TRANSACTIONS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 352, Number 3, Pages 1203–1215
S 0002-9947(99)02290-4
Article electronically published on September 20, 1999
ASYMPTOTICS TOWARD THE PLANAR RAREFACTION WAVE
FOR VISCOUS CONSERVATION LAW
IN TWO SPACE DIMENSIONS
MASATAKA NISHIKAWA AND KENJI NISHIHARA
Abstract. This paper is concerned with the asymptotic behavior of the solution toward the planar rarefaction wave r( xt ) connecting u+ and u− for the
scalar viscous conservation law in two space dimensions. We assume that the
initial data u0 (x, y) tends to constant states u± as x → ±∞, respectively.
Then, the convergence rate to r( xt ) of the solution u(t, x, y) is investigated
without the smallness conditions of |u+ − u− | and the initial disturbance. The
proof is given by elementary L2 -energy method.
1. Introduction
We consider the Cauchy problem for the scalar viscous conservation law in two
space dimensions:
(1.1)
(1.2)
ut + f (u)x + g(u)y = µ∆u,
(t, x, y) ∈ R+ × R2 ,
u(0, x, y) = u0 (x, y),
where f and g are smooth functions, and µ is a positive constant. We assume that
f is convex, i.e.,
(1.3)
f 00 (u) ≥ α > 0 for u ∈ R,
and that the initial data is asymptotically constant:
(1.4)
u0 (x, y) → u±
as x → ±∞ for any fixed y ∈ R,
where u± are constants satisfying u− < u+ . The asymptotic behavior as t → ∞ of
the solution is closely related to that of the Riemann problem for the corresponding
hyperbolic conservation law in one space dimension:
(1.5)
(1.6)
(t, x) ∈ (−1, ∞) × R,
(
for x < 0,
u−
R
r(−1, x) = r0 (x) ≡
u+
for x > 0.
rt + f (r)x = 0,
Received by the editors July 8, 1996 and, in revised form, October 14, 1997.
1991 Mathematics Subject Classification. Primary 35L65, 35L67, 76L05.
Key words and phrases. Nonlinear stable, viscous conservation law, planar rarefaction wave,
L2 -energy method.
c
1999
American Mathematical Society
1203
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1204
MASATAKA NISHIKAWA AND KENJI NISHIHARA
The entropy solution r(t, x) of (1.5), (1.6)


for

u−
x
0 −1
)
for
r(t, x) = (f ) (
(1.7)
t+1


u
for
+
is given by
x < f 0 (u− )(t + 1),
f 0 (u− )(t + 1) ≤ x ≤ f 0 (u+ )(t + 1),
f 0 (u+ )(t + 1) < x.
The function (t, x, y) → r(t, x) is called the planar rarefaction wave. In a one
dimensional case, the asymptotic behaviors of solutions were originally investigated
by Il’in and Oleinik [3]. Harabetian [1] obtained the convergence rate toward the
rarefaction wave. Hattori and Nishihara [2] showed more precise behaviors of the
solution for the Burgers equation, employing the Hopf-Cole transformation. See
also [5], [6], [7], [8], [10].
In a two dimensional case, Xin [9] has first investigated the stability of the planar
rarefaction wave. Ito [4] has recently shown the convergence rate toward the planar
rarefaction wave. In both papers, the smallness of initial disturbance is essentially
assumed. In [4], the rarefaction wave is also assumed to be weak.
Our main purpose in this paper is to show that the solution u(t, x, y) asymptotically behaves as r(t, x) with the same rate as that in [4] without smallness
conditions, which improves their results.
∂
∂
2
2
Denote R+
= {(x, y) ∈ R2 ; x > 0}, R−
= {(x, y) ∈ R2 ; x < 0} and D = ( ∂x
, ∂y
).
Then, our main theorem is as follows.
2
2
) ∩ L1 (R±
) and Dα u0 (x, y) ∈
Theorem 1. Suppose that u0 (x, y) − u± ∈ L2 (R±
H 1 (R2 ), |α| = 1. Then the problem (1.1),(1.2) has a unique global solution u(t, x, y)
satisfying
(1.8)
1
sup ku(t, ·, y) − r(t, ·)kL2 (Rx ) ≤ C(1 + t)− 4 log(2 + t),
y∈R
where C is a positive constant depending on u0 .
