A singular limit problem for the Rosenau-Korteweg-de Vries-regulared long wave and Rosenau-korteweg-de Viers equation

0.2 MB·English

by  G. M. Coclite

A SINGULAR LIMIT PROBLEM
FOR THE ROSENAU-KORTEWEG-DE VRIES
-REGULARIZED LONG WAVE
AND ROSENAU-KORTEWEG-DE VRIES EQUATION.
GIUSEPPE MARIACOCLITEANDLORENZODIRUVO
Abstract. We consider the Rosenau-Korteweg-de Vries-regularized long wave and Rosenau-
5
Korteweg-de Vries equations, which contain nonlinear dispersive effects. We prove that, as
1
thediffusionparameter tends tozero, thesolutions ofthedispersiveequations converge tothe
0
unique entropy solution of a scalar conservation law. The proof relies on deriving suitable a
2
priori estimates together with an application of the compensated compactness method in the

n Lp setting.
a
J

9
2
1. Introduction

]
P The dynamics of shallow water waves that is observed along lake shores and beaches
A
has beena research area for thepast fewdecades in thearea of oceanography (see [1,21]).
h. There are several models proposed in this context: Korteweg-de Vries (KdV) equation,
t Boussinesq equation, Peregrine equation, regularized long wave (RLW) equation, Kawa-
a
m hara equation, Benjamin-Bona-Mahoney equation, Bona-Chen equation and several oth-
ers. These models are derived from first principles under various different hypothesis and
[
approximations. They are all well studied and very well understood.
1
The dynamics of dispersive shallow water waves, on the other hand, is captured with
v
1 slightly differentmodelslikeRosenau-Kawahara, Rosenau-KdV,andRosenau-KdV-RLW
7 equations [2, 8, 9, 10, 14].
3
In particular, the Rosenau-KdV-RLW equation is
7
0 (1.1) ∂ u+a∂ u+k∂ un+b ∂3 u+b ∂3 u+c∂5 u= 0, a, k, b , b , c R.
. t x x 1 xxx 2 txx txxxx 1 2 ∈
1
0 Here u(t,x) is the nonlinear wave profile. The first term is the linear evolution one, while
5 a is the advection (or drifting) coefficient. The two dispersion coefficients are b and b .
1 2
1
The higher order dispersion coefficient is c, while the coefficient of nonlinearity is k where
:
v n is the nonlinearity parameter. These are all known and given parameters.
i
X In [14], the authors analyzed (1.1). They got solitary waves, shock waves and singular
r solitons along with conservation laws.
a
In the case n = 2, a = 0, k = 1, b = 1, b = 1, c= 1, we have
1 2
−
(1.2) ∂ u+∂ u2+∂3 u ∂3 u+∂5 u = 0.
t x xxx txx txxxx
−
Choosing n = 2, a = 0, k = 1, b = b = 0, c= 1, (1.1) reads
2 1
(1.3) ∂ u+∂ u2+∂5 u= 0,
t x txxxx
Date:January30,2015.
2000 Mathematics Subject Classification. 35G25, 35L65,35L05.
Key words and phrases. Singular limit, compensated compactness, Rosenau-KdV-RLW, equation, entropy
condition.
TheauthorsaremembersoftheGruppoNazionaleperl’AnalisiMatematica,laProbabilita`eleloroApplicazioni
(GNAMPA)oftheIstitutoNazionalediAltaMatematica(INdAM).
1
2 G.M.COCLITEANDL.DIRUVO
which is known as Rosenau equation (see [16, 17]). Existence and uniqueness of solutions
for (1.3) has been proved in [13].
Finally, if n = 2, a = 0, k = 1, b = 1, b = 0, c= 1, (1.1) reads
1 2
(1.4) ∂ u+∂ u2+∂3 u+∂5 u= 0,
t x xxx txxxx
which is known as Rosenau-KdV equation.
In[20], theauthor discussedthesolitary wave solutions and(1.4). In[9], a conservative
linear finite difference scheme for the numerical solution for an initial-boundary value
problem of Rosenau-KdV equation was considered. In [7, 15], the authors discussed
the solitary solutions for (1.4) with solitary ansatz method. The authors also gave two
invariants for (1.4). In particular, in [15], the authors studied the two types of soliton
solutions, one is a solitary wave and the other is a singular soliton. In [19], the authors
proposedanaveragelinearfinitedifferenceschemeforthenumericalsolutionoftheinitial-
boundary value problem for (1.4).
If n= 2, a = 0, k = 1, b = 0, b = 1, c = 1, (1.1) reads
1 2
−
(1.5) ∂ u+∂ u2 ∂3 u+∂5 u= 0,
t x txx txxxx
−
which is the Rosenau-RLW equation.
In this paper, we analyze (1.2) and (1.5). Arguing as [5], we re-scale the equations as
follows
(1.6) ∂ u+∂ u2+β∂3 u β∂3 u+β2∂5 u=0,
t x xxx txx txxxx
−
(1.7) ∂ u+∂ u2 β∂3 u+β2∂5 u=0,
t x txx txxxx
−
where β is the diffusion parameter.
