On the decoupling of the improved Boussinesq equation into two uncoupled Camassa-Holm equations

0.18 MB·

by  H. A. Erbay

ON THE DECOUPLING OF THE IMPROVED BOUSSINESQ
EQUATION INTO TWO UNCOUPLED CAMASSA-HOLM EQUATIONS
H.A. Erbay∗ and S. Erbay
7 Department of Natural and Mathematical Sciences, Faculty of Engineering,
1
Ozyegin University,
0
Cekmekoy 34794,Istanbul, Turkey
2
n A. Erkip
a
J Faculty of Engineering and Natural Sciences,

2 Sabanci University,
1 Tuzla 34956,Istanbul, Turkey

]
P
Abstract
A
. Werigorouslyestablishthat,inthelong-waveregimecharacterized
h
by the assumptions of long wavelength and small amplitude, bidirec-
t
a
tional solutions of the improved Boussinesq equation tend to associ-
m
ated solutions of two uncoupled Camassa-Holm equations. We give a
[
precise estimate for approximation errors in terms of two small posi-

1
tive parameters measuring the effects of nonlinearity and dispersion.
v
1 Our results demonstrate that, in the present regime, any solution of
9 the improved Boussinesq equation is split into two waves propagat-
4
ing in opposite directions independently, each of which is governed
3
0 by the Camassa-Holm equation. We observe that the approximation
. error for the decoupled problem considered in the present study is
1
0 greater than the approximation error for the unidirectional problem
7
characterized by a single Camassa-Holm equation. We also consider
1
: lower order approximations and we state similar error estimates for
v
both the Benjamin-Bona-Mahony approximation and the Korteweg-
i
X
de Vries approximation.
r
a
1 Introduction
In this study, we consider the improved Boussinesq (IB) equation
2 2
u u δ u ǫ(u ) = 0, (1)
tt xx xxtt xx
− − −
which appears as a relevant model in various areas of physics (see, e.g. [15,
18,8]forsolidmechanics), andweproceedalongouranalysisoftheCamassa-
Holm (CH) approximation of the IB equation initiated in [11]. In (1), u =
1
u(x,t)isareal-valuedfunction, andǫandδ aretwo smallpositive parameters
measuring the effects of nonlinearity and dispersion, respectively. In [11], by
a proper choice of initial data, we restricted our attention to the right-going
solutionsoftheIBequationandshowed that, forsmall amplitude longwaves,
they are well approximated by associated solutions of a single CH equation
[4]. In the present study we remove the assumption about the solutions
being unidirectional and consider solutions traveling in both directions with
general initial disturbances. We then show that, in the long-wave regime,
solutions of the IB equation can be split into two counter-propagating parts
up to a small error. To be more precise, it is shown that any solution of the
IB equation is well approximated by the sum w+ + w− of solutions of two
uncoupled CH equations
3 5 3
+ + + + 2 + 2 + 2 + + + +
w +w +ǫw w δ w δ w ǫδ (2w w +w w ) = 0,
t x x − 4 xxx − 4 xxt − 4 x xx xxx
(2)
3 5 3
w− w− ǫw−w− + δ2w− δ2w− + ǫδ2(2w−w− +w−w− ) = 0,
t − x − x 4 xxx − 4 xxt 4 x xx xxx
(3)
where w+ and w− denote the right and left going waves, respectively. We
mainly establish existence, consistency and convergence results for the CH
approximation of the IB equation in the decoupled case. We prove the de-
composition and give the convergence rate between bounded solutions of the
IB equation and the sum of two counter-propagating solutions of uncoupled
CH equations. We observe that the approximation errors remain small in
suitable norms over an arbitrarily long time interval. We also give error esti-
mates for the Benjamin-Bona-Mahony (BBM) and Korteweg-de Vries (KdV)
approximations of the IB equation in the same setting, where w+ and w− are
solutions of the two uncoupled BBM equations [2] or KdV equations [13].
The KdV, BBM and CH equations arise as formal asymptotic models for
unidirectional propagation of weakly nonlinear and weakly dispersive waves
inavariety ofphysical situations. Recently, therehasbeen agrowing interest
to rigorously relate solutions of the asymptotic equations to solutions of the
parent equations of original physical problem. For instance, in the context of
water waves, the KdV, BBM and CH equations have been rigorously justified
as unidirectional asymptotic models of the water wave equations in [7], [1]
and [5], respectively (the reader is referred to [14] for a detailed discussion of
the water waves problem). In the case of bidirectional propagation of small
2
amplitude long waves, an uncoupled system of two KdV equations, one for
waves moving to the left and one for waves moving to the right, appears
as the simplest asymptotic model of the underlying physical problem. In
[16], [3], [20], it was proven that bidirectional, small amplitude, long-wave
solutions of the water wave problem are well approximated by combinations
of solutions of two uncoupled KdV equations. In [21], a similar justification
framework was used for anuncoupled system of two CH equations once again
in the water wave setting.
In this paper, attention is given to the IB equation that describes the
time evolution of nonlinear dispersive waves in many practically important
situations. In [17] and [19], the validity of the uncoupled KdV system was
established as a leading order approximation for long wavelength solutions
