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

Authors: Wenming Hu
Journal: Advances in Mathematical Physics
 (2020)

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## 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.


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## Body

## 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.(1)∂tρ+divρu=0conservationofmass,∂tρu+divρu⊗u+∇p=−αtρuconservationofmomentum,∂tρe+12ρu2+divuρe+12ρu2+p=−αtρu2conservationofenergy,ρ0,x=ρ0x,u0,x=u0x,S0,x=S0x,where x∈ℝnn≥1, ∇=∂1,⋯,∂n, and ρ,u=u1,⋯,un, p=pρ,S, e=eρ,S, and S stand for the density, velocity, pressure, inner energy, and entropy, respectively. The coefficient of the damping term −αtρu in the second equation of (1) verifies αt≥0 (when αt≡0, it means that there is no damping in the system (1)). The inner energy eρ,S satisfies
(2)de=TdS+pρ2dρ,where T>0 is the temperature. The equation of state is given by
(3)pρ,S=p0ργeS/cν,with p0>0 and the adiabatic exponent γ=cp/cν>1 (γ≈1.4 for the air). The positive constants cp and cν are the specific heat at constant pressure and volume, respectively. For convenience, let cν=1.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 αt≡1 and γ∈2∪3,∞. 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 αt≥0.Next, we study the pressureless Euler-Poisson system with time-dependent damping inℝnn≥2.
(4)∂tρ+divρu=0,∂tρu+divρu⊗u=δρ∇Φ−αtρu,ΔΦ=ρ−mtotal,ρ0,x=ρ0x,u0,x=u0x,where Φt,x and mtotal=∫ρ0xdx are the electrostatic potential and total mass, respectively. δ=0 coincides with the pressureless Euler system. δ=1 stands for repulsive forces and attractive forces with δ=−1.The results for existence theories can be found in [11–14]. For the blowup results for compressible Euler-Poisson equations without time-dependent damping αt≡0, 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 αt≥0 and attractive forces, under suitable initial conditions.At last, we are concerned with the classical solutions to the 2D ideal compressible transverse MHD flow. Letx∈ℝ3, and the 3D compressible isentropic MHD system reads
(5)∂tϱ+divxϱu=0,∂tϱu+divxϱu⊗u+∇xPϱ=curlxB×B=−12∇xB+B⋅∇xB,∂tB=curlxu×B=−u⋅∇xB−Bdivxu+B⋅∇xu,divxB=0,ϱ0,x=ϱ0x,u0,x=u0x,B0,x=B0x,where ϱ, u=u1,u2,u3, p=pϱ=p0ϱγ, and B=B1,B2,B3 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:
(6)ρt,x=ρt,x1,x2,ut,x=u1t,x1,x2,u2t,x1,x2,0,Bt,x=0,0,Bt,x1,x2.The divergence-free condition of the magnetic fieldB is naturally fulfilled. Let x=x1,x2, ∇=∂1,∂2, and u=u1,u2; then, (5) with the structural assumption (6) can be simplified as the following 2D hyperbolic system:
(7)∂tρ+divρu=0,∂tρu+divρu⊗u+∇pρ+12B2=0,∂t+u⋅∇B+Bdivu=0,ρ0,x=ρ0x,u0,x=u0x,B0,x=B0x.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 Section2, 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.HypothesisHE.
There existsa1∈ℝn such that
(8)∑0≤ℓ≤2∇ℓρ0γ−1a1+∑ℓ=1,2∇ℓS0a1=0,S0a1<∞,divu0a1≤−n∫0t∗e−∫0ταsdsdτ−1,forsomet∗>0,∂iu0j−∂ju0ia1=0,i,j=1,⋯,n.Define the first blowup timetE∗ by
(9)∫0tE∗e−∫0ταsdsdτ=infa1−ndivu0a1,where the infimum taking over all the a1 verifies the Hypothesis HE. We assume that the infimum is reached at a~1, and for convenience, we omit the tilde above a1. The aforementioned two conventions also work in the following part. It is easy to check that tE∗ 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 tE∗>0.Theorem 1.
