Line Solutions for the Euler and Euler-Poisson Equations with Multiple Gamma Law

Abstract

In this paper, we study the Euler and Euler-Poisson equations in $R^{N}$,
with multiple $\gamma$-law for pressure function: \begin{equation}
P(\rho)=e^{s}\sum_{j=1}^{m}\rho^{\gamma_{j}}, \end{equation} where all
$\gamma_{i+1}>\gamma_{i}\geq1$, is the constants. The analytical line solutions
are constructed for the systems. It is novel to discover the analytical
solutions to handle the systems with mixed pressure function. And our solutions
can be extended to the systems with the generalized multiple damping and
pressure function.

Full text

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arXiv:1005.3651v1 [math-ph] 20 May 2010
Line Solutions for the Euler and Euler-Poisson
Equations with Multiple Gamma Law
Ling Hei Yeung∗
Manwai Yuen†
Department of Applied Mathematics, The Hong Kong Polytechnic University,
Hung Hom, Kowloon, Hong Kong
Revised 20-May-2010
Abstract
In this paper, we study the Euler and Euler-Poisson equations in RN, with multiple γ-law
for pressure function:
P(ρ) = es
m
?
j=1
ργj,
(1)
where all γi+1 > γi ≥ 1, is the constants. The analytical line solutions are constructed for
the systems. It is novel to discover the analytical solutions to handle the systems with mixed
pressure function. And our solutions can be extended to the systems with the generalized
multiple damping and pressure function.
Key Words: Multiple Gamma Law, Euler Equations, Euler-Poisson Equations, Analyt-
ical Solutions, Navier-Stokes Equations, Global Solutions, External forces, Free Boundary,
Multiple Damping
∗E-mail address: lightisgood2005@yahoo.com.hk
†E-mail address: nevetsyuen@hotmail.com
1

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Yeung and Yuen
1Introduction
The N-dimensional Euler equations can be formulated as the follows:























ρt+∇ · (ρ? u) =0,
ρ[? ut+ (? u · ∇)? u] + ∇P(ρ)= −δρ ▽ Φ − ρ?F(t),
St+ ? u · ∇S = 0,
∆Φ(t,? x) = α(N)ρ,
(2)
with ? x = (x1,x2,...,xN) ∈ RN,
and ρ = ρ(t,? x), ? u(t,? x) ∈ RNand S = S(t) are the density, the velocity and the entropy respectively.
And α(N) is a constant related to the unit ball in RN: α(1) = 2; α(2) = 2π and for N ≥ 3,
α(N) = N(N − 2)V ol(N) = N(N − 2)
πN/2
Γ(N/2 + 1), (3)
where V ol(N) is the volume of the unit ball in RNand Γ is a Gamma function.
When δ = 1, the system can model fluids that are self-gravitating , such as gaseous stars. For
N = 3, the equations (2) are the classical (non-relativistic) descriptions of a galaxy in astrophysics.
See [1] and [2], for details about the systems.
When δ = −1, the system is the compressible Euler-Poisson equations with repulsive forces. The
equation (2)4is the Poisson equation through which the potential with repulsive forces is deter-
mined by the density distribution of the electrons. In this case, the system can be viewed as a
semiconductor model. See [3] and [10] for detailed analysis of the system.
When δ = 0, the potential forces are ignored. The system is called the Euler equations. See [4]
and [13] for detailed analysis of the system.
Here P = P(ρ) is the pressure, the γ-law on the pressure for the single gas, i.e.
P(ρ) = esργ, (4)
is a universal hypothesis. The constant γ = cP/cv≥ 1, where cpand cvare the specific heats per
unit mass under constant pressure and constant volume respectively, is the ratio of the specific
heats. In particular, the fluid is called isothermal if γ = 1. More generally, the pressure function

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Line Solutions to Euler and Euler-Poisson Equations
3
of mixed gases, can be expressed by the multiple γ-law ([11], [14], and [15]):
P(ρ) = P1+ P2+ ...PN= es
m
?
j=1
ργj, (5)
where all γj+1> γj≥ 1, is the constant, m is a positive integer.
The system with the multiple γ-law can reflect the better approximation of the real situations.
For example, the fluids in the stars mix with many types of gases in models of gaseous stars in
astrophysics [15].
Here the time-dependent external force?F(t) = (F1(t),F2(t),...,FN(t)) ∈ C0are coupled in the
systems.
The system (2) can be rewritten in scalar form,















∂ρ
∂t+?N
+
∂xi
k=1uk
∂ρ
∂xk+ ρ?N
j=1ργj?
k=1uk
k=1
∂uk
∂xk=0,
ρ
?
∂ui
∂t+?N
k=1uk∂ui
∂xk
?
∂
?
St+?N
eS?m
= −ρδ∂
∂xiΦ + ρFi(t), for i = 1,2,...N,
∂S
∂xk= 0.
(6)
For the Euler equations (6), (δ = 0), with?F(t) = 0, in radial symmetry:
ρ(t,? x) = ρ(t,r) and ? u =? x
rV (t,r) :=? x
rV,
(7)
with r =
??N
i=1x2
i
?1/2
,
there exists a family of solutions for the Euler equations (6), (δ = 0), without the external force
(?F = 0), for γ > 1, [9]























