# Global Existence of Solutions for a Nonstrictly Hyperbolic System

Authors: De-yin Zheng; Yun-guang Lu; Guo-qiang Song; Xue-zhou Lu
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

We obtain the global existence of weak solutions for the Cauchy problem of the nonhomogeneous, resonant system. First, by using the technique given in Tsuge (2006), we obtain the uniformly boundedL
∞ estimates z
(
ρ
δ
,
ε
,
u
δ
,
ε
)
≤
B
(
x
) and w
(
ρ
δ
,
ε
,
u
δ
,
ε
)
≤
β when a
(
x
) is increasing (similarly, w
(
ρ
δ
,
ε
,
u
δ
,
ε
)
≤
B
(
x
) and z
(
ρ
δ
,
ε
,
u
δ
,
ε
)
≤
β when a
(
x
) is decreasing) for the ε-viscosity and δ-flux approximation solutions of nonhomogeneous, resonant system without the restriction z
0
(
x
)
≤
0 or w
0
(
x
)
≤
0 as given in Klingenberg and Lu (1997), where z and w are Riemann invariants of nonhomogeneous, resonant system; B
(
x
)
>
0 is a uniformly bounded function of x depending only on the function a
(
x
) given in nonhomogeneous, resonant system, and β is the bound of B
(
x
). Second, we use the compensated compactness theory, Murat (1978) and Tartar (1979), to prove the convergence of the approximation solutions.


---

## Body

## 1. Introduction

The following system(1)
(
a
ρ
)
t
+
(
a
ρ
u
)
x
=
0
,
(
a
ρ
u
)
t
+
(
a
(
ρ
u
2
+
P
(
ρ
)
)
)
x
=
a
x
P
(
ρ
)
,
a
t
=
0
describes the evolution of an isothermal fluid in a nozzle with discontinuous cross-sectional area a
(
x
)
>
0, where ρ and u stand for the density and the particle velocity of the fluid under consideration, respectively, and P
(
ρ
) denotes the pressure function (See [1]). The existence of global weak solutions for the Cauchy problem or the initial boundary value problem of system (1) has been studied in [1–3]. In [4–6], the authors showed the global existence of BV entropy solutions to a more general class of nonhomogeneous, resonant system by the generalized Glimm scheme.The Riemann problem for a more general resonant system ofn
+
1 equations,
(2)
a
t
=
0
,
u
t
+
F
(
u
,
a
)
x
=
0
,
was resolved in [7], where u
∈
R
n and F
:
R
n
×
R
→
R
n is a smooth function.To study the existence of entropy solutions of the Cauchy problem (1), the main difficulty is to establish boundedness of solutions because the equations are not in conservative form and the Conley-Chuey-Smoller principle of invariant regions does not apply (See [1] for the details about the physical background of system (1) and its difficulty in analysis). For the polytropic gas and the adiabatic exponent γ
∈
(
1
,
5
/
3
], the definition of a finite energy solution (unbounded) is given and its existence is obtained by using the compensated compactness method in [1].For smooth solution, system (1) is equivalent to the following conservation laws of three equations:
(3)
a
t
=
0
,
(
a
ρ
)
t
+
(
a
ρ
u
)
x
=
0
,
u
t
+
(
1
2
u
2
+
∫
0
ρ
P
′
(
s
)
s
d
s
)
x
=
0
or the system of two equations
(4)
ρ
t
+
(
ρ
u
)
x
=
A
(
x
)
ρ
u
,
u
t
+
(
1
2
u
2
+
∫
0
ρ
P
′
(
s
)
s
d
s
)
x
=
0
,
where A
(
x
)
=
-
(
a
′
(
x
)
/
a
(
x
)