Our plan in this paper is as follows. In the next section, we construct a smooth
rarefaction wave, which is different from that in [4], and reformulate our problem. In
the last two sections, we give the proofs of theorems for the reformulated problems.
2. Smooth approximation and preliminaries
We first introduce the function w̃(t, x) as a solution to the problem:
(2.1)
w̃t + w̃ w̃x = µw̃xx ,
(2.2)
r̃0R (x)
w̃(−1, x) =
(t, x) ∈ (−1, ∞) × R,
≡ f 0 (r0R (x)).
The Hopf-Cole transformation gives the information of the properties of w̃. Using
w̃(t, x), we define “the smooth rarefaction wave” w(t, x) as
(2.3)
w(t, x) = (f 0 )−1 (w̃(t, x))|t≥0 .
According to (1.3), w(t, x) satisfies
(2.4)
(2.5)
f 000 (w) 2
w ,
f 00 (w) x
w(0, x) = w0 (x) ≡ f 0 (w̃(0, x)).
wt + f (w)x = µwxx + µ
(t, x) ∈ R+ × R,
The properties of the smooth rarefaction wave w(t, x) are stated in the following
lemma. From now on, we denote several constants by C or c without confusion.
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ASYMPTOTICS TOWARD THE PLANAR RAREFACTION WAVES
1205
Lemma 1 (Hattori and Nishihara [2]). The smooth rarefaction wave w(t, x) given
by (2.3) satisfies the following properties:
(i) |w(t, x) − u± | ≤ C exp(−c|x|2 ),
(ii) wx (t, x) > 0,
1
(iii) kwx (t, ·)kLp (R) ≤ (1 + t)−1+ p , kwxx (t, ·)kLp (R) ≤ (1 + t)−1 ,
(iv) kw(t, ·) − r(t, ·)kLp (R) ≤ C(1 + t)−
p−1
2p
.
000
(w) 2
Since there is a “forcing term” ff 00 (w)
wx in the equation (2.4), we further introduce the smooth rarefaction wave U (t, x) approximate to w, which satisfies
(2.6)
(t, x) ∈ R+ × R,
Ut + f (U )x = Uxx ,
U (0, x) = U0 (x) ≡ f 0 (w̃(0, x)).
(2.7)
The monotonicity in x of U (t, x) was obtained by Xin [9], which is important in
the a priori estimates in §4.
Lemma 2 (Xin [9]). Suppose that U0 (x) is monotonically increasing:
d
U0 (x) > 0,
dt
(2.8)
x ∈ R.
Then, the solution U (t, x) of (2.6),(2.7) satisfies
d
U (t, x) > 0,
dx
(2.9)
(t, x) ∈ R+ × R.
Thus, setting
u(t, x, y) − r(t, x) = {w(t, x) − r(t, x)} + {U (t, x) − w(t, x)} + {u(t, x, y) − U (t, x)}
≡ {w(t, x) − r(t, x)} + v(t, x) + V (t, x, y),
we have reached two reformulated problems:
(2.10)
(2.11)
f 000 (w) 2
w ,
(t, x) ∈ R+ × R,
f 00 (w) x
v(0, x) = U0 (x) − w(0, x) ≡ v0 (x),
vt + {f (w + v) − f (w)}x = µvxx − µ
and
(2.12) Vt + {f (U + V ) − f (U )}x + g(U + V )y = µ∆V,
(2.13)
(t, x, y) ∈ R+ × R2 ,
V (0, x, y) = u0 (x, y) − U0 (x) ≡ V0 (x, y).
The perturbations v and V satisfy the following theorems, respectively.
Theorem 2 (Decay estimate). Suppose that v0 ∈ H 2 (R) ∩ L1 (R). Then the problem (2.10),(2.11) has a unique global solution v(t, x) satisfying
v ∈ C 0 ([0, ∞); H 2 (R)) ∩ C 0 ([0, ∞); L1 (R)),
vx ∈ L2 (0, T ; H 2(R)),
and
(2.14)
1
kv(t, ·)kL2 (R) ≤ C(1 + t)− 4 log(2 + t).