We are interested in the no high frequency limit, we send β 0 in (1.6) and (1.7). In
→
this way we pass from (1.6) and (1.7) to the equation
(1.8) ∂ u+∂ u2 = 0
t x
which is a scalar conservation law. We prove that, when β 0, the solutions of (1.6) and
→
(1.7) converge to the unique entropy solution (1.8).
The paper is organized in three sections. In Section 2, we prove the convergence of
(1.6) to (1.8), while in Section 3, we show how to modify the argument of Section 2 and
prove the convergence of (1.7) to (1.8).
2. The Rosenau-KdV-RLW equation.
In this section, we consider (1.6) and augment it with the initial condition
(2.1) u(0,x) = u (x),
0
on which we assume that
(2.2) u L2(R) L4(R).
0
∈ ∩
We study the dispersion-diffusion limit for (1.6). Therefore, we consider the following
fifth order approximation
∂ u +∂ u2 +β∂3 u β∂3 u
t ε,β x ε,β xxx ε,β − txx ε,β
(2.3) +β2∂5 u = ε∂2 u , t > 0, x R,
 txxxx ε,β xx ε,β
∈
uε,β(0,x) = uε,β,0(x), x R,
∈


A SINGULAR LIMIT PROBLEM OF ROSENAU-KDV-RLW TYPE 3
where u is a C∞ approximation of u such that
ε,β,0 0
u u in Lp (R), 1 p < 4, as ε, β 0,
ε,β,0 → 0 loc ≤ →
u 2 + u 4 +(β+ε2) ∂ u 2 C , ε,β > 0,
k ε,β,0kL2(R) k ε,β,0kL4(R) k x ε,β,0kL2(R) ≤ 0
(2.4)
(βε+βε2+β2) ∂2 u 2 +(β2ε2+β3) ∂3 u 2 C , ε,β > 0,
xx ε,β,0 L2(R) xxx ε,β,0 L2(R) ≤ 0
β4 ∂4 u 2(cid:13) C(cid:13), ε,β > 0, (cid:13) (cid:13)
xxxx ε,β,0 L(cid:13)2(R) ≤ 0(cid:13) (cid:13) (cid:13)
and C is(cid:13)a constant(cid:13)independent on ε and β.
0 (cid:13) (cid:13)
The main result of this section is the following theorem.
Theorem 2.1. Assume that (2.2) and (2.4) hold. If
(2.5) β = (ε4),
O
then, there exist two sequences εn n∈N, βn n∈N, with εn,βn 0, and a limit function
{ } { } →
u L∞(R+;L2(R) L4(R)),
∈ ∩
such that
(2.6) u u strongly in Lp (R+ R), for each 1 p < 4;
εn,βn → loc × ≤
(2.7) u is the unique entropy solution of (1.8).
Let us prove some a priori estimates on u , denoting with C the constants which
ε,β 0
depend only on the initial data.
Lemma 2.1. For each t > 0,
u (t, ) 2 +β ∂ u (t, ) 2
k ε,β · kL2(R) k x ε,β · kL2(R)
(2.8) t
+β2 ∂2 u (t, ) 2 +2ε ∂ u (s, ) 2 ds C .
xx ε,β · L2(R) k x ε,β · kL2(R) ≤ 0
Z0
(cid:13) (cid:13)
In particular, we have (cid:13) (cid:13)
−1
(2.9) kuε,β(t,·)kL∞(R) ≤C0β 4,
−3
(2.10) k∂xuε,β(t,·)kL∞(R) ≤C0β 4.
Proof. Multiplying (2.3) by u , we have
ε,β
u ∂ u +2u2 ∂ u +βu ∂3 u
ε,β t ε,β ε,β x ε,β ε,β xxx ε,β
(2.11)
βu ∂3 u +β2u ∂5 u = εu ∂2 u .
ε,β txx ε,β ε,β txxxx ε,β ε,β xx ε,β
−
Since
1 d
u ∂ u dx = u (t, ) 2 ,
R ε,β t ε,β 2dt k ε,β · kL2(R)
Z
2 u2 ∂ u dx =0,
ε,β x ε,β
R
Z
β u ∂3 u dx = ∂ u ∂2 u dx = 0,
ε,β xxx ε,β x ε,β xx ε,β
R R
Z Z
β d
β u ∂3 u dx = ∂ u (t, ) 2 ,
− R ε,β txx ε,β 2 dt k x ε,β · kL2(R)
Z
β2 u ∂5 u dx = β2 ∂ u ∂4 u dx
ε,β txxxx ε,β x ε,β txxx ε,β
R − R
Z Z
4 G.M.COCLITEANDL.DIRUVO
β2 d
= ∂2 u (t, ) 2 ,
2 dt xx ε,β · L2(R)
(cid:13) (cid:13)
ε u ∂2 u dx = ε ∂(cid:13)u (t, ) 2(cid:13) .
