Oct 5, 2010 - arXiv:1001.0380v3 [math.AP] 5 Oct 2010. Blowup for the Euler ... On the other hand, the Poisson equation (2)3 can be solved as. Î¦(t, x)...

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Blowup for the Euler and Euler-Poisson Equations with Repulsive Forces Manwai Yuen∗ Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong Revised 05-Oct-2010

Abstract In this paper, we study the blowup of the N -dim Euler or Euler-Poisson equations with repulsive forces, in radial symmetry. We provide a novel integration method to show that the non-trivial classical solutions (ρ, V ), with compact support in [0, R], where R > 0 is a positive constant and in the sense which ρ(t, r) = 0 and V (t, r) = 0 for r ≥ R, under the initial condition H0 =

Z

R

rV0 dr > 0,

(1)

0

blow up on or before the finite time T = R3 /(2H0 ) for pressureless fluids or γ > 1. The main contribution of this article provides the blowup results of the Euler (δ = 0) or Euler-Poisson (δ = 1) equations with repulsive forces, and with pressure (γ > 1), as the previous blowup papers ([1] [2], [3] and [4]) cannot handle the systems with the pressure term, for C 1 solutions. Key Words: Euler Equations, Euler-Poisson Equations, Integration Method, Blowup, Repulsive Forces, With Pressure, C 1 Solutions, No-Slip Condition ∗ E-mail

address: [email protected]

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M.W.Yuen

Introduction

The isentropic Euler (δ = 0) or Euler-Poisson (δ = ±1) equations can be written in the following form:

ρt +∇ · (ρu) =0,

ρ[ut + (u · ∇)u]+∇P =ρ∇Φ, ∆Φ(t, x) = δα(N )ρ,

(2)

where α(N ) is a constant related to the unit ball in RN : α(1) = 1, α(2) = 2π and α(3) = 4π. And as usual, ρ = ρ(t, x) ≥ 0 and u = u(t, x) ∈ RN are the density and the velocity respectively. P = P (ρ) is the pressure function. The γ-law can be applied on the pressure term P (ρ), i.e. P (ρ) = Kργ ,

(3)

which is a common hypothesis. If the parameter is set as K > 0, we call the system with pressure; if K = 0, we call it pressureless. The constant γ = cP /cv ≥ 1, where cP , cv are the specific heats per unit mass under constant pressure and constant volume respectively, is the ratio of the specific heats, that is, the adiabatic exponent in the equation (3). In particular, the fluid is called isothermal if γ = 1. If K > 0, we call the system with pressure; if K = 0, we call it pressureless. In the above systems, the self-gravitational potential field Φ = Φ(t, x) is determined by the density ρ itself, through the Poisson equation (2)3 . When δ = −1, the system can model fluids that are self-gravitating , such as gaseous stars. In addition, the evolution of the simple cosmology can be modelled by the dust distribution without pressure term. This describes the stellar systems of collisionless and gravitational n-body systems [5]. And the pressureless Euler-Poisson equations can be derived from the Vlasov-PoissonBoltzmann model with the zero mean free path [6]. For N = 3 and δ = −1, the equations (2) are the classical (non-relativistic) descriptions of a galaxy in astrophysics. See [7] and [8], for details about the systems. When δ = 1, the system is the compressible Euler-Poisson equations with repulsive forces. The equation (2)3 is the Poisson equation through which the potential with repulsive forces is determined by the density distribution of the electrons. In this case, the system can be viewed as a

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Blowup for Euler and EP Equations

semiconductor model. See [9] and [10] for detailed analysis of the system. On the other hand, the Poisson equation (2)3 can be solved as Φ(t, x) = δ

Z

G(x − y)ρ(t, y)dy,

(4)

RN

where G is Green’s function for the Poisson equation in the N -dimensional spaces defined by |x|, N = 1; . (5) G(x) = log |x|, N = 2; −1 |x|N −2 , N ≥ 3. Usually, the Euler-Poisson equations can be rewritten in the scalar form: N N ∂ρ k ∂ρ Σ uk ∂x + ρ Σ ∂u =0, ∂t + k=1 k k=1 ∂xk N ∂ui ∂P ∂Φ i ρ ∂u + ∂x = ρ ∂x , for i = 1, 2, ...N. ∂t + Σ uk ∂xk i i

(6)

k=1

For the construction of the analytical solutions for the systems, interested readers should refer

to [11], [12], [13], [14] and [15]. The results for local existence theories can be found in [16], [17] and [18]. The analysis of stabilities for the systems may be referred to [19], [20], [21], [1], [2], [3], [22], [13], [23], [24], [4] and [25]. We seek the radial symmetry solutions ρ(t, ~x) = ρ(t, r) and ~u = with the radius r =

P

N i=1

x2i

1/2

~x ~x V (t, r) =: V , r r

(7)

.