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1206
MASATAKA NISHIKAWA AND KENJI NISHIHARA
Theorem 3 (Decay estimate). Suppose that V0 ∈ H 2 (R2 ) ∩ L1 (R2 ). Then, the
problem (2.12),(2.13) has a unique global solution V (t, x, y) satisfying
V ∈ C 0 ([0, ∞); H 2 (R)),
∇V ∈ L2 (0, ∞; H 2 (R)),
and
(2.15)
3
sup kV (t, ·, y)kL2 (Rx ) ≤ C(1 + t)− 4 .
y∈R
Theorem 2, Theorem 3 and Lemma 1 (iv) yield the desired estimate (1.8). In
the next two sections, we devote ourselves to the proofs of Theorems 2 and 3,
respectively.
3. Decay estimates for the perturbation v
We begin with the Cauchy problem
(3.1)
f 000 (w) 2
w ,
(t, x) ∈ R+ × R,
f 00 (w) x
v(0, x) = U (0, x) − w(0, x) ≡ v0 (x).
vt + {f (w + v) − f (w)}x = µvxx − µ
(3.2)
We shall show that the problem (3.1),(3.2) has a unique global solution in the
solution space X(0, ∞), where
 
 ψ ∈ C 0 ([0, T ]; H 2(R)), ψx ∈ L2 (0, T ; H 2 (R)) 
.
XM (0, T ) = ψ and
sup kψ(t, ·)kH 2 ≤ M
 
[0,T ]
In what follows, we often abbreviate the domain R of H 2 (R), etc.
Proposition 1 (Local existence). Suppose that v0 ∈ H 2 (R). For any M > 0,
there exists a positive constant T0 depending on M such that if kv0 kH 2 ≤ M, then
the problem (3.1), (3.2) has a unique solution v(t, x) ∈ X2M (0, T0 ).
Proposition 1 can be proved in a standard way. So we omit the proof. Next, we
show a priori estimates of v.
Proposition 2 (A priori estimate). Suppose that v is a solution of (3.1),(3.2) in
XM (0, T ) for positive constants T and M . Then there exists a positive constant C0
such that
Z tZ
Z t
2
2
(3.3)
wx v dxdτ +
kvx (τ )k2H 2 dτ ≤ C0 (kv0 k2H 2 + 1).
kv(t)kH 2 +
0
R
0
Proof. Multiplying (3.1) by v, we have
(3.4)
Z
Z
Z
Z
f 000 (w) 2
1 d
wx dx.
v 2 dx +
v{f (w + v) − f (w)}x dx + µ
vx2 dx = −
v 00
2 dt R
R
R
R f (w)
The second term of (3.4) is estimated by the following:
Z
Z
v{f (w + v) − f (w)}x dx = −
vx {f (w + v) − f (w)}dx
R
R
Z
Z
w+v
0
(3.5) =
−
f (y)dy − f (w)v
+ {f (w + v) − f (w) − f (w)v}wx dx
R
w
x
Z
α
≥
wx v 2 dx.
2 R
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ASYMPTOTICS TOWARD THE PLANAR RAREFACTION WAVES
1207
The right hand side is estimated as follows:
Z
Z
Z
Z
000
α
2
2
v f (w) wx2 dx ≤ C
w
|v|dx
≤
w
|v|
dx
+
C
wx3 dx.
x
x
00 (w)
f
4
R
R
R
R
Integrating (3.4) over [0,t] and using Lemma 1 (iii), we get
Z tZ
Z t
2
2
(3.6)
wx v dxdτ +
kvx (τ )k2 dτ ≤ C0 (kv0 k2 + 1).
kv(t)k +
0
R
0
2
Here and later, by k·k we denote the L -norm in R or R2 without confusions. Next,
we derive the higher order estimates. Multiplying (3.1) by (−vxx ), we have
Z
Z
Z
1 d
2
2
v dx −
vxx {f (w + v) − f (w)}x dx + µ
vxx
dx
2 dt R x
R
R
Z
(3.7)
f 000 (w) 2
w dx.
=µ
vxx 00
f (w) x
R
The right-hand side is estimated as
Z
000
vxx f (w) w2 dx ≤ 1 kvxx k2 + Ckwx k4 4 .
