentropic Euler equations with damping,” Journal of Mathematical Analysis and Applications, vol. 432, no. 2, pp. 715–724, 2015.

T. C. Sideris, “Formation of singularities in three-dimensional compressible fluids,” Communications in Mathematical Physics, vol. 101, no. 4, pp. 475–485, 1985.

T. C. Sideris, B. Thomases, and D. Wang, “Long time behavior of solutions to the 3D compressible Euler equations with damping,” Communications in Partial Differential Equations, vol. 28, no. 3-4, pp. 795–816, 2003.

Z. P. Xin, “Blowup of smooth solutions to the compressible Navier-Stokes equation with compact density,” Communications on Pure and Applied Mathematics, vol. 51, no. 3, pp. 229–240, 1998.

T. Suzuki, “Irrotational blowup of the solution to compressible Euler equation,” Journal of Mathematical Fluid Mechanics, vol. 15, no. 3, pp. 617–633, 2013.

Z. Lei, Y. Du, and Q. T. Zhang, “Singularities of solutions to compressible Euler equations with vacuum,” Mathematical Research Letters, vol. 20, no. 1, pp. 41–50, 2013.

T. Li and D. Wang, “Blowup phenomena of solutions to the Euler equations for compressible fluid flow,” Journal of Differential Equations, vol. 221, no. 1, pp. 91–101, 2006.

D. Li, C. X. Miao, and X. Y. Zhang, “On the isentropic compressible Euler equation with adiabatic index γ = 1,” Pacific Journal of Mathematics, vol. 262, no. 1, pp. 109–128, 2013.