). When a is a constant, system (3) or system (4) itself has many different physical backgrounds. For instance, it is a scaling limit system of Newtonian dynamics with long-range interaction for a continuous distribution of mass in R (cf. [8, 9]) and also a hydrodynamic limit for the Vlasov equation (cf. [10]). Its global weak solution was obtained by using the random choice method [2] in [11] and by using the compensated compactness theory in [12, 13].By simple calculations, two eigenvalues of system (4) are
(5)
λ
1
=
u
-
P
′
(
ρ
)
,
λ
2
=
u
+
P
′
(
ρ
)
,
with corresponding Riemann invariants
(6)
z
(
u
,
ρ
)
=
∫
c
ρ
‍
P
′
(
s
)
s
d
s
-
u
,
w
(
u
,
ρ
)
=
∫
c
ρ
‍
P
′
(
s
)
s
d
s
+
u
,
where c is a constant.The existence of weak solutions for the Cauchy problem (4) with bounded initial data
(7)
(
ρ
(
x
,
0
)
,
u
(
x
,
0
)
)
=
(
ρ
0
(
x
)
,
u
0
(
x
)
)
,
ρ
0
(
x
)
≥
0
was first studied in [12], where a
(
x
) is a smooth function and the technical condition z
(
u
0
(
x
)
,
ρ
0
(
x
)
)
≤
0 or w
(
u
0
(
x
)
,
ρ
0
(
x
)
)
≤
0 on the initial data is imposed for obtaining thea-priori, uniform L
∞ estimate of z or w.Without the conditionz
0
(
x
)
≤
0 or w
0
(
x
)
≤
0, the reasonable estimate, depending on the variable x
:
z
(
x
,
t
)
≤
C
x
-
(
2
(
γ
-
1
)
/
(
γ
+
1
)
), C
>
0 was first obtained in [2] for system (1) when P
(
ρ
)
=
ρ
γ, 1
<
γ
<
5
/
3 by using a modified Godunov scheme, and in [14] for general pressure function P
(
ρ
) and smooth function a
(
x
) by using the compensated compactness.In this paper, using the vanishing viscosity method and the maximum principle coupled with the flux approximation proposed in [15] for the homogeneous system of isentropic gas dynamics, we extend the results in [2, 12] to the Cauchy problem (4)–(7) for any bounded initial data and for the function a
(
x
) satisfying the conditions 0
<
c
1
≤
a
(
x
)
≤
c
2, a
′
(
x
)
∈
L
1
(
R
).We first construct the sequence of hyperbolic systems(8)
ρ
t
+
(
u
(
ρ
-
δ
)
)
x
=
A
ε
1
(
x
)
u
(
ρ
-
δ
)
,
u
t
+
(
1
2
u
2
+
P
1
(
ρ
,
δ
)
)
x
=
0
to approximate system (4), where δ
>
0 denotes the flux approximation constant and the approximation pressure
(9)
P
1
(
ρ
,
δ
)
=
∫
δ
ρ
‍
t
-
δ
t
2
P
′
(
t
)
d
t
,
A
ε
1
(
x
)
=
-
(
a
ε
1
(
x
)
′
/
a
ε
1
(
x
)
), and a
ε
1
=
a
(
x
)
*
G
ε
1 is the smooth approximation of a
(
x
), G
ε
1 being a mollifier. If a
(
x
) is a monotonic function, 0
<
c
1
≤
a
(
x
)
≤
c
2 as required in Theorem 2, and ɛ and δ converge to zero much faster than ε
1, then it is easy to prove that A
ε
1
(
x
) and a
ε
1 satisfy
(10)
lim
⁡
ε
1
→
0
A
ε
1
(
x
)
=
A
(
x
)
,
a
.
e
.
on
⁡
R
,
lim
⁡
ε
1
,
δ
→
0
δ
A
ε
1
(
x
)
=
0
,
a
.
e
.
on
⁡
R
,
A
ε
1
(
x
)
is
uniformly
bounded
in
L
1
(
R
)
,
lim
⁡
ε
1
,
ɛ
→
0
(
ɛ
a
ε
1
(
x
)
′
,
ɛ
a
ε
1
(
x
)
′′
)
=
(
0,0
)
,
a
.
e
.
on
⁡
R
.
Second, we add the viscosity terms to the right-hand side of (8) to obtain the following parabolic system:
(11)
ρ
t
+
(
u
(
ρ
-
δ
)
)
x
=
A
ε
1
(
x
)
u
(
ρ
-
δ
)
+
ɛ
ρ
x
x
,
u
t
+
(
1
2
u
2
+
P
1
(
ρ
,
δ
)
)
x
=
ɛ
u
x
x
,
with initial data
(12)
(
ρ
δ
,
ɛ
(
x
,
0
)
,
u
δ
,
ɛ
(
x
,
0
)
)
=
(
ρ
0
(
x
)
+
δ
,
u
0
(
x
)
)
,
where (
ρ
0
(
x
)
,
u
0
(
x
)
) are given in (7).Lemma 1.