L
4
00 (w) x
f
R
The second term of (3.7) is estimated as
Z
Z
2
vxx {f (w + v) − f (w)}x dx ≤ µ kvxx k2 + C kvx (t)k2 +
wx v dx .
4
R
R
Here, the maximum principle for a parabolic equation has been employed. Hence,
we have
Z t
2
(3.8)
kvxx (τ )k2 dτ ≤ C0 (kv0 k2H 1 + 1).
kvx (t)k +
0
Differentiating (3.1) twice in x, and multiplying it by vxx , we have
Z
1 d
2
kvxx (t)k +
vxx {f (w + v) − f (w)}xxx dx + µkvxxx (t)k2
2 dt
R
000
Z
f (w) 2
w
= −µ
vxx
dx,
f 00 (w) x xx
R
which yields
(3.9)
2
kvxx (t)k +
Z
0
t
kvxxx (τ )k2 dτ ≤ C0 (kv0 k2H 2 + 1).
Thus, the proof of Proposition 3 is complete.
Combining Proposition 1 with Proposition 2, we obtain the global result.
Theorem 4 (Global existence). Suppose that v0 (x) ∈ H 2 (R). Then the problem
(3.1),(3.2) has a unique global solution v(t, x) satisfying
v ∈ C 0 ([0, ∞); H 2 (R)),
vx ∈ L2 (0, ∞; H 2 (R)),
and the estimate (3.3).
In order to obtain the decay order of v, we further assume that v0 ∈ L1 (R).
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1208
MASATAKA NISHIKAWA AND KENJI NISHIHARA
Lemma 3. Suppose that v0 ∈ L1 (R) ∩ H 2 (R). Then the solution v(t, x) also satisfies
kv(t)kL1 ≤ kv0 kL1 + C1 log(1 + t),
(3.10)
where C1 is a constant depending on |u+ − u− |.
Proof. The L1 -estimate (3.10) of v can be proved by the same method as that in
[4]. So we omit the proof.
Theorem 5 (Decay estimate). Suppose that v0 ∈ H 2 (R) ∩ L1 (R). Then, for any
0 < ε < 12 , the solution v(t, x) of (3.1),(3.2) satisfies
1
(3.11)
(1 + t)k+ 2 +ε k∂xk v(t)k2
Z
Z t
1
+
(1 + τ )k+ 2 +ε
wx |∂xk v(τ )|2 dx + k∂xk vx (τ )k2 dτ
0
≤ CIk (1 + t)ε ρk (t),
(3.12)
R
k = 0, 1,
(1 + t)2+ε k∂x2 v(t)k2
Z
Z t
+
(1 + τ )2+ε
wx |∂x2 v(τ )|2 dx + k∂x2 vx (τ )k2 dτ
0
≤ CI2 (1 + t)ε ρ2 (t),
R
where
Ik = (kv0 kL1 + kv0 kH k + 1)2 , k = 0, 1, ρ0 = log2 (2 + t), ρ1 = log10 (2 + t),
and
70
I2 = (kv0 kL1 + kv0 kH 2 + 1) 3 , ρ2 = log6 (2 + t).
Remark. The estimate (3.11) with k = 0 shows (2.14) in Theorem 2.
Proof. The proof is similar to one in Ito [4]. However, the smooth rarefaction wave
w(t, x) in [4] is different from ours and its estimates are done for the linearized
equation around w(t, x). Hence, we give the outline of the proof.
First, we show (3.11) with k = 0. From (3.4) and Lemma 1 (iii), we have
Z
d
kv(t)k2 +
(3.13)
wx v 2 dx + kvx (t)k2 ≤ C(1 + t)−2 .
dt
R
1
Multiplying (3.13) by (1 + t) 2 +ε , we have
(3.14)
d
1
1
{(1 + t) 2 +ε kv(t)k2 } + (1 + t) 2 +ε
dt
1
Z
wx v
R
− 32 +ε
≤ C{(1 + t)− 2 +ε kv(t)k2 + (1 + t)
4
dx + kvx (t)k
}.