ρ(t,r) =







y(r/a(t))1/(γ−1)
a(t)N
, for y(
r
a(t)) ≥ 0;
0, for y(
r
a(t)) < 0
, V (t,r) =
·a(t)
a(t)r, S(t,r) = lnK,
··a(t) =
−λ
a(t)1+N(γ−1), a(0) = a0> 0,
·a(0) = a1,
y(x) =(γ−1)λ
2γKx2+ αθ−1,
;(8)
where K is a positive number,
for γ = 1, [18]















ρ(t,r) =ey(r/a(t))
a(t)N ,V (t,r) =
·a(t)
a(t)r, S(t,r) = lnK
··a(t) =
−λ
a(t),a(0) = a0> 0,
·a(0) = a1,
y(x) =
λ
2Kx2+ α,
(9)

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4
Yeung and Yuen
where λ, α, a0and a1are constants.
The separation method for analytical solutions, were used to handle other similar systems with
single γ functions:
P(ρ) = esργ, (10)
or without pressure function, in [7], [5], [9], [12], [16], [17], [18], [19] and [20].
It is very natural to extend the results for the system with multiple γ function (6):
P(ρ) = es
m
?
j=1
ργj.(11)
In this article, we have obtained a class of line solutions to the Euler equations (6) (δ = 0) and
Euler-Poisson equations (6) (δ = ±1), with multiple Gamma function, in the following theorems:
Theorem 1 For the Euler equations with multiple Gamma function, (δ = 0) (6), we have the
family of the solutions,

















ρ(t,? x) = f
?N
Σ
i=1Ci(xi− ai(t))
?
,
? u(t,? x) = (˙ a1(t), ˙ a2(t),..., ˙ aN(t)),
S(t,? x) = ln
?
g
?N
Σ
i=1Ci(xi− ai(t))
??
,
(12)
where





¨ ai(t) = Fi(t) + Ciξ, for i = 1,2,...N,
a(0) = a0, ˙ a(0) = a1,
(13)
(1)for γ1> 1, with







ξ + ˙ g(z)?m
j=1f (z)γj−1+ g(z)?m
g(z) > 0, for z ∈ (−∞,∞),
j=1γjf (z)γi−2˙f (z) = 0,
;(14)
where C1, C2,..., CN, ξ, g0and g1are arbitrary constants; and f ≥ 0 is an arbitrary C1function;
(2)for γ1= 1, with







ξ + ρCi˙ g(z) + g(z)
˙f(z)
f(z)Ci+ ˙ g(z)?m
g(z) > 0, for z ∈ (−∞,∞),
j=2f (z)γj−1+ g(z)?m
j=2γjf (z)γi−2˙f (z) = 0,
(15)
f > 0 is an arbitrary C1function.

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Line Solutions to Euler and Euler-Poisson Equations
5
We notice that the velocity ? u(t,? x) of the solutions (12) and (18), are only time-dependent
functions:
? u(t,? x) = (˙ a1(t), ˙ a2(t),..., ˙ aN(t)). (16)
It is different from the conventional velocity for the analytical solutions:
? u(t,? x) =
·a(t)
a(t)? x. (17)
Moreover, we need some modification to have the corresponding results for the Euler-Poisson
equations.
Theorem 2 For the Euler-Poisson equations with multiple Gamma function, (δ = ±1), (6), we
have the family of the solutions,

















ρ(t,? x) =
N
Σ
α(N)¨f
i=1C2
i
?N
Σ
i=1Ci(xi− ai(t))
?
,
? u(t,? x) = (˙ a1(t), ˙ a2(t),..., ˙ aN(t)),
S(t,? x) = ln
?
g
?N
Σ
i=1Ci(xi− ai(t))
??
(18)
where







¨ ai(t) = Fi(t) + Ciξ + di(t), for i = 1,2,...N,
a(0) = a0, ˙ a(0) = a1,
(19)
(1)for γ1> 1, with









ξ +
α(N)
N
Σ
i=1C2
i
˙ g(z)?m
j=1f (z)γj−1+
α(N)
N
Σ
i=1C2
i
g(z)?m
j=1γjf (z)γi−2˙f (z) + δ˙f(z) = 0,
g(z) > 0, for z ∈ (−∞,∞),
(20)
where C1, C2,..., CN, ξ, g0and g1are arbitrary constants with
N
Σ
i=1C2
i> 0;¨f ≥ 0 is an arbitrary
C3function; and d1,d2,....,dN are arbitrary C0functions;
(2)for γ1= 1, with



















ξ +
α(N)
N
Σ
i=1C2
i
˙ g(z) +
α(N)
N
Σ
i=1C2
i
g(z)
...
f (z)
f(z)
+α(N)
N
Σ
i=1C2
i
˙ g(z)?m
j=2¨f (z)γj−1+
α(N)
N
Σ
i=1C2
i
g(z)?m
j=2γj¨f (z)γj−2...
f (z) + δ˙f(z) = 0,
g(z) > 0, for z ∈ (−∞,∞),
(21)
...
f > 0 is an arbitrary C1function.