R ε,β xx ε,β − k x ε,β · kL2(R)
Z
Integrating (2.11) on R, we get
d d
u (t, ) 2 +β ∂ u (t, ) 2
dt k ε,β · kL2(R) dt k x ε,β · kL2(R)
(2.12)
d
+β2 ∂2 u (t, ) 2 +2ε ∂ u (t, ) 2 = 0.
dt xx ε,β · L2(R) k x ε,β · kL2(R)
(2.8) follows from (2.4), (2.12) and(cid:13)an integratio(cid:13)n on (0,t).
(cid:13) (cid:13)
We prove (2.9). Due to (2.8) and the H¨older inequality,
x
u2 (t,x) =2 u ∂ u dx 2 u ∂ u dx
ε,β ε,β x ε,β ε,β x ε,β
−∞ ≤ R| || |
Z Z
−1
≤2kuε,β(t,·)kL2(R)k∂xuε,β(t,·)kL2(R) ≤ C0β 2.
Therefore,
−1
uε,β(t,x) C0β 4,
| | ≤
which gives (2.9).
Finally, we prove (2.10). Thanks to (2.8) and the H¨older inequality,
x
∂ u2 (t,x) =2 ∂ u ∂2 u dx 2 ∂ u ∂2 u dx
x ε,β x ε,β xx ε,β x ε,β xx ε,β
−∞ ≤ R| || |
Z Z
≤2k∂xuε,β(t,·)kL2(R) ∂x2xuε,β(t,·) L2(R) ≤ C0β−12C0β−1 ≤ C0β−23.
Hence, (cid:13) (cid:13)
(cid:13) (cid:13)
−3
k∂xuε,β(t,·)kL∞(R) ≤ C0β 4,
that is (2.10). (cid:3)
Following [3, Lemma 2.2], or [5, Lemma 2.2], or [6, Lemma 4.2], we prove the following
result.
Lemma 2.2. Assume (2.5). For each t > 0,
i) the family u is bounded in L∞(R+;L4(R));
ε,β ε,β
{ }
ii) the families ε∂ u , √βε∂2 u , ε√β∂2 u , εβ∂3 u ,
x ε,β ε,β xx ε,β ε,β xx ε,β ε,β xxx ε,β ε,β
β√β∂3 u { , β}∂4 {u are}boun{ded in L∞(R}+;L2{(R)); }
xxx ε,β ε,β xxxx ε,β ε,β
{ } { }
iii) the families √βε∂2 u , β√ε∂3 u , β√βε∂4 u ,
tx ε,β ε,β txx ε,β ε,β txxx ε,β ε,β
{ } { } { }
εβ√β∂3 u , √εu ∂ u , ε√ε∂2 u are bounded in
xxx ε,β ε,β ε,β x ε,β ε,β xx ε,β ε,β
{ } { } { }
L2(R+ R).
×
Moreover,
t
(2.13) β ∂ u (s, )∂2 u (s, ) ds C ε2, t > 0,
x ε,β · xx ε,β · L1(R) ≤ 0
Z0
(cid:13) t (cid:13)
(2.14) (cid:13) β2 ∂2 u (t, )(cid:13)2 ds C ε5, t > 0.
xx ε,β · L2(R) ≤ 0
Z0
(cid:13) (cid:13)
Proof. Let A, B, C be some posit(cid:13)ive constants(cid:13)which will be specified later. Multiplying
(2.3) by
u3 Aβε∂3 u Bε2∂2 u +Cβ2∂4 u ,
ε,β txx ε,β xx ε,β xxxx ε,β
− −
A SINGULAR LIMIT PROBLEM OF ROSENAU-KDV-RLW TYPE 5
we have
u3 Aβε∂3 u Bε2∂2 u +Cβ2∂4 u ∂ u
ε,β txx ε,β xx ε,β xxxx ε,β t ε,β
− −
(cid:0) +2 u3ε,β Aβε∂t3xxuε,β Bε2∂x2xuε,β +Cβ(cid:1)2∂x4xxxuε,β uε,β∂xuε,β
− −
+β(cid:0)u3ε,β Aβε∂t3xxuε,β Bε2∂x2xuε,β +Cβ2∂x4xxxuε,β(cid:1) ∂x3xxuε,β
(2.15) − −
β(cid:0)u3ε,β Aβε∂t3xxuε,β Bε2∂x2xuε,β +Cβ2∂x4xxxuε,β(cid:1)∂t3xxuε,β
− − −
+β2(cid:0) u3ε,β Aβε∂t3xxuε,β Bε2∂x2xuε,β +Cβ2∂x4xxxuε,β(cid:1) ∂t5xxxxuε,β
− −
= ε u3ε,(cid:0)β Aβε∂t3xxuε,β Bε2∂x2xuε,β +Cβ2∂x4xxxuε,β ∂x2(cid:1)xuε,β.
− −
We observe that (cid:0) (cid:1)
u3 Aβε∂3 u Bε2∂2 u +Cβ2∂4 u ∂ u dx
ε,β txx ε,β xx ε,β xxxx ε,β t ε,β
R − −
Z
(cid:0) 1 d (cid:1)
(2.16) = u (t, ) 4 +Aβε ∂2 u (t, ) 2
4dt k ε,β · kL4(R) tx ε,β · L2(R)
Bε2 d (cid:13) Cβ2 d (cid:13)
+ ∂ u (t, ) 2 (cid:13)+ ∂(cid:13)2 u (t, ) 2 .