By the Gagliardo-Nirenberg inequality
(3.15)
2
2
kv(t)k2 ≤ Ckv(t)kL3 1 (R) kvx (t)k 3 ,
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2
ASYMPTOTICS TOWARD THE PLANAR RAREFACTION WAVES
we obtain
1
1
d
{(1 + t) 2 +ε kv(t)k2 } + (1 + t) 2 +ε
dt
4
1
Z
R
2
wx v dx + kvx (t)k
2
2
1209
3
≤ C{(1 + t)− 2 +ε kv(t)kL3 1 kvx (t)k 3 + (1 + t)− 2 +ε }
1
3
1
≤ (1 + t) 2 +ε kvx (t)k2 + C{(1 + t)−1+ε kv(t)k2L1 + (1 + t)− 2 +ε }
2
1
1
3
≤ (1 + t) 2 +ε kvx (t)k2 + C{(1 + t)−1+ε (kv0 k2L1 + C1 log2 (1 + t)) + (1 + t)− 2 +ε };
2
that is,
Z
1
1
d
+ε
2
+ε
2
2
2
2
{(1 + t)
kv(t)k } + (1 + t)
wx v dx + kvx (t)k
dt
(3.16)
R
3
≤ C{(1 + t)−1+ε (kv0 k2L1 + C1 log2 (1 + t)) + (1 + t)− 2 +ε }.
Integrating (3.16) over [0,t] in t, we obtain (3.11) with k = 0.
Next, we derive (3.11) with k = 1. From (3.7), we have
Z
1 d
kvx (t)k2 −
(3.17)
vxx {f (w + v) − f (w)}x dx + µkvxx (t)k2 ≤ C(1 + t)−3 .
2 dt
R
Here
Z
vxx {f (w + v) − f (w)}x dx
Z
Z 1
1 3
vx − vxx {f 0 (w + v) − f 0 (w)}wx dx.
=
f 00 (w + v)wx vx2 dx +
2 R
R 2
−
R
Hence, due to (1.3), we have
Z
d
2
kvx (t)k + α
wx vx2 dx + kvxx (t)k2
dt
R
Z
Z
(3.18)
≤C
|vxx ||v||wx |dx +
|vx |3 dx + (1 + t)−3
R
R
Z
1
2
2 2
≤ kvxx (t)k + C
wx v dx + kvx (t)k3L3 + (1 + t)−3 .
2
R
3
Multiplying (3.18) by (1 + t) 2 +ε , we have
(3.19)
Z
d
3
3
3
{(1 + t) 2 +ε kvx (t)k2 } + α(1 + t) 2 +ε
wx vx2 dx + (1 + t) 2 +ε kvxx (t)k2
dt
ZR
1
3
≤ C (1 + t) 2 +ε kvx (t)k2 + (1 + t) 2 +ε
wx2 v 2 dx
R
o
3
3
+(1 + t) 2 +ε kvx (t)k3L3 + (1 + t)− 2 +ε .
Noting that
3
(1 + t) 2 +ε
Z
Z
3
wx2 v 2 dx ≤ (1 + t) 2 +ε kwx (t)kL∞
wx v 2 dx
R
R
Z
1
+ε
2
wx v 2 dx,
≤ C(1 + t)
R
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1210
MASATAKA NISHIKAWA AND KENJI NISHIHARA
and making use of (3.11) with k = 0 and the Gagliardo-Nirenberg inequality
7
5
kvx (t)k3L3 (R) ≤ Ckvxx (t)kL4 2 (R) kv(t)kL4 2 (R) ,
(3.20)
we obtain
Z
Z t
3
3
(1 + τ ) 2 +ε α
wx vx2 dx + kvxx (τ )k2 dτ
(1 + t) 2 +ε kvx2 (t)k2 +
R
0
Z t
3
(1 + τ ) 2 +ε kv(τ )k10
dτ
≤ C I0 (1 + t)ε ρ0 +
L2 (R)
0
Z t
3
ε
+ε
− 12
5
2
(1 + τ )
(I0 (1 + τ ) ρ0 ) dτ ,
≤ C I0 (1 + t) ρ0 +
0
which yields (3.11) with k = 1. Finally, multiply (3.9) by (1 + t)2+ε and use (3.11).
After several calculations, we can obtain the desired estimate (3.12). Though the
5
details are omitted, we cannot multiply (3.9) by (1 + t) 2 +ε in our method. Because
5
we have the decay order kwxx (t)k2 = O(t−2 ), not O(t− 2 ) (cf. Ito [4]).
Thus the proof is complete.