Let0
<
c
1
≤
a
(
x
)
≤
c
2 or equivalently 0
<
c
1
≤
a
ε
1
(
x
)
≤
c
2 for two constants c
1 and c
2. If a
(
x
) is increasing or equivalently a
ε
1
(
x
)
′
≥
0, then we can choose a function B
(
x
) satisfying B
′
(
x
)
≤
0, 0
<
β
0
≤
B
(
x
)
≤
β, and
(13)
2
A
ε
1
(
x
)
B
(
x
)
≤
B
′
(
x
)
≤
A
ε
1
(
x
)
B
(
x
)
,
where the positive constants β
0, and β depend on c
1, and c
2, but are independent of ε
1.The proof of Lemma1 is trivial.By applying the maximum principle to the Cauchy problem (11)-(12), we first obtain the L
∞ estimates z
(
ρ
δ
,
ɛ
,
u
δ
,
ɛ
)
≤
B
(
x
) and w
(
ρ
δ
,
ɛ
,
u
δ
,
ɛ
)
≤
β when a
(
x
) is increasing (similarly w
(
ρ
δ
,
ɛ
,
u
δ
,
ɛ
)
≤
B
(
x
) and z
(
ρ
δ
,
ɛ
,
u
δ
,
ɛ
)
≤
β when a
(
x
) is decreasing) for a suitable positive, bounded function B
(
x
) given in Lemma 1; then by using the compensated compactness theory and the already existed compact frameworks given in [12, 13], we give the following global existence theorem of weak solutions.Theorem 2.
Let(
P
′
(
ρ
)
/
ρ
)
′
≥
0 and 0
<
c
1
≤
a
(
x
)
≤
c
2 for two positive constants c
1 and c
2.
(A) Leta
(
x
) be increasing and we choose B
(
x
)
∈
C
2
(
R
) satisfying all conditions in Lemma 1 and, moreover, B
′′
(
x
)
≤
0 or B
′′
(
x
)
=
B
1
+
B
2, where B
1
≤
0 and |
2
c
B
2
|
≤
|
B
(
x
)
B
′
(
x
)
| for a sufficiently small constant c. Then the Riemann invariants z and w of system (4) with respect to the approximated solutions of the Cauchy problem (11)-(12) satisfy the estimate
(14)
z
(
ρ
δ
,
ɛ
,
u
δ
,
ɛ
)
=
∫
δ
ρ
δ
,
ɛ
‍
P
′
(
s
)
s
d
s
-
u
δ
,
ɛ
≤
B
(
x
)
,
if z
(
ρ
δ
,
ɛ
(
x
,
0
)
,
u
δ
,
ɛ
(
x
,
0
)
)
≤
B
(
x
) and
(15)
w
(
ρ
δ
,
ɛ
(
x
,
t
)
,
u
δ
,
ɛ
(
x
,
t
)
)
=
∫
δ
ρ
δ
,
ɛ
‍
P
′
(
s
)
s
d
s
+
u
δ
,
ɛ
≤
β
,
if w
(
ρ
δ
,
ɛ
(
x
,
0
)
,
u
δ
,
ɛ
(
x
,
0
)
)
≤
β.
(B) For such functionA
(
x
) and the initial data satisfying the conditions in Part (
A
), if either P
(
ρ
)
=
(
1
/
γ
)
ρ
γ, γ
>
3, or (
P
′
(
ρ
)
/
ρ
)
′
≥
d, where d is a positive constant, then there exists a subsequence of (
ρ
δ
,
ɛ
(
x
,
t
)
,
u
δ
,
ɛ
(
x
,
t
)
), which converges pointwise to a pair of bounded functions (
ρ
(
x
,
t
)
,
u
(
x
,
t
)
) as δ, and ɛ tend to a zero, and the limit is a weak entropy solution of the Cauchy problem (4)–(7).Definition 3.