For the solutions in spherical symmetry, the Poisson equation (2)3 is transformed to rN −1 Φrr (t, x) + (N − 1) rN −2 Φr =α (N ) δρrN −1 ,

Φr =

α (N ) δ rN −1

Z

(8)

r

ρ(t, s)sN −1 ds.

(9)

0

By standard computation, the Euler-Poisson equations in radial symmetry can be written in the following form: ρt + V ρr + ρVr + N − 1 ρV = 0, r ρ (Vt + V Vr ) + Pr (ρ) = ρΦr (ρ) .

(10)

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M.W.Yuen

Historically, Makino, Ukai and Kawashima initially defined the tame solutions [1] for outside the compact of the solutions Vt + V Vr = 0.

(11)

Following this, Makino and Perthame considered the tame solutions for the system with gravitational forces [2]. After that Perthame discovered the blowup results for 3-dimensional pressureless system with repulsive forces [3] (δ = 1). In short, all the results above rely on the solutions with radial symmetry: α(N )δ Vt + V Vr = N −1 r

Z

r

ρ(t, s)sN −1 ds.

(12)

0

And the Emden ordinary differential equations were deduced on the boundary point of the solutions with compact support: D2 R δM ˙ = N −1 , R(0, R0 ) = R0 ≥ 0, R(0, R0 ) = 0, Dt2 R where

dR dt

(13)

:= V and M is the mass of the solutions, along the characteristic curve. They showed

the blowup results for the C 1 solutions of the system (10). Recently, Chae and Tadmor [4] showed the finite time blowup, for the pressureless Euler-Poisson equations with attractive forces (δ = −1), under the initial condition, S := { a ∈ RN ρ0 (a) > 0, Ω0 (a) = 0, ∇ · u(0, x(0) < 0} 6= φ,

(14)

where Ω is the rescaled vorticity matrix (Ω0 ij ) = 12 (∂i uj0 −∂j ui0 ) with the notation u = (u1 , u2 , ...., uN ) in their paper and some point x0 . They use the analysis of spectral dynamics to show the Racatti differential inequality, 1 D div u ≤ − (div u)2 . Dt N

(15)

The solution for the inequality (15) blows up on or before T = −N/(∇ · u(0, x0 (0)). However, their method cannot be applied to the system with repulsive forces to obtain the similar blowup result. On the other hand, in [24], we have the blowup results if the solutions with compact support under the condition, 2

Z

Ω(t)

2

(ρ |u| + 2P )dx < M 2 − ǫ,

(16)

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Blowup for Euler and EP Equations

where M is the mass of the solution. In this article, the alternative approach is adopted to show that there is no global existence of C 1 solutions for the system, (6) (δ = 0 or δ = 1), with compact support without the condition (14). We notice that the conditions in our result are different from the works of Engerlberg et. al [26]. Theorem 1 Consider the N -dimensional Euler (δ = 0) or Euler-Poisson equations with repulsive forces (δ = 1) (2). The non-trivial classical solutions (ρ, V ), in radial symmetry, with compact support in [0, R], where R > 0 is a positive constant (which ρ(t, r) = 0 and V (t, r) = 0 for r ≥ R) and the initial velocity such that: H0 =

Z

R

rV0 dr > 0,

(17)

0

blow up on or before the finite time T = R3 /(2H0 ), for pressureless fluids (K = 0) or γ > 1. The solutions (ρ, u) may lose their regularity, for example the velocity function V ∈ C 0 only or the shock waves appear on or before the finite time T .

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Integration Method

In this section, we present the proof of Theorem 1. The technique of the proof was selected simply to deduce the partial differential equations to the Racatti equation, to show the blowup result. However, we note our integration method is novel to the studies of blowup for this kind of the systems. Proof. In general, we show that the ρ(t, x(t; x)) preserves its positive nature as the mass equation (6)1 can be converted to be Dρ + ρ∇ · u = 0, Dt

(18)

∂ D = + (u · ∇) . Dt ∂t

(19)

Z t ∇ · u(t, x(t; 0, x0 ))dt ≥ 0, ρ(t, x) = ρ0 (x0 (0, x0 )) exp −

(20)

with the material derivative,

We integrate the equation (18):

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for ρ0 (x0 (0, x0 )) ≥ 0, along the characteristic curve. We use the momentum equation (10)2 with the non-trivial solutions in radial symmetry, ρ0 6= 0, to have: Vt + V Vr + Kγργ−2ρr = Φr , ∂ 1 2 ( V ) + Kγργ−2ρr = Φr , ∂r 2 ∂ 1 rVt + r ( V 2 ) + Kγrργ−2 ρr = rΦr , ∂r 2 Vt +