4. Decay estimates for the perturbation V
In this section, we consider the Cauchy problem in two space dimension:
(4.1)
Vt + {f (U + V ) − f (U )}x + g(U + V )y = ∆V,
(4.2)
V (0, x, y) = V0 (x, y) ≡ u0 (x, y) − U0 (x).
The solution space is
 
 ψ ∈ C 0 ([0, T ]; H 2 (R2 )), ∇ψ ∈ L2 (0, T ; H 2 (R2 )) 
X̃M (0, T ) = ψ and
,
sup kψ(t, ·, ·)kH 2 ≤ M
 
[0,T ]
with T > 0. Then we have
Proposition 3 (Local existence). Suppose that V0 ∈ H 2 (R2 ). For any M > 0,
there exists a positive constant T0 depending on M such that if kV0 kH 2 ≤ M, then
the problem (4.1), (4.2) has a unique solution V (t, x, y) ∈ X̃2M (0, T0 ).
Proposition 3 can be proved in a standard way. So we omit the proof. Next, we
show a priori estimates of V .
Proposition 4 (A priori estimate). Suppose that V is a solution of (4.1),(4.2) in
X̃M (0, T ) for positive constants T and M . Then there exists a positive constant C1
depending on V0 such that
Z tZ
Z t
(4.3)
Ux V 2 dxdydτ +
k∇V (τ )k2H 2 dτ ≤ C1 kV0 k2H 2 .
kV (t)k2H 2 +
0
R2
0
Proof. Multiplying (4.1) by V and integrating the resultant equation over R2 , we
have
Z
1 d
kV (t)k2 +
V {f (U + V ) − f (U )}x dxdy
2 dt
2
ZR
(4.4)
V g(U + V )y dxdy + µk∇V (t)k2 = 0.
+
R2
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ASYMPTOTICS TOWARD THE PLANAR RAREFACTION WAVES
1211
The second and third terms are, respectively, estimated as follows:
Z
Z
V {f (U + V ) − f (U )}x dxdy = −
Vx {f (U + V ) − f (U )}dxdy
R2
R2
"
!
Z
Z
−
=
R2
U +V
U
f (y)dy − f (U )V
x
#
0
+ {f (U + V ) − f (U ) − f (U )V }Ux dxdy
(4.5)
α
≥
2
Z
R2
Ux V 2 dxdy.
Since U is independent of y,
Z
Z
V g(U + V )y dxdy = −
R2
R2
Vy g(U + V )dxdy
Z
Z
=−
(4.6)
R2
∂y
!
U +V
g(ξ)dξ
dxdy = 0.
U
Using (4.5) and (4.6), we have the basic estimate
Z tZ
Z t
kV (t)k2 +
(4.7)
Ux V 2 dxdydτ +
k∇V (τ )k2 dτ ≤ CkV0 k2 .
0
R2
0
The estimates of the derivatives in x, y of V can be obtained similarly to those in
Proposition 2. We omit the details.
The combination of Propositions 3 and 4 gives the global result.
Theorem 6 (Global existence). Suppose that V0 (x) ∈ H 2 (R2 ). Then the problem
(4.1),(4.2) has a unique global solution V (t, x, y) satisfying
V ∈ C 0 ([0, ∞); H 2 (R)),
∇V ∈ L2 (0, ∞; H 2 (R)).
and the estimate (4.3).
We now show the decay estimates on V . As in Lemma 3, the following L1 estimate plays an important roll.
Lemma 4 (Ito [4]). Suppose further, in Theorem 6, that V0 ∈ L1 (R2 ). Then the
solution V (t, x, y) also satisfies
kV (t)kL1 (R2 ) ≤ kV0 kL1 (R2 ) .
(4.8)
Applying Lemma 4, we have the following theorem.