Letρ,u,S be the C3-smooth solution to the system (1) with γ>1, αt≥0, and the initial data ρ0,u0,S0. Suppose that ρ0,u0,S0 satisfies the Hypothesis HE; then, the classical solution will blow up before or at tE∗.Remark 2.
Now, we give the conditions ondivu0 in Hypothesis HE and the blowup time tE∗ for some special choices of αt. It is not hard to check that
(10)αt≡0,divu0a1<0,tE∗=infa1−ndivu0a1,αt≡1,divu0a1<−n,tE∗=−suplna11+ndivu0a1.
The condition ondivu0 and the blowup time in [10] are divu0a1<−n−1 and T∗≤infa1nn+1/divu0a1. Choose infa1divu0a1=−2n; then, our bound is strictly less than their bound tE∗=ln2<n+1/2. In addition, for αt=α/1+t with nonnegative constant α, we achieve
(11)α<1,divu0a1<0,tE∗=infa11−n1−αdivu0a11/1−α−1,α=1,divu0a1<0,tE∗=infa1e−n/divu0a1−1,α>1,divu0a1<−nα−1,tE∗=infa11+nα−1divu0a1−1/α−1−1.We also make some assumptions on the initial data of the Euler-Poisson system (4).HypothesisHEP.
There existsa2∈ℝn such that
(12)ρ0a2>0,divu0a2<−n12α0+m0,∂iu0j−∂ju0ia2=0,i,j,=1,⋯,n,tEP∗≔12m0infa2lnnm0−n/2αt−divu0a2−nm0−n/2αt−divu0a2>0,where the constant m0 is defined by
(13)m0≔1nmtotal+14α20>0.The second result in the present paper is stated in the following.Theorem 3.
Letρ,u,Φ be the smooth solution to the system (4) with δ=−1, αt≥0, α′t≤0, and the initial data ρ0,u0. Suppose that ρ0,u0 satisfies the Hypothesis HEP; then, singularity will be developed before or at tEP∗.At last, we turn to our last result. The initial data of the MHD system (7) is assumed to verify.HypothesisHMHD.
There existsa3∈ℝ2 such that
(14)B0xρ0x<∞,∀x∈ℝ2,∑0≤ℓ≤2∇ℓρ0a3+∑ℓ=1,2∇ℓρ0γ−1a3+∇ℓB0a3ρ0a3=0,∂1u02−∂2u01a3=0,divu0a3<0,tMHD∗≔infa3−ndivu0a3>0.Theorem 4.
Letρ,u,B be the smooth solution to the system (7) with the initial data ρ0,u0,B0. Suppose that ρ0,u0,B0 satisfies the Hypothesis HMHD with γ>1; then, the solution will blowup before or at tMHD∗.Remark 5.
The special caseB≡0 in (5) coincides with the isentropic Euler system (1).Remark 6.
Ifρt,x>0, the first and third equations in (7) derive the following well-known “frozen” law:
(15)∂t+u⋅∇Bρ=0.
Under the assumptionB0x/ρ0x<∞, in the Hypothesis HMHD, it is natural to define Bt,x=0 when ρt,x=0. For the special case B0x/ρ0x≡1, Bt,x≡ρt,x holds for any t>0 when the smooth solution exists.Remark 7.
We give some examples of the HypothesisHMHD. Near the point a3
(16)ρ0x=x−a33γ−1/2,γ≥3,x−a36/γ−1,1<γ<3.Remark 8.
The sign conditions ofdivu0 in (8) and (14) are necessary. If the spectrum of ∇u0 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 Theorem1. Before the proof, a new reformulation of (1) is achieved as follows.Lemma 9.
Under the assumption thatρ,u,S∈C1, the equations in (1) can be rewritten as
(17)Dtρ+ρdivu=0,Dtu+1ρ∇p=−αtu,DtS=0,where Dt≔∂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
(18)∂t12ρu2+div12ρuu2+u⋅∇p=−αtu2.Subtracting the third equation in (1) by the above identity, then
(19)∂tρe+divρue+pdivu=0.This, together with (2), derives
(20)ρ∂t+u⋅∇S+pρ∂t+u⋅∇ρ+ρdivu=ρ∂t+u⋅∇S=0,which implies DtS=0 in t,xρt,x>0. Noting the C1-smoothness of S, it is natural to define DtS=0 in t,xρt,x=0. This completes the proof of Lemma 9.Proof of Theorem 1.