2 dt k x ε,β · kL2(R) 2 dt xx ε,β · L2(R)
We have that (cid:13) (cid:13)
(cid:13) (cid:13)
2 u3 Aβε∂3 u Bε2∂2 u +Cβ2∂4 u u ∂ u dx
ε,β txx ε,β xx ε,β xxxx ε,β ε,β x ε,β
R − −
Z
(cid:0) (cid:1)
(2.17) = 2Aβε u ∂ u ∂3 u dx 2Bε2 u ∂ u ∂2 u dx
ε,β x ε,β txx ε,β ε,β x ε,β xx ε,β
− R − R
Z Z
2Cβ2 (∂ u )2∂3 u dx 2Cβ2 u ∂2 u ∂3 u dx.
x ε,β xxx ε,β ε,β xx ε,β xxx ε,β
− R − R
Z Z
Since
2Cβ2 (∂ u )2∂3 u 2Cβ2 u ∂2 u ∂3 u dx
x ε,β xxx ε,β ε,β xx ε,β xxx ε,β
− R − R
Z Z
(2.18) =5Cβ2 (∂x2xuε,β)2∂xuε,βdx
R
Z
5β2
= (∂ u )2∂3 u dx,
− 2 R x ε,β xxx ε,β
Z
it follows from (2.17) and (2.18) that
2 u3 Aβε∂3 u Bε2∂2 u +Cβ2∂4 u u ∂ u dx
ε,β txx ε,β xx ε,β xxxx ε,β ε,β x ε,β
R − −
Z
(cid:0) (cid:1)
= 2Aβε u ∂ u ∂3 u dx 2Bε2 u ∂ u ∂2 u dx
ε,β x ε,β txx ε,β ε,β x ε,β xx ε,β
(2.19) − R − R
Z Z
5Cβ2
(∂ u )2∂3 u dx
− 2 R x ε,β xxx ε,β
Z
.
We observe
β u3 Aβε∂3 u Bε2∂2 u +Cβ2∂4 u ∂3 u dx
ε,β txx ε,β xx ε,β xxxx ε,β xxx ε,β
R − −
(2.20) Z
(cid:0) (cid:1)
= 3β u2 ∂ u ∂2 u dx Aβ2ε ∂3 u ∂3 u dx.
ε,β x ε,β xx ε,β xxx ε,β txx ε,β
− R − R
Z Z
6 G.M.COCLITEANDL.DIRUVO
We get
β u3 Aβε∂3 u Bε2∂2 u +Cβ2∂4 u ∂3 u dx
ε,β txx ε,β xx ε,β xxxx ε,β txx ε,β
− R − −
Z
(cid:0) (cid:1)
(2.21) = 3β u2 ∂ u ∂2 u dx+Aβ2ε ∂3 u (t, ) 2
R ε,β x ε,β tx ε,β txx ε,β · L2(R)
Z
Bβε2 d C(cid:13)β3 d (cid:13)
+ ∂2 u (t, ) 2 + (cid:13) ∂3 u(cid:13) (t, ) 2 .
2 dt xx ε,β · L2(R) 2 dt xxx ε,β · L2(R)
(cid:13) (cid:13) (cid:13) (cid:13)
We have that (cid:13) (cid:13) (cid:13) (cid:13)
β2 u3 Aβε∂3 u Bε2∂2 u +Cβ2∂4 u ∂5 u dx
ε,β txx ε,β xx ε,β xxxx ε,β txxxx ε,β
R − −
Z
(cid:0) (cid:1)
(2.22) = 3β2 u2 ∂ u ∂4 u dx+Aβ3ε ∂4 u (t, ) 2
− R ε,β x ε,β txxx ε,β txxx ε,β · L2(R)
Z
Bβ2ε2 d Cβ4(cid:13)d (cid:13)
+ ∂3 u (t, ) 2 + (cid:13) ∂4 u ((cid:13)t, ) 2 .
2 dt xxx ε,β · L2(R) 2 dt xxxx ε,β · L2(R)
(cid:13) (cid:13) (cid:13) (cid:13)
Moreover, (cid:13) (cid:13) (cid:13) (cid:13)
ε u3 Aβε∂3 u Bε2∂2 u +Cβ2∂4 u ∂2 u dx
ε,β txx ε,β xx ε,β xxxx ε,β xx ε,β
R − −
Z
(cid:0) Aβε d (cid:1)
(2.23) = 3ε u (t, )∂ u (t, ) 2 ∂2 u (t, ) 2
− k ε,β · x ε,β · kL2(R)− 2 dt xx ε,β · L2(R)
ε3B ∂2 u (t, ) 2 β2εC ∂3 u(cid:13) (t, ) 2 (cid:13).