Theorem 7 (Decay estimate). Suppose that V0 (x, y) ∈ H 2 (R2 ) ∩ L1 (R2 ) and let
V (t, x, y) be the solution of (4.1),(4.2). Then, for any ε > 0, there exists a constant
C > 0 such that the following decay estimates hold:
(4.9)
(1 + t)1+ε kV (t)k2
Z
Z t
1+ε
+
(1 + τ )
0
2
R2
Ux |V (τ )| dxdy + k∇V (τ )k
≤ C(1 + t)ε (kV0 kL1 + kV0 k)2 ,
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2
dτ
1212
MASATAKA NISHIKAWA AND KENJI NISHIHARA
15
(4.10)
(1 + t) 8 +ε kVx (t)k2
Z
Z t
15
+
(1 + τ ) 8 +ε
0
R2
4
ε
Ux |Vx (τ )|2 dxdy + k∇Vx (τ )k2 dτ
≤ C(1 + t) log (2 + t)(kV0 kL1 + kV0 kH 1 )2 ,
(4.11)
(1 + t)2+ε kVy (t)k2
Z
Z t
2+ε
+
(1 + τ )
2
R2
0
Ux |Vy (τ )| dxdy + k∇Vy (τ )k
2
dτ
≤ C(1 + t)ε (kV0 kL1 + kV0 kH 1 )2 ,
39
(4.12)
(1 + t) 16 +ε kVxx (t)k2
Z
Z t
39
+ε
16
+
(1 + τ )
2
R2
0
Ux |Vxx (τ )| dxdy + k∇Vxx (τ )k
2
dτ
≤ C(1 + t)ε log8 (2 + t)(kV0 kL1 + kV0 kH 2 )2 ,
23
(4.13)
(1 + t) 8 +ε kVxy (t)k2
Z
Z t
23
+ε
+
(1 + τ ) 8
R2
0
2
Ux |Vxy (τ )| dxdy + k∇Vxy (τ )k
2
dτ
≤ C(1 + t)ε log8 (2 + t)(kV0 kL1 + kV0 kH 2 )2 ,
(4.14)
(1 + t)3+ε kVyy (t)k2
Z
Z t
3+ε
+
(1 + τ )
R2
0
2
Ux |Vyy (τ )| dxdy + k∇Vyy (τ )k
2
dτ
≤ C(1 + t)ε (kV0 kL1 + kV0 kH 2 )2 .
Remark. From (4.9) and (4.11), the estimate (2.15) in Theorem 3 is obtained as
sup kV (t, ·, y)k2 ≤ CkV (t, ·, ·)kkVy (t, ·, ·)k
y
1
3
≤ C(1 + t)− 2 −1 = C(1 + t)− 2 .
Proof. From (4.4)–(4.6), we get
(4.15)
1
1 d
kV (t)k2 + α
2 dt
2
Z
R2
Ux V 2 dxdy + µk∇V (t)k2 ≤ 0.
Multiplying (4.15) by 2(1 + t)1+ε , we have
(4.16)
Z
d
1+ε
2
1+ε
2
2
{(1 + t) kV (t)k } + (1 + t)
α
Ux V dxdy + 2k∇V (t)k
dt
R2
≤(1 + ε)(1 + t)ε kV (t)k2 .
By the Gagliardo-Nirenberg inequality
kV (t)k2L2 (R2 ) ≤ CkV (t)kL1 (R2 ) k∇V (t)kL2 (R2 ) ,
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ASYMPTOTICS TOWARD THE PLANAR RAREFACTION WAVES
we obtain
1213
Z
d
{(1 + t)1+ε kV (t)k2 } + (1 + t)1+ε α
Ux V 2 dxdy + 2k∇V (t)k2
dt
R2
≤ C(1 + t)
(4.17)
1+ε
2
1+ε
≤ (1 + t)
k∇V (t)k(1 + t)
ε−1
2
kV (t)kL1 (R2 )
2
k∇V (t)k + C(1 + t)ε−1 kV (t)k2L1 (R2 )
≤ (1 + t)1+ε k∇V (t)k2 + C(1 + t)ε−1 kV0 k2L1 (R2 ) .
Integrating (4.17) over [0, t] in t, we obtain (4.9). Next, we estimate Vy and Vx .
∂
(4.1) by Vy , we have
First, multiplying ∂y
Z
1 d
kVy (t)k2 +
Vy {f (U + V ) − f (U )}xy dxdy
2 dt
2
ZR
(4.18)
Vy g(U + V )yy dxdy + µk∇Vy (t)k2 = 0.
+
R2
The integration by parts gives:
The second and third terms of (4.18)
Z
Z
1
1
00
2
=
f (U + V )Ux Vy dxdy +
{f 00 (U + V )Vx Vy2 + g 00 (U + V )Vy3 }dxdy.
