A Complete Global Solution to the Pressure Gradient Equation

0.18 MB·English

by  Zhen Lei

A Complete Global Solution to the Pressure
Gradient Equation
Zhen Lei Yuxi Zheng
∗ †
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May 6, 2006
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a
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Abstract
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We study the domain of existence of a solution to a Riemann problem for the

] pressure gradient equation in two space dimensions. The Riemann problem is
P
A the expansion of a quadrant of gas of constant state into the other three vacuum
. quadrants. The global existence of a smooth solution was established in Dai and
h
Zhang[Arch.Rational Mech. Anal., 155(2000), 277-298] uptothefreeboundary
t
a
of vacuum. We prove that the vacuum boundary where the system is degenerate
m
is the trivial coordinate axes.
[

1 Keyword: Regularity, vacuum boundary, two dimensional Riemann problem, charac-
v
teristic decomposition.
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0 AMS subject classification: Primary: 35L65, 35J70, 35R35; Secondary: 35J65.
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3
.
1 1 Introduction
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0 The pressure gradient system
:
v
i u +p = 0,
X t x
r v +p = 0, (1.1)
a  t y

 E +(pu) +(pv) = 0,
t x y

where E = (u2 + v2)/2 + p, appeared first in the flux-splitting method of Li and
Cao [5] and Agarwal and Halt [1] in numerical computation of the Euler system of a
∗School of Mathematical Sciences, Fudan University, Shanghai 200433, P. R. China; School of
Mathematics and Statistics, Northeast Normal University, Changchun 130024, P. R. China; Key
Laboratory of Mathematics for Nonlinear Sciences (Fudan University), Ministry of Education, P. R.
China, email: [email protected].
†Department of Mathematics, Pennsylvania State University, State college, PA 16802, email:
[email protected].
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compressible gas. Later, an asymptotic derivation was given in Zheng [10, 13] from the
two-dimensional full Euler system for an ideal fluid
ρ + (ρU) = 0,
t
∇·
(ρU) + (ρU U +pI) = 0,
 t
∇· ⊗

 (ρE) + (ρEU +pU) = 0,
t
∇·
where U = (u,v),E = (u2+v2)/2+p/((γ 1)ρ), and γ > 1 is a gas constant. We refer

−
thereader tothebooksofZheng [11]andLiet. al. [7]formorebackgroundinformation,
and the papers [10, 9, 2, 12, 6, 13] for recent studies. After being decoupled from the
pressure gradient system (1.1), the pressure satisfies the following second order quasi-
linear hyperbolic equation
p
t
∆ p = 0. (1.2)
(x,y)
p t −
(cid:16) (cid:17)
Dai and Zhang [2] studied a Riemann problem for system (1.1), see also Yang and
Zhang [8] by the hodograph method. In the self similar variables ξ = x/t,η = y/t, the
value of the pressure variable of the Riemann data is
p(ξ,η) = ξ2, for 0 < ξ,η √p , (ξ √p )2 +η2 = p ,
1 1 1
≤ −
(1.3)

p(ξ,η) = η2, for 0 < ξ,η √p , ξ2 +(η √p )2 = p .
 1 1 1
≤ −
Here p is any positive number. They showed that the Goursat problem for system
1 
(1.2) admits a global solution in the self-similar plane, which is smooth with a possible
vacuum near the origin (see Figure 1, where a = √p ). We are interested in the size of
1
the vacuum boundary (ξ,η) p(ξ,η) = 0 where the pressure gradient system (1.2) is
{ | }
degenerate. Somewhat surprisingly, our result shows that the vacuum bubble is trivial
and the entire vacuum boundary is the trivial coordinate axes in the self-similar plane
(see Figure 2, where a = √p ), which is stated in our main theorem at the end of
1
Section 3.
Further motivation for the study of the current problem is that the study of bound-
aries such as a sonic curve is important in establishing the global existence of a solution
to a general two dimensional Riemann problem of the pressure gradient system. In ad-
dition, the solution of the current problem covers wave interaction problems in which
only some fractions of the plane waves are involved. Wave interactions of these kinds
are common in two dimensional Riemann problems. Finally, the study of the pressure
gradient system has motivated work on two dimensional full Euler systems, see Li [4]
and Zheng [14, 15].
2 Integration along characteristics
In the self-similar variables ξ = x/t, η = y/t, the pressure gradient equation (1.2)
takes the form
(ξ∂ +η∂ )2p (ξ∂ +η∂ )p ((ξ∂ +η∂ )p)2
ξ η ξ η ξ η
∆ p+ = 0. (2.1)
p − (ξ,η) p − p2
2
η
+
a
0
=
p
O ξ
a
Figure 1. Solution with a presumed vacuum bubble.
Figure 1:
3
η
+
a
O ξ
a
Figure 2. The global solution.
Figure 2:
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In the polar coordinates
r = ξ2 +η2,
θ = parctan η,
 ξ
equation (2.1) can be decomposed along the characteristics into the following form (see

