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

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.


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 timedependent velocity damping in these three systems.
At first, we consider the Cauchy problem to the full compressible Euler system with time-dependent damping.
The global existence theories for the Euler equations with damping can be found in [1][2][3][4][5][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 αðtÞ ≡ 1 and γ ∈ f2g ∪ ½3,∞Þ. 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 γ > 1 and the general damping αðtÞ ≥ 0.
Next, we study the pressureless Euler-Poisson system with time-dependent damping in ℝ n ðn ≥ 2Þ.
At last, we are concerned with the classical solutions to the 2D ideal compressible transverse MHD flow. Let x ∈ ℝ 3 , and the 3D compressible isentropic MHD system reads where ϱ, u = ðu 1 , u 2 , u 3 Þ, p = pðϱÞ = p 0 ϱ γ , and B = ðB 1 , B 2 , B 3 Þ 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 B is naturally fulfilled. Let x = ðx 1 , x 2 Þ, ∇ = ð∂ 1 , ∂ 2 Þ, and u = ðu 1 , u 2 Þ; 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.

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 a 1 ∈ ℝ n such that Define the first blowup time t * E by where the infimum taking over all the a 1 verifies the Hypothesis HE. We assume that the infimum is reached atã 1 , and for convenience, we omit the tilde above a 1 . The aforementioned two conventions also work in the following part. It is easy to check that t * E 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 t * E > 0.

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Remark 2. Now, we give the conditions on div u 0 in Hypothesis HE and the blowup time t * E for some special choices of αðtÞ. It is not hard to check that The condition on div u 0 and the blowup time in [10] are div u 0 ða 1 Þ < −n − 1 and T * ≤ inf Choose inf a 1 div u 0 ða 1 Þ = −2n; then, our bound is strictly less than their bound t * E = ln 2 < ððn + 1Þ/2Þ. In addition, for αðtÞ = ðα/ð1 + tÞÞ with nonnegative constant α, we achieve We also make some assumptions on the initial data of the Euler-Poisson system (4).
Hypothesis H EP . There exists a 2 ∈ ℝ n such that where the constant m 0 is defined by The second result in the present paper is stated in the following.
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 a 3 ∈ ℝ 2 such that Theorem 4. Let ðρ, u, BÞ be the smooth solution to the system (7) with the initial data ðρ 0 , u 0 , B 0 Þ. Suppose that ðρ 0 , u 0 , B 0 Þ satisfies the Hypothesis HMHD with γ > 1; then, the solution will blowup before or at t * MHD .
Remark 8. The sign conditions of div u 0 in (8) and (14) are necessary. If the spectrum of ∇u 0 is nonnegative, global existence was given in [21,22].

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.

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Lemma 9. Under the assumption that ðρ, u, SÞ ∈ C 1 , the equations in (1) can be rewritten as where D t ≔ ∂ t + u ⋅ ∇ denotes the material derivative.
Proof. The first two equations in (17) are trivial, and we focus on the last equation of the entropy S. Multiplying the second equation in (17) by ρu, it implies that Subtracting the third equation in (1) by the above identity, then This, together with (2), derives which implies D t S = 0 in fðt, xÞjρðt, xÞ > 0g. Noting the C 1 -smoothness of S, it is natural to define D t S = 0 in fðt, xÞjρðt, xÞ = 0g. This completes the proof of Lemma 9.
Proof of Theorem 1. Denote U ≔ ∇u and apply ∇ to the second equation in (17); then, we achieve Let U ± ≔ ð1/2ÞðU ± U T Þ, and we deduce from (21) that where ∇ðð1/ρÞ∇pÞ ± = ð1/2Þ½∇ðð1/ρÞ∇pÞ ± ð∇ðð1/ρÞ∇pÞÞ T . Now, we turn to the gradient of the pressure. According to (3) with γ > 1, we find that Define the flow line Xðt, aÞ starting from a by Then, we conclude from the first and third equations in where c 0 is a positive constant. On the other hand, due to the Hypothesis HE on U ð0, a 1 Þ and the fact that equation (23) is homogenous in U , we obtain U ðt, Xðt, a 1 ÞÞ = 0. Thereafter, we have proved that This, together with Let y 1 ðtÞ ≔ −ð1/nÞe Ð t 0 αðτÞdτ div uðt, Xðt, a 1 ÞÞ,; then, we find that

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This means y 1 ðtÞ ≥ y 1 ð0Þ > 0 and Note that there exists Cðt * E Þ > 0 such that for any Then, we infer that div uðt, Xðt, a 1 ÞÞ blows up before or at t * E . This completes the Proof of Theorem 1.

Blowup of the Euler-Poisson System with Time-Dependent Damping
In the present section, we turn our attention to the Proof of Theorem 3.

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 B/ρ 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 d dt div u t, X t, a 3 ð Þ ð Þ≤ − 1 n div u ð Þ 2 t, X t, a 3 ð Þ ð Þ , ð43Þ 5 Advances in Mathematical Physics which yields div u t, X t, a 3 ð Þ ð Þ≤ n div u 0 a 3 ð Þ n + t div u 0 a 3 ð Þ ⟶ −∞,

Data Availability
No data were used to support this study.

Conflicts of Interest
The authors declare that they have no conflicts of interest.