− xx ε,β · L2(R)− xxx (cid:13)ε,β · L2(R)(cid:13)
(cid:13) (cid:13) (cid:13) (cid:13)
It follows from (2.16), (2.19(cid:13)), (2.20), (2.(cid:13)21), (2.22), (2.(cid:13)23), and an i(cid:13)ntegration of (2.15)
on R that
d 1 Aβε+Bβε2+Cβ2
u (t, ) 4 + ∂2 u (t, ) 2
dt 4k ε,β · kL4(R) 2 xx ε,β · L2(R)
(cid:0) (cid:1) !
(cid:13) (cid:13)
d Bε2 Bβ2ε2(cid:13)+Cβ3 (cid:13)
+ ∂ u (t, ) 2 + ∂3 u (t, ) 2
dt 2 k x ε,β · kL2(R) 2 xxx ε,β · L2(R)
(cid:0) (cid:1) !
(cid:13) (cid:13)
Cβ4 d (cid:13) (cid:13)
+ ∂4 u (t, ) 2 +Aβε ∂2 u (t, )
2 dt xxxx ε,β · L2(R) tx ε,β · L2(R)
+Aβ2ε ∂(cid:13)3 u (t, ) 2 (cid:13) +Aβ3ε ∂(cid:13)4 u (t, )(cid:13)2
(cid:13)txx ε,β · L2((cid:13)R) (cid:13)txxx ε,β · (cid:13)L2(R)
+3ε u(cid:13) (t, )∂ u (t(cid:13), ) 2 +ε3B(cid:13) ∂2 u (t, )(cid:13)2
(2.24) k ε(cid:13),β · x ε,β (cid:13)· kL2(R) (cid:13) xx ε,β ·(cid:13)L2(R)
+β2εC ∂3 u (t, ) 2 (cid:13) (cid:13)
xxx ε,β · L2(R) (cid:13) (cid:13)
= 2Aβε (cid:13)(cid:13)uε,β∂xuε,β∂t3x(cid:13)(cid:13)xuε,βdx+2Bε2 uε,β∂xuε,β∂x2xuε,βdx
R R
Z Z
5Cβ2
(∂ u )2∂3 u dx 3β u2 ∂ u ∂2 u dx
2 R x ε,β xxx ε,β − R ε,β x ε,β xx ε,β
Z Z
+Aβ2ε ∂3 u ∂3 u 3β u2 ∂ u ∂2 u dx
xxx ε,β txx ε,β ε,β x ε,β tx ε,β
R − R
Z Z
+3β2 u2 ∂ u ∂4 u dx.
ε,β x ε,β txxx ε,β
R
Z
A SINGULAR LIMIT PROBLEM OF ROSENAU-KDV-RLW TYPE 7
Due to the Young inequality,
2Aβε u ∂ u ∂3 u dx Aε 2u ∂ u β∂3 u dx
ε,β x ε,β txx ε,β ε,β x ε,β txx ε,β
R ≤ R| || |
(cid:12)Z (cid:12) Z
(cid:12) (cid:12) Aβ2ε
(2.25) ≤(cid:12)(cid:12) 2Aεkuε,β(t,·)∂xuε,β(t,(cid:12)(cid:12)·)k2L2(R) + 2 ∂t3xxuε,β(t,·) 2L2(R),
(cid:13) (cid:13)
2Bε2 uε,β∂xuε,β∂x2xuε,βdx ε12uε,β∂xuε(cid:13),β ε232B∂x2xu(cid:13)ε,β dx
R ≤ R
(cid:12)(cid:12)(cid:12)Zε u (t, )∂ u (t, )(cid:12)(cid:12)(cid:12) 2 Z (cid:12)(cid:12)+2B2ε3 ∂2 (cid:12)(cid:12)u(cid:12)(cid:12) (t, ) 2 (cid:12)(cid:12).
≤(cid:12) 2 k ε,β · x ε,β ·(cid:12)kL2(R)(cid:12) xx(cid:12)(cid:12)ε,β · L2(R)(cid:12)
(cid:13) (cid:13)
Hence, from (2.24),
(cid:13) (cid:13)
d 1 Aβε+Bβε2+Cβ2
u (t, ) 4 + ∂2 u (t, ) 2
dt 4k ε,β · kL4(R) 2 xx ε,β · L2(R)
(cid:0) (cid:1) !
(cid:13) (cid:13)
d Bε2 Bβ2ε2(cid:13)+Cβ3 (cid:13)
+ ∂ u (t, ) 2 + ∂3 u (t, ) 2
dt 2 k x ε,β · kL2(R) 2 xxx ε,β · L2(R)
(cid:0) (cid:1) !