[6, 3])
∂ ∂ p = mp ∂ p,
+ r
− − (2.2)

∂ ∂ p = mp ∂ p,
 + r +
− −
provided that

p < r2,
where
λr4
m = ,
2p2
and
∂ = ∂ 1∂ ,
± θ ± λ r
(2.3)

λ = p .
 r2(r2 p)
−
q
Note that the equation is invariant under the following scaling transformation

ξ η p
(ξ,η,p) ( , , ). (p > 0)
1
−→ √p √p p
1 1 1
Thus, without loss of generality, the corresponding boundary condition (1.3) in the
polar coordinates can be set as the form
p = ξ2 = r2cos2θ = 4cos4θ, on r = 2cosθ, π/4 θ π/2;
≤ ≤
(2.4)

p = η2 = r2sin2θ = 4sin4θ, on r = 2sinθ, 0 θ π/4.

≤ ≤
The solution exists in the interaction zone up to a possible vacuum bubble. See Figure
3.
The characteristic form (2.2) of the pressure gradient equation enjoys a number of
useful properties. For example, the quantities ∂ p keep their positivities/negativities
±
along characteristics of a plus/minus family, and the sign-persevering quantities yield
monotonicity of the primary variable p (see [6], also [3] where the authors propose to
call them Riemann sign-persevering variables), and the fact that a state adjacent to a
constant state for the pressure gradient system must be a simple wave in which p is
constant along the characteristics of a plus/minus family [6].
The characteristic decomposition (2.2) has played an important role and was a
powerful tool for building the existence of smooth solutions in the work of Dai and
Zhang [2]. We are interested in the size of the vacuum boundary (r,θ) p(r,θ) = 0
{ | }
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θ
π/2
d
b
+
π/4
p=0
c
−
1/2
a 2 r
0
Figure 3. Presumed vacuum in the polar coordinates.
Figure 3:
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where the pressure gradient equation is degenerate. For this purpose, we rewrite (2.2)
as
∂ ∂ p = q∂ p∂ p q(∂ p)2,
+ +
− − − − (2.5)

∂ ∂ p = q∂ p∂ p q(∂ p)2,
 + + +
− − −
where
 r2
q = . (2.6)
4p(r2 p)
−
Define the characteristic curves ra(θ) and rb (θ) as
+
−
dr−a(θ) = 1 , dr+b(θ) = 1 ,
dθ −λ r−a(θ),θ and dθ λ r+b(θ),θ (2.7)
 
ra(θ ) = 2sinθ , rb (θ ) = 2cosθ .
 a (cid:0) a (cid:1)  + b (cid:0) b(cid:1)
−
We point out here that for convenience, we also use the notation ra(r,θ) (rb (r,θ),
+
respectively) which represents the characteristic passing through the−point (r,θ) and
intersecting the lower (upper, respectively) boundary at point a (b, respectively). See
Figure 3.
Now let us rewrite the system (2.5) as the following form
∂ 1 exp θ q∂ p rb (φ),φ dφ = qexp θ q∂ p rb (φ),φ dφ,
+ ∂−p θb + + θb + +
 (cid:16) (cid:17)
∂ 1 expRθ q∂ p(cid:0)ra(φ),φ(cid:1)dφ = qexpRθ q∂ p(cid:0)ra(φ),φ(cid:1)dφ.
 − ∂+p θa − − θa − −
(cid:16) (cid:17)
R (cid:0) (cid:1) R (cid:0) (cid:1)
Integratethe above equations along the positive and negative characteristics rb (θ) and
+
ra(θ) from θ and θ to θ, respectively, with respect to θ, one can get the iterative
b a
ex−pressions of ∂ p and ∂ p:
+
−
1 exp θ q∂ p rb (φ),φ dφ = 1 2cosθ ,θ
∂−p θb + + ∂−p b b
 + θRq rb (ψ)(cid:0),ψ exp (cid:1)ψq∂ p rb(cid:0)(φ),φ dφd(cid:1)ψ,
 θb + θb + + (2.8)