(21) (22) (23)

with multiplying r on the both sides. We take integration with respect to r, to the above equation, for γ > 1 or K ≥ 0: Z R Z R d 1 2 γ−2 rVt dr + Kγrρ ρr dr = r ( V )+ rΦr dr, dr 2 0 0 0 0 Z R Z R Z R Z R Z α(N )δr r d 1 2 Kγr γ−1 N −1 r ( V )+ dρ = rVt dr + ρ(t, s)s ds dr, dr 2 rN −1 0 0 γ−1 0 0 0 Z R Z R Z R Kγr γ−1 d 1 2 dρ ≥ 0, rVt dr + r ( V )+ dr 2 γ −1 0 0 0 Z

R

Z

R

(24) (25) (26)

for δ ≥ 0. It follows with integration by part: Z

0

R

rVt dr−

1 2

Z

R

V 2 dr+ 0

1 RV (t, R)2 − 0 · V (t, 0)2 − 2

Z

R

0

Kγ γ−1 Kγ γ−1 Rρ (t, R) − 0 · ργ−1 (t, 0) ≥ 0. ρ dr+ γ−1 γ−1 (27)

The above inequality with the boundary compact condition of V (t, R) = 0 and ρ(t, R) = 0, becomes Z

R

rVt dr −

0

1 2

Z

R

V 2 dr −

0

Z

R 0

Kγ γ−1 ρ dr = 0. γ−1

(28)

As r and t are independent variables and V is C 1 in the domain [0, R] in the assumption of the theorem, we may change the differentiation and the integration as the following: Z Z Z R d R 1 R 2 Kγ γ−1 ρ dr ≥ 0, rV dr − V dr − dt 0 2 0 γ −1 0 Z R Z Z d 1 R Kγ γ−1 1 R 1 2 2 2 V dr ≥ ρ dr ≥ 0, V dr − dt 2 0 2 0 2r γ −1 0

(29) (30)

for γ > 1 or K = 0. For the non-trivial initial condition ρ0 ≥ 0, we have the following differential inequality: d 1 dt 2

Z

0

R

V dr2 −

1 2

Z

0

R

1 2 2 V dr ≥ 0, 2r

(31)

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d dt

Z

R

0

R

Z R 1 1 2 2 V dr ≥ V 2 dr2 , 2r 2R 0 0 Z R Z R 1 d V dr2 ≥ V 2 dr2 . dt 0 2R 0 Z

V dr2 ≥

(32) (33)

By denoting H := H(t) =

R

Z

1 rV dr = 2

0

R

Z

V dr2 ,

(34)

0

and with the Cauchy-Schwarz inequality, Z R V · 1dr2 ≤ 0

!1/2

V 2 dr2

0

R R 0 V dr2 R

R

Z

R

Z

≤

V 2 dr2

0

4H 2 ≤ R2

Z

R

Z

R

1dr2

0

!1/2

!1/2

,

,

V 2 dr2 ,

(35)

(36)

(37)

0

1 2H 2 ≤ R3 2R

Z

R

V 2 dr2 ,

(38)

0

the inequality (32) becomes d 1 H≥ dt 2R

Z

R

V 2 dr2 ≥

0

2H 2 , R3

2H 2 d H≥ 3 . dt R With the initial condition: H0 =

RR 0

(39) (40)

rV0 dr > 0, we can obtain H≥

−R3 H0 . 2H0 t − R3

(41)

Therefore, the solutions blow up on or before the finite time T = R3 /(2H0 ). This completes the proof. Remark 2 For controlled experiments in engineering, fluids are kept in a fixed ball solid container with a radial R. Therefore, it requires the compact support condition for t ≥ 0, ρ(t, r) = 0 and V (t, r) = 0,

(42)

with r ≥ R. This corresponding condition is called no-slip condition (solid boundary condition) [27] and [28]. On the other hand, in computing simulations, the systems are usually coupled with the similar

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boundary conditions for real applications. Therefore, the condition for compact support (no-slip condition) is reasonable in modelings. But for free boundary problems, fluids may not be bounded by a fixed volume for all time. Therefore, further research is needed to study the corresponding result in future works. Remark 3 It is still an open question whether or not there exists time-local C 1 -solution with compact support for any initial condition with compact support. On the other hand, if the global solutions with compact support whose radii expand unboundedly as time tends to infinity, the discussion of this paper can offer no information about this case.

Remark 4 This article has shed new light on situations with the pressure term. In particular, it provides the blowup results of the Euler (δ = 0) or Euler-Poisson (δ = 1) equations with repulsive forces, and with pressure (γ > 1). This is the main contribution of the article, as the previous blowup papers ([1] [2], [3] and [4]) cannot handle the systems with the pressure term, for C 1 solutions. A further refinement for the non-radial symmetry is expected in future studies.