For integrable functiona
′
(
x
)
/
a
(
x
)
∈
L
1
(
R
), a pair of bounded measurable functions (
ρ
(
x
,
t
)
,
u
(
x
,
t
)
) is called a weak entropy solution of the Cauchy problem (4)–(7), if
(16)
∫
0
∞
‍
∫
-
∞
∞
‍
ρ
ϕ
t
+
(
ρ
u
)
ϕ
x
-
a
′
(
x
)
a
(
x
)
(
ρ
u
)
ϕ
d
x
d
t
+
∫
-
∞
∞
‍
ρ
0
(
x
)
ϕ
(
x
,
0
)
d
x
=
0
,
∫
0
∞
‍
∫
-
∞
∞
‍
u
ϕ
t
+
(
1
2
u
2
+
∫
0
ρ
‍
P
′
(
s
)
s
d
s
)
ϕ
x
d
x
d
t
+
∫
-
∞
∞
‍
u
0
(
x
)
ϕ
(
x
,
0
)
d
x
=
0
hold for all test function ϕ
∈
C
0
1
(
R
×
R
+
) and
(17)
∫
0
∞
∫
-
∞
∞
η
(
ρ
,
u
)
ϕ
t
+
q
(
ρ
,
u
)
ϕ
x
000000
-
a
′
(
x
)
a
(
x
)
η
(
ρ
,
u
)
ρ
ρ
u
ϕ
d
x
d
t
≥
0
holds for any nonnegative test function ϕ
∈
C
0
∞
(
R
×
R
+
-
{
t
=
0
}
), where (
η
,
q
) is a pair of convex entropy-entropy flux of system (4).We can easily construct many functionsA
(
x
), and B
(
x
) satisfying the conditions in Theorem 2.Example 4.
Let(18)
A
(
x
)
=
-
1
1
+
x
2
,
and choose
(19)
B
(
x
)
=
e
-
∫
-
∞
x
‍
(
1
/
(
1
+
s
2
)
)
d
s
.
Then (13) is satisfied since B
′
(
x
)
=
A
(
x
)
B
(
x
) and B
(
x
) satisfies all the conditions in Theorem 2. In fact, we may choose B
1
=
0 and then B
2
=
B
′′
(
x
) and
(20)
|
B
′′
(
x
)
|
=
|
B
′
(
x
)
A
(
x
)
+
B
(
x
)
A
′
(
x
)
|
ggggggg
=
|
B
′
(
x
)
A
(
x
)
+
A
′
(
x
)
A
(
x
)
B
′
(
x
)
|
=
|
1
+
2
x
1
+
x
2
B
′
(
x
)
|
ggggggg
≤
M
|
B
′
(
x
)
|
for a positive constant M.We are going to prove Theorem2 in the next section.


## 2. Proof of Theorem2

By simple calculations, two eigenvalues of system (8) are
(21)
λ
1
δ
=
u
-
ρ
-
δ
ρ
P
′
(
ρ
)
,
λ
2
δ
=
u
+
ρ
-
δ
ρ
P
′
(
ρ
)
,
with corresponding right eigenvectors
(22)
r
1
=
(
1
,
u
-
P
′
(
ρ
)
)
T
,
r
2
=
(
1
,
u
+
P
′
(
ρ
)
)
T
,
and Riemann invariants
(23)
z
(
u
,
ρ
)
=
∫
δ
ρ
‍
P
′
(
s
)
s
d
s
-
u
,
w
(
u
,
ρ
)
=
∫
δ
ρ
‍
P
′
(
s
)
s
d
s
+
u
which are similar to the Riemann invariants of system (4) given by (6).We multiply (11) by (
w
ρ
,
w
u
) and (
z
ρ
,
z
u
), respectively, to obtain
(24)
w
t
+
λ
2
δ
w
x
=
ɛ
w
x
x
-
ɛ
(
P
′
(
ρ
)
ρ
)
′
ρ
x
2
+
A
ε
1
(
x
)
(
ρ
-
δ
)
u
P
′
(
ρ
)
ρ
,
(25)
z
t
+
λ
1
δ
z
x
=
ɛ
z
x
x
-
ɛ
(
P
′
(
ρ
)
ρ
)
′
ρ
x
2
+
A
ε
1
(
x
)
(
ρ
-
δ
)
u
P
′
(
ρ
)
ρ
.