 1 exRp θ(cid:0)q∂ p ra(cid:1)(φ),φRdφ = (cid:0)1 2sinθ(cid:1),θ
∂+p θa − − ∂+p a a
 + θRq ra(ψ)(cid:0),ψ exp (cid:1)ψ q∂ p ra(cid:0)(φ),φ dφd(cid:1)ψ.
 θa − θa − −


On the otherhand, nRotin(cid:0)g the bou(cid:1)ndarRy condit(cid:0)ion (2.4),(cid:1)a straight forward calcu-
lation shows that
θ
exp q∂ p rb (φ),φ dφ (2.9)
+ +
Zθb
1 θ (cid:0)1 1(cid:1)
= exp + ∂ p rb (φ),φ dφ
4 p r2 p + +
Zθb (cid:16) − (cid:17)
(cid:0) (cid:1)
= p41 r+b (θ),θ exp 1 p r+b(θ),θ 1 dp .
√2cosθ 4 (cid:0) (cid:1) r2(rb ) p 1
(cid:0) b (cid:1) n Zp(2cosθb,θb) + − 1 o
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Similarly, one has
θ
exp q∂ p ra(φ),φ dφ (2.10)
Zθa − −
(cid:0) (cid:1)
p41 ra(θ),θ 1 p r−a(θ),θ 1
= √2−sinθ exp 4 (cid:0) (cid:1) r2(ra) p dp1 .
(cid:0) a (cid:1) n Zp(2sinθa,θa) − − 1 o
Then, we have
θ ψ
q rb (ψ),ψ exp q∂ p rb (φ),φ dφ dψ (2.11)
+ + +
Zθb n Zθb o
(cid:0) 1 (cid:1)θ r2p 3 (cid:0) (cid:1)
= −4 rb (ψ),ψ
4√2cosθ r2 p +
b Zθb −
(cid:0) (cid:1)
1 p r+b(ψ),ψ 1
exp dp dψ,
× 4 (cid:0) (cid:1) r2(rb ) p 1
n Zp(2cosθb,θb) + − 1 o
and
θ ψ
q ra(ψ),ψ exp q∂ p ra(φ),φ dφdψ (2.12)
Zθa − Zθa − −
(cid:0) 1 (cid:1)θ r2p 3 (cid:0) (cid:1)
= −4 ra(ψ),ψ
4√2sinθa Zθa r2 −p −
(cid:0) (cid:1)
1 p r−a(ψ),ψ 1
exp dp dψ.
× 4 (cid:0) (cid:1) r2(ra) p 1
n Zp(2sinθa,θa) − − 1 o
Finally, by substituting (2.10) and (2.12) into the second equality of (2.8), we arrive
at a new iterative expression for ∂ p :
+
∂ p r,θ (2.13)
+
(cid:0) (cid:1) exp θ q∂ p ra(φ),φ dφ
= θa − −
∂+1p 2sinθa,θa + θθaqRr−a(ψ),(cid:0)ψ exp θψ(cid:1)a q∂−p r−a(φ),φ dφdψ
(cid:0) (cid:1)4p1R(r,θ)(cid:0)exp 1 (cid:1)p r−a(θR),θ 1(cid:0) dp (cid:1)
= 4 4 p((cid:0)2sinθa,θ(cid:1)a) r2(r−a)−p1 1 .
n o
1 + θ r2p−43 ra(ψ)R,ψ exp 1 p r−a(ψ),ψ 1 dp dψ
2√2sin2θacosθa θa r2−p − 4 p((cid:0)2sinθa,θa(cid:1)) r2(r−a)−p1 1
n o
R (cid:0) (cid:1) R
Similarly, by substituting (2.9) and (2.11) into the first equality of (2.8), one obtains
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the new iterative formula for ∂ p :
−
∂ p(r,θ) (2.14)
− −
exp θ q∂ p rb (φ),φ dφ
= − θb + +
1 2cosθ ,θ + θ q Rrb (ψ),ψ(cid:0) exp ψ(cid:1)q∂ p rb (φ),φ dφdψ
∂−p b b θb + θb + +
(cid:0) (cid:1)4p1(Rr,θ)(cid:0)exp 1 (cid:1)p r+b(θR),θ 1(cid:0) dp (cid:1)
= 4 4 p((cid:0)2cosθb,θ(cid:1)b) r2(r+b)−p1 1 .
n o
1 θ r2p−43 rb (ψ)R,ψ exp 1 p r+b(ψ),ψ 1 dp dψ
2√2cos2θbsinθb − θb r2−p + 4 p((cid:0)2cosθb,θb(cid:1)) r2(r+b)−p1 1
n o
R (cid:0) (cid:1) R
3 The boundary of the vacuum bubble
In this section, we use the iterative formulas (2.13) and (2.14) to prove that the vacuum
bubble (r,θ) p(r,θ) = 0,θ (0,π/2) is in fact the trivial origin (0,0) in the self-
{ | ∈ } { }
similar plane. We use the method of contradiction. Assume to the contrary that there
is abubble with boundaryr (θ) 0for allθ (0, π), and r (θ) > 0 forsome θ (0, π).
0 ≥ ∈ 2 0 ∈ 2
That means p r (θ),θ = 0 for θ (0, π), and the solution is smooth in the domain
0 ∈ 2