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Acknowledgement

The author would like to thank the comments of reviewers to improve the article.

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[3] B. Perthame, Nonexistence of Global Solutions to Euler-Poisson Equations for Repulsive Forces, Japan J. Appl. Math. 7 (1990), 363–367. [4] D. H. Chae and E. Tadmor, On the Finite Time Blow-up of the Euler-Poisson Equations in RN , Commun. Math. Sci. 6 (2008), 785–789. [5] H. H. Fliche and R. Triay, Euler-Poisson-Newton Approach in Cosmology, Cosmology and Gravitation, 346–360, AIP Conf. Proc., 910, Amer. Inst. Phys., Melville, NY, 2007. [6] R. T. Glassey, The Cauchy Problem in Kinetic Theory, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1996. [7] J. Binney and S. Tremaine, Galactic Dynamics, Princeton Univ. Press, 1994. [8] S. Chandrasekhar, An Introduction to the Study of Stellar Structure, Univ. of Chicago Press, 1939. [9] C. F. Chen, Introduction to Plasma Physics and Controlled Fusion, Plenum, New York (1984). [10] P.L. Lions, Mathematical Topics in Fluid Mechanics. Vols. 1, 2, 1998, Oxford: Clarendon Press, 1998. [11] P. Goldreich and S. Weber, Homologously Collapsing Stellar Cores, Astrophys, J. 238 (1980), 991-997. [12] T. Makino, Blowing up Solutions of the Euler-Poission Equation for the Evolution of the Gaseous Stars, Transport Theory and Statistical Physics 21 (1992), 615–624. [13] Y.B. Deng, J.L. Xiang and T. Yang, Blowup Phenomena of Solutions to Euler-Poisson Equations, J. Math. Anal. Appl. 286 (2003), 295–306. [14] T.H. Li, Some Special Solutions of the Multidimensional Euler Equations in RN , Comm. Pure Appl. Anal. 4 (2005), 757–762. [15] M.W. Yuen, Analytical Blowup Solutions to the 2-dimensional Isothermal Euler-Poisson Equations of Gaseous Stars, J. Math. Anal. Appl. 341 (2008), 445–456.

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[16] T. Makino, On a Local Existence Theorem for the Evolution Equation of Gaseous Stars, Patterns and waves, 459–479, Stud. Math. Appl., 18, North-Holland, Amsterdam, 1986. [17] M. Bezard, Existence locale de solutions pour les equations d’Euler-Poisson (Local Existence of Solutions for Euler-Poisson Equations), Japan J. Indust. Appl. Math. 10 (1993), 431–450 (in French). [18] P. Gamblin, Solution reguliere a temps petit pour l’equation d’Euler–Poisson (Small-time Regular Solution for the Euler-Poisson Equation), Comm. Partial Differential Equations 18 (1993), 731–745 (in French). [19] T.C. Sideris, Formation of Singularities in Three-dimensional Compressible Fluids, Comm. Math. Phys. 101 (1985), 475–485. [20] S. Alinhac, Blowup for Nonlinear Hyperbolic Equations. Progress in Nonlinear Differential Equations and their Applications, 17. Birkh¨aser Boston, Inc., Boston, MA, 1995. [21] S. Engelberg, Formation of Singularities in the Euler and Euler-Poisson Equations, Phys. D, 98, 67–74, 1996. [22] Y.B. Deng, T.P. Liu, T. Yang and Z.A. Yao, Solutions of Euler-Poisson Equations for Gaseous Stars, Arch. Ration. Mech. Anal. 164 (2002) 261–285. [23] J. Jang, Nonlinear Instability in Gravitational Euler-Poisson Systems for γ = 6/5, Arch. Ration. Mech. Anal. 188 (2008), 265–307. [24] M.W. Yuen, Stabilities for Euler-Poisson Equations in Some Special Dimensions, J. Math. Anal. Appl. 344 (2008), 145–156. [25] D. H. Chae and S. Y. Ha, On the Formation of Shochs to the Compressible Euler Equations, Commun. Math. Sci. 7 (2009), 627–634. [26] S. Engelberg, H.L. Liu and E. Tadmor, Critical Thresholds in Euler-Poisson Equations, Indiana Univ. Math. J. 50 (2001), 109–157.

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[27] M. A. Day, The No-slip Condition of Fluid Dynamics, Erkenntnis 33 (1990), 285–296. [28] Y. A. Cengel, R. H. Truner and J. M. Cimbala, Fundamentals of thermal-Fluid Sciences, 3rd edition, McGraw-Hill Higher Education, New York, 2008.