Let v
=
w
-
B
(
x
). Then
(26)
v
t
+
(
u
+
ρ
-
δ
ρ
P
′
(
ρ
)
)
(
v
x
+
B
′
(
x
)
)
=
ɛ
v
x
x
+
ɛ
B
′′
(
x
)
-
ɛ
(
P
′
(
ρ
)
ρ
)
′
ρ
x
2
+
A
ε
1
(
x
)
ρ
-
δ
ρ
P
′
(
ρ
)
×
(
B
(
x
)
+
v
-
∫
δ
ρ
‍
P
′
(
ρ
)
ρ
d
ρ
)
,
or
(27)
v
t
+
(
u
+
ρ
-
δ
ρ
P
′
(
ρ
)
)
v
x
+
B
′
(
x
)
(
B
(
x
)
+
v
-
∫
δ
ρ
‍
P
′
(
ρ
)
ρ
d
ρ
)
+
(
ρ
-
δ
)
B
′
(
x
)
P
′
(
ρ
)
ρ
=
ɛ
v
x
x
+
ɛ
B
′′
(
x
)
-
ɛ
(
P
′
(
ρ
)
ρ
)
′
ρ
x
2
+
A
ε
1
(
x
)
B
(
x
)
(
ρ
-
δ
)
P
′
(
ρ
)
ρ
+
A
ε
1
(
x
)
(
ρ
-
δ
)
P
′
(
ρ
)
ρ
v
-
A
ε
1
(
x
)
(
ρ
-
δ
)
P
′
(
ρ
)
ρ
∫
δ
ρ
‍
P
′
(
ρ
)
ρ
d
ρ
,
or
(28)
v
t
+
a
(
x
,
t
)
v
x
+
b
(
x
,
t
)
v
+
(
1
2
B
′
(
x
)
B
(
x
)
-
ɛ
B
′′
(
x
)
)
-
B
′
(
x
)
∫
δ
ρ
‍
P
′
(
ρ
)
ρ
d
ρ
+
1
2
B
(
x
)
B
′
(
x
)
+
[
B
′
(
x
)
-
A
ε
1
(
x
)
B
(
x
)
]
(
ρ
-
δ
)
P
′
(
ρ
)
ρ
+
A
ε
1
(
x
)
(
ρ
-
δ
)
P
′
(
ρ
)
ρ
∫
δ
ρ
‍
P
′
(
ρ
)
ρ
d
ρ
=
ɛ
v
x
x
-
ɛ
(
P
′
(
ρ
)
ρ
)
′
ρ
x
2
,
where a
(
x
,
t
)
=
u
+
(
(
ρ
-
δ
)
/
ρ
)
P
′
(
ρ
) and b
(
x
,
t
)
=
B
′
(
x
)
-
A
ε
1
(
x
)
(
ρ
-
δ
)
(
P
′
(
ρ
)
/
ρ
).Using the first equation of (11), we have thea-priori estimate ρ
≥
δ. Since the conditions on B
′′
(
x
) in Theorem 2, the following two terms on the left-hand side of (28):
(29)
1
2
B
′
(
x
)
B
(
x
)
-
ɛ
B
′′
(
x
)
≥
0
.
Now, we consider the other terms
(30)
L
=
-
B
′
(
x
)
∫
δ
ρ
‍
P
′
(
ρ
)
ρ
d
ρ
+
1
2
B
(
x
)
B
′
(
x
)
+
[
B
′
(
x
)
-
A
ε
1
(
x
)
B
(
x
)
]
(
ρ
-
δ
)
P
′
(
ρ
)
ρ
+
A
ε
1
(
x
)
(
ρ
-
δ
)
P
′
(
ρ
)
ρ
∫
δ
ρ
‍
P
′
(
ρ
)
ρ
d
ρ
on the left-hand side of (28).First, we have fromB
′
(
x
)
≥
B
(
x
)
A
ε
1
(
x
) that
(31)
L
≥
-
B
′
(
x
)
∫
δ
ρ
‍
P
′
(
ρ
)
ρ
d
ρ
+
1
2
B
(
x
)
B
′
(
x
)
+
A
ε
1
(
x
)
(
ρ
-
δ
)
P
′
(
ρ
)
ρ
∫
δ
ρ
‍
P
′
(
ρ
)
ρ
d
ρ
.