DenoteU≔∇u and apply ∇ to the second equation in (17); then, we achieve
(21)DtU=−UU−∇1ρ∇p−αtU.
LetU±≔1/2U±UT, and we deduce from (21) that
(22)DtU+=−U+U+−U−U−−∇1ρ∇p+−αtU+,(23)DtU−=−U+U−−U−U+−∇1ρ∇p−−αtU−,where ∇1/ρ∇p±=1/2∇1/ρ∇p±∇1/ρ∇pT. Now, we turn to the gradient of the pressure. According to (3) with γ>1, we find that
(24)∇1ρ∇p=∇1ρ∇p0ργeS=p0∇eS1ρ∇ργ+p0∇ργ−1∇eS=p0γ−1∇eS∇ργ−1+p0∇eS∇ργ−1+p0ργ−1∇2eS=p0eSγ−1∇2ργ−1+p0γγ−1∇eS∇ργ−1+p0ργ−1∇2eS.
Define the flow lineXt,a starting from a by
(25)ddtXt,a=ut,Xt,a,X0,a=a.
Then, we conclude from the first and third equations in (17) that
(26)∑0≤ℓ≤2∇ℓργ−1t,Xt,a1≤∑0≤ℓ≤2∇ℓρ0γ−1a1ec0∑1≤ℓ≤3∫0t∇ℓuτ,Xτ,a1dτ,∑ℓ=1,2∇ℓSt,Xt,a1≤∑ℓ=1,2∇ℓS0a1ec0∑1≤ℓ≤2∫0t∇ℓuτ,Xτ,a1dτ,where c0 is a positive constant. On the other hand, due to the Hypothesis HE on U_0,a1 and the fact that equation (23) is homogenous in U_, we obtain U_t,Xt,a1=0.
Thereafter, we have proved that(27)ddtU+t,Xt,a1≤−U+U+t,Xt,a1−αtU+t,Xt,a1.
This, together with(28)traceU+U+≥1ntraceU+2=1ndivu2,implies
(29)ddtdivut,Xt,a1≤−1ndivu2t,Xt,a1−αtdivut,Xt,a1.
Lety1t≔−1/ne∫0tατdτdivut,Xt,a1,; then, we find that
(30)y1′t≥e−∫0tατdτy12t≥0.
This meansy1t≥y10>0 and
(31)y1t≥y101−y10∫0te−∫0ταsdsdτ⟶+∞,t⟶tE∗t<tE∗.
Note that there existsCtE∗>0 such that for any t∈0,tE∗
(32)1CtE∗≤e∫0tατdτ≤CtE∗.
Then, we infer thatdivut,Xt,a1 blows up before or at tE∗. 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 Theorem3.Proof of Theorem 3.
Similar to (22) and (23), we can obtain
(33)DtU+=−U+U+−U−U−−∇2Φ−αtU+,(34)DtU−=−U+U−−U−U+−αtU−.
By taking the trace of the two sides of equation (33), we find that
(35)Dtdivu=−traceU+U+−traceU_U_−ΔΦ−αtdivu≤−1ndivu2−traceU_U_−ρ−mtotal−αtdivu,where we have used (28) again. We conclude from the Hypothesis HEP and (34) that U_t,Xt,a2=0.
Lety2t≔1/ndivut,Xt,a2+1/2αt; then, it is easy to check that
(36)y2′t≤−y22t+1nmtotal+14α2t+12α′t≤−y22t+1nmtotal+14α20=−y22t+m02,where we have used the fact α′t≤0, and m0 is defined in (13). Now, we make the ansatz that before the formation of singularity
(37)y2t<−m0<0,which derives y22t>m02. Thereafter, we find that
(38)ddtlnm0−y2t−m0−y2t=2m0y2′ty22t−m02≤−2m0,
Integrating the above differential inequality yields(39)1+2m0−m0−y2t=m0−y2t−m0−y2t≤e−2m0tm0−y20−m0−y20.This, together with m0−y20/−m0−y20=1+2nm0/−nm0−n/2α0−divu0a2>1 which follows from the Hypothesis HEP, implies
(40)−m0−y2t≥2m0e−2m0tm0y20/−m0y20−1⟶+∞,t⟶tEP∗t<tEP∗.
According to the boundedness ofsup0,tEP∗αt, we achieve that divut,Xt,a2 blows up before or at tEP∗. On the other hand, we also conclude from (40) that −m0−y2t>0. 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 Theorem4.Proof of Theorem 4.
The Proof of Theorem4 is analogous with that of Theorem 1. Instead of (22) and (23), we achieve
(41)DtU+=−U+U+−U_U_−∇1ρ∇p+12B2+,DtU_=−U+U_−U_U+−∇1ρ∇p+12B2_.
According to the “frozen” law, it is convenient to regardB/ρ as a new unknown. Consequently, similar to (24), we find that
(42)∇1ρ∇p+12B2=p0γ−1∇2ργ−1+∇1ρ∇B22ρ2ρ2=p0γ−1∇2ργ−1+∇B2ρ2ρ+∇ρBρ∇Bρ=p0γ−1∇2ργ−1+B2ρ2∇2ρ+3Bρ∇ρ∇Bρ+ρ∇Bρ2+B∇2Bρ.
By an analogous argument as in Section2 and the Hypothesis HMHD, we finally derive
(43)ddtdivut,Xt,a3≤−1ndivu2t,Xt,a3,which yields
(44)divut,Xt,a3≤ndivu0a3n+tdivu0a3⟶−∞,t⟶tMHD∗=infa3−ndivu0a3.



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