(cid:13) (cid:13)
Cβ4 d (cid:13) (cid:13)
+ ∂4 u (t, ) 2 +Aβε ∂2 u (t, ) 2
2 dt xxxx ε,β · L2(R) tx ε,β · L2(R)
Aβ2ε (cid:13) (cid:13) (cid:13) (cid:13)
+ ∂(cid:13)3 u (t, ) 2 (cid:13) +Aβ3ε ∂(cid:13)4 u (t, )(cid:13)2
2 txx ε,β · L2(R) txxx ε,β · L2(R)
(2.26) + 5 (cid:13)(cid:13)2A ε u (t,(cid:13)(cid:13))∂ u (t, ) 2(cid:13)(cid:13) +β2εC ∂(cid:13)(cid:13)3 u (t, ) 2
2 − k ε,β · x ε,β · kL2(R) xxx ε,β · L2(R)
(cid:18) (cid:19)
+ B 2B2 ε3 ∂2 u (t, ) 2 (cid:13)(cid:13) (cid:13)(cid:13)
− xx ε,β · L2(R)
5C(cid:0)β2 (cid:1) (cid:13) (cid:13)
(∂ u (cid:13))2 ∂3 u d(cid:13)x+3β u2 ∂ u ∂2 u dx
≤ 2 R x ε,β | xxx ε,β| R ε,β| x ε,β| xx ε,β|
Z Z
+Aβ2ε ∂3 u ∂3 dx+3β u2 ∂ u ∂2 u dx
xxx ε,β txx ε,β x ε,β tx ε,β
R| || | R | || |
Z Z
+3β2 u2 ∂ u ∂4 u dx.
ε,β x ε,β txxx ε,β
R | || |
Z
From (2.5), we have
(2.27) β D2ε4,
≤
where D is a positive constant which will be specified later. It follows from (2.10), (2.27)
and the Young inequality that
5Cβ2 5
2 ZR(∂xuε,β)2|∂x3xxuε,β|dx = β2CZR 2ε12(∂xuε,β)2(cid:12)ε12∂x3xxuε,β(cid:12)
25Cβ2 Cβ2ε (cid:12) (cid:12)
(∂ u )4dx+ ∂3 u (t, ) 2(cid:12) (cid:12)
≤ 8ε R x ε,β 2 xxx ε,β · L2(R)
Z
25C (cid:13) (cid:13) Cβ2ε
(2.28) β2 ∂ u (t, ) 2 ∂ u(cid:13) (t, ) 2 (cid:13)+ ∂3 u (t, ) 2
≤ 8ε k x ε,β · kL∞(R)k x ε,β · kL2(R) 2 xxx ε,β · L2(R)
C0β21 ∂ u (t, ) 2 + Cβ2ε ∂3 u (t, ) 2 (cid:13)(cid:13) (cid:13)(cid:13)
≤ ε k x ε,β · kL2(R) 2 xxx ε,β · L2(R)
Cβ2ε (cid:13) (cid:13)
C Dε ∂ u (t, ) 2 + (cid:13)∂3 u (t, )(cid:13)2 .
≤ 0 k x ε,β · kL2(R) 2 xxx ε,β · L2(R)
(cid:13) (cid:13)
(cid:13) (cid:13)
8 G.M.COCLITEANDL.DIRUVO
Due to (2.9), (2.27) and the Young inequality,
3β u2 ∂ u ∂2 u dx 3β u (t, ) 2 ∂ u ∂2 u dx
R ε,β| x ε,β| xx ε,β| ≤ k ε,β · kL∞(R) R| x ε,β| xx ε,β|
Z Z
(2.29) C0β21 ∂xuε,β ∂x2xuε,β dx ε12∂xuε,β C0Dε23∂x2xuε,β dx
≤ R| || | ≤ R
Z Z (cid:12) (cid:12)(cid:12) (cid:12)
ε ∂ u (t, ) 2 +D2C2ε3 ∂2(cid:12)u (t, )(cid:12)(cid:12)2 . (cid:12)
≤ k x ε,β · kL2(R) 0 x(cid:12)x ε,β · (cid:12)(cid:12)L2(R) (cid:12)
(cid:13) (cid:13)
(cid:13) (cid:13)
Thanks to the Young inequality,
1
Aβ2ε ∂3 u ∂3 u dx = Aβ2ε 2∂3 u ∂3 u dx
R| xxx ε,β|| txx ε,β| R xxx ε,β 2 txx ε,β
(2.30) Z Z (cid:12) (cid:12)
Aβ2(cid:12)ε (cid:12)(cid:12) (cid:12)
≤ 2Aβ2ε ∂x3xxuε,β(t,·) 2L2(R)+ 8 (cid:12) ∂t3xxuε,β(cid:12)((cid:12)(cid:12)t,·) 2L2(R)(cid:12)(cid:12).
(cid:13) (cid:13) (cid:13) (cid:13)
(cid:13) (cid:13) (cid:13) (cid:13)
It follows from (2.9), (2.27) and the Young inequality that
3u2 ∂ u
3βZRu2ε,β|∂xuε,β||∂t2xuε,β|dx = βZR(cid:12)(cid:12) εε,β12Ax21 ε,β(cid:12)(cid:12)(cid:12)ε12A21∂t2xuε,β(cid:12)dx
9β βε(cid:12)A (cid:12)(cid:12) (cid:12)
u4 (∂ u )2dx+ (cid:12) ∂2 u (t(cid:12),(cid:12)) 2 (cid:12)
≤ 2εA R ε,β x ε,β 2(cid:12) tx ε,β (cid:12) · L2(R)
Z
9 (cid:13) (cid:13)
β u (t, ) 2 u (t, )(cid:13)∂ u (t, ) (cid:13)2
(2.31) ≤ 2εA k ε,β · kL∞(R)k ε,β · x ε,β · kL2(R)
βεA
+ ∂2 u (t, ) 2
2 tx ε,β · L2(R)
C0β12 (cid:13)(cid:13)u (t, )∂ u(cid:13)(cid:13) (t, ) 2 + βεA ∂2 u (t, ) 2
≤ εA k ε,β · x ε,β · kL2(R) 2 tx ε,β · L2(R)
C0Dε u (t, )∂ u (t, ) 2 + βεA(cid:13)(cid:13)∂2 u (t, )(cid:13)(cid:13)2 .