Second, we have from (
P
′
(
ρ
)
/
ρ
)
′
≥
0, A
ε
1
(
x
)
≥
0 that
(32)
A
ε
1
(
x
)
(
ρ
-
δ
)
P
′
(
ρ
)
ρ
∫
δ
ρ
‍
P
′
(
ρ
)
ρ
d
ρ
≥
A
ε
1
(
x
)
(
∫
δ
ρ
P
′
(
ρ
)
ρ
d
ρ
)
2
and so
(33)
L
≥
-
B
′
(
x
)
∫
δ
ρ
‍
P
′
(
ρ
)
ρ
d
ρ
+
1
2
B
(
x
)
B
′
(
x
)
+
A
ε
1
(
x
)
(
∫
δ
ρ
P
′
(
ρ
)
ρ
d
ρ
)
2
=
A
ε
1
(
x
)
(
∫
δ
ρ
P
′
(
ρ
)
ρ
d
ρ
-
B
′
(
x
)
2
A
ε
1
(
x
)
)
2
+
1
2
B
(
x
)
B
′
(
x
)
-
(
B
′
(
x
)
)
2
4
A
ε
1
(
x
)
≥
0
,
since B
′
(
x
)
≤
2
B
(
x
)
A
(
x
).Thus, (28) is reduced to the following inequality about v:
(34)
v
t
+
a
(
x
,
t
)
v
x
+
b
(
x
,
t
)
v
≤
ɛ
v
x
x
and we can prove that v
≤
0 or w
≤
B
(
x
) if applying for the maximum principle to (34).To prove the estimate ofz, we have from (25) that
(35)
z
t
+
a
1
(
x
,
t
)
z
x
≤
ɛ
z
x
x
+
A
(
x
)
(
ρ
-
δ
)
u
P
′
(
ρ
)
ρ
,
where a
1
(
x
,
t
)
=
u
-
(
ρ
-
δ
)
(
P
′
(
ρ
)
/
ρ
).Let(36)
z
=
X
+
β
+
N
(
x
2
+
L
h
e
3
t
)
L
2
,
where β
≥
B
(
x
)
>
0 is the upper bound of z
0
(
x
) and N, and h are the bounds of |
z
| and |
a
1
(
x
,
t
)
| obtained from the local solution. Then
(37)
X
(
x
,
0
)
=
z
0
(
x
)
-
β
-
N
(
x
2
+
L
h
)
L
2
<
0
,
X
(
±
L
,
t
)
=
z
(
±
L
,
t
)
-
β
-
N
(
L
2
+
L
h
e
3
t
)
L
2
<
0
.We have from (35)–(37) that
(38)
X
(
x
,
t
)
<
0
,
on
⁡
(
-
L
,
L
)
×
(
0
,
T
)
.
We argue by assuming that (38) is violated for X at a point (
x
,
t
) in (
-
L
,
L
)
×
(
0
,
T
). Let t
¯ be the least upper bound of values of t at which X
<
0. Then, by the continuity we see that X
=
0 at some points (
x
¯
,
t
¯
)
∈
(
-
L
,
L
)
×
(
0
,
T
). So X
t
≥
0, X
x
=
0, and -
X
x
x
≥
0 at (
x
¯
,
t
¯
); that is,
(39)
X
t
+
a
1
(
x
,
t
)
X
x
-
ɛ
X
x
x
≥
0
at
(
x
¯
,
t
¯
)
.
But from (35) and (36),
(40)
X
t
+
a
1
(
x
,
t
)
X
x
-
ɛ
X
x
x
≤
A
(
x
)
(
ρ
-
δ
)
u
P
′
(
ρ
)
ρ
-
(
3
N
h
L
e
3
t
L
2
+
a
1
(
x
,
t
)
2
N
x
L
2
-
ɛ
2
N
L
2
)
.
Since X
=
0 on (
x
¯
,
t
¯
); then
(41)
z
=
β
+
N
(
x
2
+
L
h
e
3
t
)
L
2
>
β
≥
w
at
(
x
¯
,
t
¯
)
,
Thus, u
<
0 at (
x
¯
,
t
¯
) from the relation of w, and z given by (23). So the right-hand side of (40) is negative, which yields a conclusion contradicting (39). So (38) is proved. Therefore for any point (
x
0
,
t
0
) in (
-
L
,
L
)
×
(
0
,
T
),
(42)
z
(
x
0
,
t
0
)
<
β
+
N
(
x
0
2
+
L
h
e
3
t
0
)
L
2
,
which yields the desired estimate
(43)
z
(
x
,
t
)
≤
β
,
if we let L
↑
∞ in (42), and, hence, complete the proof of Part (A) in Theorem 2.For the homogeneous case (a
(
x
)
=
0), the convergence of (
ρ
δ
,
ɛ
,
u
δ
,
ɛ
)
→
(
ρ
,
u
) as δ, and ɛ tend to zero in the Part (B) was given in [13] when P
(
ρ
)
=
(
1
/
γ
)
ρ
γ, γ
>
3, and given in [12] when (
P
′
(
ρ
)
/
ρ
)
′
≥
d
>
0 by using the compensated compactness theory [16, 17] coupled with some basic ideas of the kinetic formulation [18, 19].Now, we are going to prove the convergence of(
ρ
δ
,
ɛ
,
u
δ
,
ɛ
)
→
(
ρ
,
u
) as δ, and ɛ tend to zero for the inhomogeneous system (11).Any entropy-entropy flux pair(
η
(
ρ
,
u
)
,
q
(
ρ
,
u
)
) of the original hyperbolic system (4) satisfies the additional system. Consider
(44)
q
ρ
=
u
η
ρ
+
P
′
(
ρ
)
ρ
η
u
,
q
u
=
ρ
η
ρ
+
u
η
u
.