2 R2
2 R2
Hence,
Z
α
1 d
kVy (t)k2 +
Ux Vy2 dxdy + µk∇Vy (t)k2
2 dt
2 R2
Z
≤C
(|Vy |3 + |Vx ||Vy |2 )dxdy.
(4.19)
R2
Since
Z
C
|Vx ||Vy |2 dxdy
Z
sup |Vx (t, x, y)|kVy (t, x, ·)k2L2 (Ry ) dx
≤C
R y∈R
Z
1
1
kVx (t, x, ·)kL2 2 (Ry ) kVxy (t, x, ·)kL2 2 (Ry ) kVy (t, x, ·)k2L2 (Ry ) dx
≤C
R
Z
2
8
µ
kVx (t, x, ·)kL3 2 (Ry ) kVy (t, x, ·)kL3 2 (Ry ) dx
≤ kVxy (t)k2L2 (R2 ) + C
4
R
µ
≤ kVxy (t)k2L2 (R2 )
4
Z
R2
2
4
4
kVx (t, x, ·)kL3 2 (Ry ) kV (t, x, ·)kL3 2 (Ry ) kVyy (t, x, ·)kL3 2 (Ry ) dx
R
Z
µ
kVx (t, x, ·)k2L2 (Ry ) kV (t, x, ·)k4L2 (Ry ) dx
≤ k∇Vy (t)k2L2 (R2 ) + C
4
R
µ
≤ k∇Vy (t)k2L2 (R2 ) + C sup kV (t, x, ·)k4L2 (Ry ) kVx (t)k2L2 (R2 )
4
x∈R
µ
2
≤ k∇Vy (t)kL2 (R2 ) + CkV (t)k2L2 (Ry ) kVx (t)k4L2 (R2 )
4
+C
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1214
MASATAKA NISHIKAWA AND KENJI NISHIHARA
and
Z
C
R2
|Vy |3 dxdy ≤
we have
(4.20)
µ
k∇Vy (t)k2L2 (R2 ) + CkV (t)k2L2 (R2 ) kVx (t)k2L2 (R2 ) kVy (t)k2L2 (R2 ) ,
4
Z
α
µ
1 d
kVy (t)k2 +
Ux Vy2 dxdy + k∇Vy (t)k2
2 dt
2 R2
2
2
2
≤ CkV (t)kL2 (R2 ) kVx (t)kL2 (R2 ) (kVx (t)k2L2 (R2 ) + kVy (t)k2L2 (R2 ) ).
Noting that kV (t)k2L2 (R2 ) ≤ C(1 + t)−1 , we multiply (4.20) by (1 + t)2+ε and
∂
(4.1) by Vx . Then,
integrating it over [0, t] to obtain (4.11). Second, multiplying ∂x
after similar calculations to the above, we have
Z
α
µ
1 d
kVx (t)k2 +
Ux |Vx |2 dxdy + k∇Vx (t)k2
2 dt
2 R2
2
(4.21)
≤ CkV (t)k2L2 (R2 ) kVx (t)k2L2 (R2 ) (kVx (t)k2L2 (R2 ) + kVy (t)k2L2 (R2 ) )
Z
+ CkUx (t)kL∞
Ux V 2 dxdy.
R2
7
Since kUx (t)kL∞ (R) ≤ kwx (t)kL∞ (R) + kvx (t)kL∞ (R) ≤ C(1 + t)− 8 log4 (2 + t) by
15
virtue of Theorem 5, we can multiply (4.21) by (1 + t) 8 +ε , not (1 + t)2+ε , to obtain
(4.10). The estimates (4.12)–(4.14) for the second derivatives of V are obtained
by more complicated calculations than those for the first derivatives. We omit the
details.
Thus the proof of Theorem 7 is complete.
Acknowledgment
The work of the second author was supported in part by Waseda University
Grant for Special Research Project 96A-199.
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Department of Mathematics, School of Science and Engineering, Waseda University,
3-4-1 Okubo, Shinjuku, Tokyo 169, Japan
E-mail address: [email protected]
School of Political Science and Economics, Waseda University Tokyo, 169-50, Japan
E-mail address: [email protected]
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