≤ A k ε,β · x ε,β · kL2(R) 2 tx ε,β · L2(R)
(cid:13) (cid:13)
(cid:13) (cid:13)
Again by (2.9), (2.27) and the Young inequality,
3β2ZRu2ε,β|∂xuε,β||∂t4xxxuε,β|dx = ZR(cid:12)(cid:12)3β12uε2ε12,βA∂21xuε,β(cid:12)(cid:12)(cid:12)β32ε12A12∂t4xxxuε,β(cid:12)dx
9β β3ε(cid:12)A (cid:12)(cid:12) (cid:12)
≤ 2εA Ru4ε,β(∂xuε,β)2dx+ 2(cid:12)(cid:12) ∂t4xxxuε,β(t,(cid:12)(cid:12)·)(cid:12) 2L2(R) (cid:12)
Z
9 (cid:13) (cid:13)
β u (t, ) 2 u (t, )∂(cid:13) u (t, ) 2 (cid:13)
(2.32) ≤ 2εA k ε,β · kL∞(R)k ε,β · x ε,β · kL2(R)
β3εA
+ ∂4 u (t, ) 2
2 txxx ε,β · L2(R)
C0β12 u(cid:13)(cid:13) (t, )∂ u ((cid:13)(cid:13)t, ) 2 + β3εA ∂4 u (t, ) 2
≤ εA k ε,β · x ε,β · kL2(R) 2 txxx ε,β · L2(R)
C D β3εA(cid:13) (cid:13)
0 ε u (t, )∂ u (t, ) 2 + (cid:13)∂4 u (t, )(cid:13)2 .
≤ A k ε,β · x ε,β · kL2(R) 2 txxx ε,β · L2(R)
(cid:13) (cid:13)
(cid:13) (cid:13)
A SINGULAR LIMIT PROBLEM OF ROSENAU-KDV-RLW TYPE 9
From (2.26), (2.28), (2.29), (2.30), (2.31) and (2.32), we get
d 1 Aβε+Bβε2+Cβ2
u (t, ) 4 + ∂2 u (t, ) 2
dt 4k ε,β · kL4(R) 2 xx ε,β · L2(R)
(cid:0) (cid:1) !
(cid:13) (cid:13)
d Bε2 Bβ2ε2(cid:13)+Cβ3 (cid:13)
+ ∂ u (t, ) 2 + ∂3 u (t, ) 2
dt 2 k x ε,β · kL2(R) 2 xxx ε,β · L2(R)
(cid:0) (cid:1) !
(cid:13) (cid:13)
Cβ4 d βεA (cid:13) (cid:13)
+ ∂4 u (t, ) 2 + ∂2 u (t, ) 2
2 dt xxxx ε,β · L2(R) 2 tx ε,β · L2(R)
3Aβ2ε (cid:13) (cid:13) Aβ3ε (cid:13) (cid:13)
(2.33) + (cid:13)∂3 u (t, ) 2(cid:13) + (cid:13)∂4 u (t, (cid:13)) 2
8 txx ε,β · L2(R) 2 txxx ε,β · L2(R)
C(cid:13) (cid:13) (cid:13) (cid:13)
+β2ε (cid:13) 2A ∂3 (cid:13)u (t, ) 2 (cid:13) (cid:13)
2 − xxx ε,β · L2(R)
(cid:18) (cid:19)
5 C D(cid:13) (cid:13)
+ 2A 0 (cid:13) ε u (t, )(cid:13)∂ u (t, ) 2
2 − − A k ε,β · x ε,β · kL2(R)
(cid:18) (cid:19)
+ B 2B2 D2C2 ε3 ∂2 u (t, ) 2
− − 0 xx ε,β · L2(R)
C(cid:0)ε ∂ u (t, ) 2 (cid:1). (cid:13) (cid:13)
≤ 0 k x ε,β · kL2(R) (cid:13) (cid:13)
We search A, B, C such that
5 C D
0
2A > 0,
2 − − A
B 2B2 D2C2 > 0,
C− − 0
 2A > 0,
2 −

that is 

4A2 5A+2C D < 0,
0
−
(2.34) 2B2 B D2C2 < 0,
 − − 0
C >4A.
We choose 

(2.35) C = 6A.
The first inequality of (2.34) admits a solution, if
25 32C D > 0,
0
−
that is
25
(2.36) D < .
