Blowup of Solutions to the Compressible Euler-Poisson and Ideal MHD Systems

Abstract

In the present paper, we study the blowup of the solutions to the full compressible Euler system and pressureless Euler-Poisson system with time-dependent damping. By some delicate analysis, some Riccati-type equations are achieved, and then, the finite time blowup results can be derived.

1. Introduction

In the present paper, we are concerned with the blowup of the solutions to three models of compressible fluids, namely, the full Euler system, pressureless Euler-Poisson system, and ideal MHD system. In addition, there can also be time-dependent velocity damping in these three systems.

At first, we consider the Cauchy problem to the full compressible Euler system with time-dependent damping. where , , and , , , and stand for the density, velocity, pressure, inner energy, and entropy, respectively. The coefficient of the damping term in the second equation of (1) verifies (when , it means that there is no damping in the system (1)). The inner energy satisfies where is the temperature. The equation of state is given by with and the adiabatic exponent ( for the air). The positive constants and are the specific heat at constant pressure and volume, respectively. For convenience, let .

The global existence theories for the Euler equations with damping can be found in [1–6] and the references therein. In [7, 8], the behavior of solutions of the system (1) was considered. For the compressible Euler equations in 3D, the finite time blowup of classical solutions was proved by Sideris in [9]. Recently, the authors in [10] have proved the blowup result of the solutions to (1) in finite time with and . In this paper, by more careful analysis on the pressure and some Riccati-type equations, we can achieve a similar blowup result of system (1) with the adiabatic exponent and the general damping .

Next, we study the pressureless Euler-Poisson system with time-dependent damping in . where and are the electrostatic potential and total mass, respectively. coincides with the pressureless Euler system. stands for repulsive forces and attractive forces with .

The results for existence theories can be found in [11–14]. For the blowup results for compressible Euler-Poisson equations without time-dependent damping , see [15–18] with repulsive forces and [17, 19, 20] with attractive forces. In this paper, we established the finite time blowup result for the Euler-Poisson equations with time-dependent damping and attractive forces, under suitable initial conditions.

At last, we are concerned with the classical solutions to the 2D ideal compressible transverse MHD flow. Let , and the 3D compressible isentropic MHD system reads where , , , and denote the density, velocity, pressure, and magnetic field. The “ideal” means that there is no viscosity or resistivity in the system (5). In this paper, we consider the 2D transverse flow as follows:

The divergence-free condition of the magnetic field is naturally fulfilled. Let , , and ; then, (5) with the structural assumption (6) can be simplified as the following 2D hyperbolic system:

In [21], under similar initial condition to that of [22], the author obtained the global existence of classical solution of the transverse MHD flow. For the large initial data in 3D and small data in 1D, Rammaha [23] proved the finite time blowup of solutions to the MHD system. In this article, under some suitable conditions, we proved the blowup results for the 2D transverse flow.

The paper is organized as follows. In Section 2, we will present our main results. In Sections 3–5, by using the spectral dynamics method as in [10, 19, 20, 24], some Riccati-type equations are achieved, and then, we will give the proof of the main results.

2. Main Results

In this section, we will introduce our main results in the paper. Now, we give the first result. To this purpose, some conditions on the initial data of the Euler system (1) are stated as follows.

Hypothesis H_E. There exists such that

Define the first blowup time by where the infimum taking over all the verifies the Hypothesis HE. We assume that the infimum is reached at , and for convenience, we omit the tilde above . The aforementioned two conventions also work in the following part. It is easy to check that is well defined, since the term in the integration on the left side of (9) is positive. On the other hand, the Hypothesis HE implies that .

Theorem 1. Let be the -smooth solution to the system (1) with , , and the initial data . Suppose that satisfies the Hypothesis HE; then, the classical solution will blow up before or at .

Remark 2. Now, we give the conditions on in Hypothesis HE and the blowup time for some special choices of . It is not hard to check that The condition on and the blowup time in [10] are and . Choose ; then, our bound is strictly less than their bound . In addition, for with nonnegative constant , we achieve

We also make some assumptions on the initial data of the Euler-Poisson system (4).

Hypothesis H_EP. There exists such that where the constant is defined by

The second result in the present paper is stated in the following.

Theorem 3. Let be the smooth solution to the system (4) with , , , and the initial data . Suppose that satisfies the Hypothesis HEP; then, singularity will be developed before or at .

At last, we turn to our last result. The initial data of the MHD system (7) is assumed to verify.

Hypothesis H_MHD. There exists such that

Theorem 4. Let be the smooth solution to the system (7) with the initial data . Suppose that satisfies the Hypothesis HMHD with ; then, the solution will blowup before or at .

Remark 5. The special case in (5) coincides with the isentropic Euler system (1).

Remark 6. If , the first and third equations in (7) derive the following well-known “frozen” law: Under the assumption , in the Hypothesis HMHD, it is natural to define when . For the special case , holds for any when the smooth solution exists.

Remark 7. We give some examples of the Hypothesis HMHD. Near the point

Remark 8. The sign conditions of in (8) and (14) are necessary. If the spectrum of is nonnegative, global existence was given in [21, 22].

3. Blowup of the Euler System with Time-Dependent Damping

In this section, we deal with the Proof of Theorem 1. Before the proof, a new reformulation of (1) is achieved as follows.

Lemma 9. Under the assumption that , the equations in (1) can be rewritten as where denotes the material derivative.

Proof. The first two equations in (17) are trivial, and we focus on the last equation of the entropy . Multiplying the second equation in (17) by , it implies that

Subtracting the third equation in (1) by the above identity, then

This, together with (2), derives which implies in . Noting the -smoothness of , it is natural to define in . This completes the proof of Lemma 9.

Proof of Theorem 1. Denote and apply to the second equation in (17); then, we achieve Let , and we deduce from (21) that where . Now, we turn to the gradient of the pressure. According to (3) with , we find that Define the flow line starting from by Then, we conclude from the first and third equations in (17) that where is a positive constant. On the other hand, due to the Hypothesis HE on and the fact that equation (23) is homogenous in , we obtain .
Thereafter, we have proved that This, together with implies Let ; then, we find that This means and Note that there exists such that for any Then, we infer that blows up before or at . This completes the Proof of Theorem 1.

4. Blowup of the Euler-Poisson System with Time-Dependent Damping

In the present section, we turn our attention to the Proof of Theorem 3.

Proof of Theorem 3. Similar to (22) and (23), we can obtain By taking the trace of the two sides of equation (33), we find that where we have used (28) again. We conclude from the Hypothesis HEP and (34) that .
Let ; then, it is easy to check that where we have used the fact , and is defined in (13). Now, we make the ansatz that before the formation of singularity which derives . Thereafter, we find that Integrating the above differential inequality yields This, together with which follows from the Hypothesis HEP, implies According to the boundedness of , we achieve that blows up before or at . On the other hand, we also conclude from (40) that . Consequently, the ansatz (37) makes sense. This completes the proof of Theorem 3.

5. Blowup of 2D Ideal Compressible Transverse MHD Flow

In the last section, we finish the proof of Theorem 4.

Proof of Theorem 4. The Proof of Theorem 4 is analogous with that of Theorem 1. Instead of (22) and (23), we achieve According to the “frozen” law, it is convenient to regard as a new unknown. Consequently, similar to (24), we find that By an analogous argument as in Section 2 and the Hypothesis HMHD, we finally derive which yields

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

Wenming Hu was supported by the NSFC (Nos. 11601236 and 11971237) and by the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (No. 19KJA320001).