bounded by the bubble r (θ) and the upper and lower characteristic boundaries. We
0
(cid:0) (cid:1)
intend to deduce contradictions, which in turn proves that r (θ) = 0 for all θ (0,π/2)
0
∈
andthevacuumbubble (r (θ),θ) isinfactthetrivialorigin (0,0) intheself-similar
0
{ } { }
plane.
Before we start the above procedure, we point out two observations which roughly
imply the nonexistence of the vacuum bubble. For presenting the observations, we
assume further that r (θ) > 0 for all θ (0, π).
0 ∈ 2
First, we can compute easily the characteristic slope
1 1
dθ/dr = λ =
− −2sinθ ∼ −2
along the upper characteristic boundary and consider θ π/2. Similar result holds
→
on the lower boundary. Now assuming there is a bubble, noting that along the plus
characteristics, by definition of λ in (2.3), we see easily that
p
dθ/dr = λ = 0
r2(r2 p) →
r −
asp 0 (near thevacuum bubble) except for(0,0)and(0, π). The above computation
→ 2
reveals that there is some kind of inconsistency for the slopes of the characteristics at
(0,0) and (0, π) in the polar coordinate plane.
2
Next, let us calculate the decay rate of p along the middle line θ = π/4. By using
the symmetry of system (2.1) (see Figure 3), we can obtain that
∂ p M p1/2
0
± ∼ ±
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asymptotically. To show the details, we propose
∂ p M0p14+δ,
± ∼ ±
as (r,θ) tends to a point of the vacuum bubble. Then, by (2.13), we have
1 p r−a(θ),θ 1 1
1
∂+p r,θ = 4p4(r,θ)exp 4 (cid:0) (cid:1) r2(ra) p dp1 2√2sin2θ cosθ
n Zp(2sinθa,θa) − − 1 o.n a a
(cid:0) (cid:1)
+ θ r2p−34 ra(ψ),ψ exp 1 p r−a(ψ),ψ 1 dp dp
Zθa (r2 −p)∂−p − n4 Zp(2(cid:0)sinθa,θa)(cid:1) r2(r−a)−p1 1o o
M (cid:0) (cid:1)
= 0p41+δ +high order terms,
4δ
which implies that δ should be 1.
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Thus, by using (2.3), we have
M
0
∂ p p
r
∼ r2
asymptotically as (r,θ) tends to a point of the vacuum bubble. Therefore,
M
0
p c exp (3.1)
∼ − r
(cid:18) (cid:19)
asymptotically as (r,θ) tends to a point of the vacuum bubble on the line θ = π/4,
which implies that there is no interior vacuum at least at θ = π.
4
We point out incidentally that Zheng’s previous numerical computation of the bub-
ble, referred to in Dai and Zhang [2], is probably caused by the fast exponential decay
(3.1).
In what follows, we concentrate on establishing the above formal intuition rigor-
ously. With the symmetry of system (2.5), let us restrict our arguments on θ (0, π].
∈ 4
Fix a point (r,θ) on the bubble boundary. Let us denote D(r,θ) the bounded
domain determined by the positive and negative characteristic curves starting from
(r,θ) and (√2, π). See Figure 4.
4
For 0 < ǫ < 1, define a curve r (θ), θ (0, π), by
ǫ ∈ 2
p r (θ),θ = ǫ. (3.2)
ǫ
From (2.13) and (2.14), it is easy to (cid:0)see that(cid:1)
∂ p(r,θ) > 0, π
+
for 0 < θ . (3.3)
(∂ p(r,θ) < 0, ≤ 4
−
Thus, by (2.3) and (3.3), we have
p (r,θ) > 0, (3.4)
r
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