32C
0
The second inequality of (2.34) admits a solution, if
1 8D2C2 > 0,
− 0
that is
√2
(2.37) D < .
4C
0
It follows from (2.36) and (2.37) that
25 √2 √2
(2.38) D < min , = .
32C 4C 4C
( 0 0) 0
10 G.M.COCLITEANDL.DIRUVO
Therefore, from (2.34), (2.35) and (2.38), we have that there exist 0 < A < A and
1 2
0 < B < B , such that choosing
1 2
(2.39) A < A< A , B < B < B , C = 6A,
1 2 1 2
(2.34) holds.
(2.33) and (2.34) give
d 1 Aβε+Bβε2+6Aβ2
u (t, ) 4 + ∂2 u (t, ) 2
dt 4 k ε,β · kL4(R) 2 xx ε,β · L2(R)
(cid:0) (cid:1) !
(cid:13) (cid:13)
d Bε2 Bβ2ε2+(cid:13)6Aβ3 (cid:13)
+ ∂ u (t, ) 2 + ∂3 u (t, ) 2
dt 2 k x ε,β · kL2(R) 2 xxx ε,β · L2(R)
(cid:0) (cid:1) !
(cid:13) (cid:13)
+3Aβ4 d ∂4 u (t, ) 2 + βεA ∂2 u (t,(cid:13)) 2 (cid:13)
dt xxxx ε,β · L2(R) 2 tx ε,β · L2(R)
3Aβ2ε (cid:13) (cid:13) Aβ3ε (cid:13) (cid:13)
+ (cid:13)∂3 u (t, ) 2(cid:13) + ∂(cid:13)4 u (t, )(cid:13) 2
8 txx ε,β · L2(R) 2 txxx ε,β · L2(R)
+β2εA ∂(cid:13)3 u (t, ) (cid:13)2 +εK u(cid:13) (t, )∂ u (t,(cid:13)) 2
(cid:13)xxx ε,β · (cid:13)L2(R) 1k ε(cid:13),β · x ε,β (cid:13)· kL2(R)
+ε3K (cid:13)∂2 u (t, ) 2(cid:13) C ε ∂ u (t, ) 2 ,
(cid:13)xx ε,β · L(cid:13)2(R) ≤ 0 k x ε,β · kL2(R)
for some K , K >(cid:13)0. (cid:13)
1 2 (cid:13) (cid:13)
(2.4), (2.8) and an integration on (0,t) give
1 Aβε+Bβε2+6Aβ2
u (t, ) 4 + ∂2 u (t, ) 2
4 k ε,β · kL4(R) 2 xx ε,β · L2(R)
(cid:0) (cid:1)
Bε2 Bβ2ε2+(cid:13)6Aβ3 (cid:13)
+ ∂ u (t, ) 2 + (cid:13) ∂3(cid:13)u (t, ) 2
2 k x ε,β · kL2(R) 2 xxx ε,β · L2(R)
(cid:0) (cid:1)
βεA t (cid:13) (cid:13)
+3Aβ4 ∂4 u (t, ) 2 + ∂2 u(cid:13) (s, ) 2 (cid:13)ds
xxxx ε,β · L2(R) 2 tx ε,β · L2(R)
Z0
3Aβ2ε(cid:13) t (cid:13) Aβ(cid:13)3ε t (cid:13)
+ (cid:13) ∂3 u ((cid:13)s, ) 2 ds+ (cid:13) ∂4 (cid:13)u (s, ) 2 ds
8 txx ε,β · L2(R) 2 txxx ε,β · L2(R)
Z0 Z0
t (cid:13) (cid:13) t (cid:13) (cid:13)
+β2εA ∂(cid:13)3 u (s, ) (cid:13)2 ds+εK u(cid:13) (s, )∂ u (t,(cid:13)) 2 ds
xxx ε,β · L2(R) 1 k ε,β · x ε,β · kL2(R)
Z0 Z0
t (cid:13) (cid:13)
+ε3K (cid:13)∂2 u (s, ) (cid:13)2 ds
2 xx ε,β · L2(R)
Z0
(cid:13)t (cid:13)
C +C ε (cid:13) ∂ u (s, (cid:13)) 2 ds C .
≤ 0 0 k x ε,β · kL2(R) ≤ 0
Z0
Hence,
u (t, ) C ,
k ε,β · kL4(R) ≤ 0
ε ∂ u (t, ) C ,
k x ε,β · kL2(R) ≤ 0
βε ∂2 u (t, ) C ,
xx ε,β · L2(R) ≤ 0
εp β(cid:13)∂2 u (t, )(cid:13) C ,
(cid:13) xx ε,β · (cid:13)L2(R) ≤ 0
pβ(cid:13)∂2 u (t, )(cid:13) C ,
(cid:13) xx ε,β · (cid:13)L2(R) ≤ 0
βε (cid:13)∂3 u (t, )(cid:13) C ,
(cid:13)xxx ε,β · (cid:13)L2(R) ≤ 0
(cid:13) (cid:13)
(cid:13) (cid:13)