Eliminating the q from (44), we have
(45)
η
ρ
ρ
=
P
′
(
ρ
)
ρ
2
η
u
u
.
Similarly, any entropy-entropy flux pair (
η
(
ρ
,
u
)
,
q
(
ρ
,
u
)
) of the approximated hyperbolic system (8) satisfies
(46)
q
ρ
=
u
η
ρ
+
(
ρ
-
δ
)
P
′
(
ρ
)
ρ
2
η
u
,
q
u
=
(
ρ
-
δ
)
η
ρ
+
u
η
u
.
By eliminating the q from (46), we have also the same entropy equation (45). Therefore, system (4) and system (8) have the same entropies.For any entropy-entropy flux pair(
η
0
(
ρ
,
u
)
,
q
0
(
ρ
,
u
)
) of system (4), by multiplying (
η
0
ρ
,
η
0
u
) to system (11), we have
(47)
η
0
t
+
q
0
x
+
δ
q
1
x
=
ɛ
η
0
x
x
-
ɛ
(
ρ
x
ɛ
,
δ
,
u
x
ɛ
,
δ
)
·
∇
2
η
0
(
ρ
ɛ
,
δ
,
u
ɛ
,
δ
)
·
(
ρ
x
ɛ
,
δ
,
u
x
ɛ
,
δ
)
T
+
η
0
ρ
A
ε
1
(
x
)
u
(
ρ
-
δ
)
,
where q
0
+
δ
q
1 is the entropy flux of the approximated system (8) corresponding to entropy η
0. Since A
ε
1
(
x
) is uniformly integrable, then the last term on the right-hand side of system (47) is compact in W
loc
⁡
-
1
,
α
(
R
×
R
+
), for some α
∈
(
1,2
), by the Sobolev embedding theorems. It is obvious that the term δ
q
1
x on the left-hand side is compact in H
loc
⁡
-
1
(
R
×
R
+
). Therefore, using the same techniques given in [12, 13] for the homogeneous system, we may prove that η
0
t
+
q
0
x is compact in H
loc
⁡
-
1
(
R
×
R
+
) and so the convergence of (
ρ
δ
,
ɛ
,
u
δ
,
ɛ
)
→
(
ρ
,
u
) as δ, and ɛ tend to zero. Furthermore, the limit (
ρ
,
u
) satisfies (16).If precisely using (10), we can prove that the limit (
ρ
,
u
) satisfies the following conservation form:
(48)
∫
0
∞
‍
∫
-
∞
∞
‍
a
(
x
)
ρ
ϕ
t
+
(
a
(
x
)
ρ
u
)
ϕ
x
000000
+
∫
-
∞
∞
‍
a
(
x
)
ρ
0
(
x
)
ϕ
(
x
,
0
)
d
x
=
0
.
In fact, we multiply the first equation in (11) by a
ε
1
(
x
) to obtain
(49)
(
a
ε
1
(
x
)
ρ
)
t
+
(
a
ε
1
(
x
)
u
(
ρ
-
δ
)
)
x
=
-
δ
a
ε
1
(
x
)
′
+
ɛ
a
ε
1
(
x
)
ρ
x
x
=
-
δ
a
ε
1
(
x
)
′
+
ɛ
(
a
ε
1
(
x
)
ρ
)
x
x
-
2
ɛ
(
a
ε
1
(
x
)
′
ρ
)
x
+
ɛ
a
ε
1
(
x
)
′′
ρ
,
which yields (48) when ε
1 goes to zero.Since both systems (4) and (8) have the same entropies, we can easily prove that the limit (
ρ
,
u
) satisfies the entropy condition (17). So we complete the proof of Theorem 2.



---

*Source: 691429-2014-03-31.xml*