of the IB equation. In the present work we extend the analysis to moderate
amplitudes by considering an uncoupled system of two CH equations as a
leading order approximation to the IB equation in the long wave regime and
provide an estimate for the approximation error. As a by-product, we also
recover both the uncoupled KdV system and the uncoupled BBM system as
the leading approximations of the IB equation. We believe that the study
of the IB equation provides a useful step in understanding long-wave limits
of the evolution equations modeling much more complicated physical situa-
tions. For a mathematical description of the long-wave limit of unidirectional
solutions of the IB equation by a single CH equation we refer to [11] (see [10]
for the formal derivation of the CH equation from the IB equation). As in
[11] the methodology used in this study adapts the techniques in [3, 5, 12].
Since the proofs in the present work are somewhat parallel with the proofs
in [11], we will present the new ingredients only.
Several points are worth emphasizing briefly. First, we remind that the
system of uncoupled CH equations (2) and (3) can be written in a more
standard form by means of the following coordinate transformations
2 3 2 3 2
x = (x t), y = (x+ t), t = t. (4)
√5 − 5 √5 5 3√5
Then, (2) and (3) become
6 9
+ + + + 2 + 2 + + + +
v + v +3ǫv v δ v ǫδ (2v v +v v ) = 0, (5)
t¯ 5 x¯ x¯ − t¯x¯x¯ − 5 x¯ x¯x¯ x¯x¯x¯
6 9
v− v− 3ǫv−v− δ2v− + ǫδ2(2v−v− +v−v− ) = 0, (6)
t¯ − 5 y¯ − y¯ − t¯y¯y¯ 5 y¯ y¯y¯ y¯y¯y¯
3
with v+(x,t) = w+(x,t) and v−(y,t) = w−(x,t), respectively. We also re-
mindthat,usingthescalingtransformationV+(X,τ) = ǫv+(x¯,t¯), V−(Y,τ) =
ǫv−(y¯,t¯), (x¯,y¯) = δ(X,Y), and t¯= δτ, we can rewrite (5) and (6) in more
standard forms with no parameters. Secondly, we observe that the approx-
imation error for the decoupled problem considered in the present study is
greater than the approximation error for the unidirectional problem charac-
terized by a single CH equation in [11]. This deterioration is partially related
to the error due to approximate splitting of the initial data for the IB equa-
tion. Another factor is due to the fact that the interaction of the right-going
and the left-going waves appears to play a major role in the residual term
that arises when we plug the solutions of the uncoupled CH equations into
the IB equation. We emphasize that the coupled models for which the inter-
action terms are not supposed to be small, provide a better description than
the decoupled ones over short time scales and that a rigorous justification of
this claim remains as an open problem.
The remainder of this paper is organized as follows. First, in Section 2,
we focus on a description of the problem setting for approximation errors.
In Section 3, we conduct a preliminary discussion of uniform estimates for
the solutions of the CH equation and we estimate the residual term that
arises when we plug the sum of solutions of the uncoupled CH equations into
the IB equation. In Section 4, using the energy estimate based on certain
commutator estimates, we obtain the convergence rate between the solutions
of the IB equation and the sum of solutions of the uncoupled CH equations.
InSection 5werecover theBBMandKdVapproximations oftheIBequation
in the decoupled case.
Our notation for function spaces is fairly standard. The notation u
Lp
k k
denotes the Lp (1 p < ) norm of u on R. The symbol u,v represents
the inner product ≤of u an∞d v in L2. The notation Hs = Hs(R) denotes the
(cid:10) (cid:11)
L2-based Sobolev space of order s on R, with the norm u = (1 +
Hs R
k k
ξ2)s u(ξ) 2dξ 1/2. We will drop the symbol R in . The symbol C will stand
R (cid:0)R
| |
for a generic positive constant. Partial differentiations are denoted by D ,
t
(cid:1) R
D etc.
x b
2 Problem Setting for Approximation Errors
In this section, we formulate the Cauchy problem for approximation errors.
For this aim we first state the following well-posedness result [6, 9] for the
4
initial-value problem of the IB equation:
Theorem 2.1. Let u ,u Hs(R), s > 1/2. Then for any pair of param-
0 1
∈
eters ǫ and δ, there is some Tǫ,δ > 0 so that the Cauchy problem for the IB
equation (1) with initial data
u(x,0) = u (x), u (x,0) = u (x), (7)
0 t 1
has a unique solution u C2 [0,Tǫ,δ],Hs(R) .
∈
The existence time Tǫ,δ ab(cid:0)ove may depen(cid:1)d on ǫ and δ and it may be
chosen arbitrarily large as long as Tǫ,δ < Tǫ,δ where Tǫ,δ is the maximal
max max
time. Furthermore, it was shown in [9] that the existence time, if it is finite,
is determined by the L∞ blow-up condition
limsup u(t) = .
k kL∞ ∞
t→Tmǫ,aδx
Let w+ and w− be two families of solutions for the Cauchy problems defined
for the CH equations (2) and (3) with initial values w+ and w−, respectively.
0 0
Given the initial data (u ,u ) for the IB equation, the first question is how
0 1
to select the corresponding initial data (w+,w−) for the CH equations (2)
0 0
and (3). Ideally we should have u = w++w− and u = w+(x,0)+w−(x,0),
0 0 0 1 t t
yet it will be convenient to choose (w+,w−) independent of the parameters
0 0
ǫ and δ. From the uncoupled CH equations we get
w+ +w− = w+ +w− + (ǫ,δ2,ǫδ2).
t t − x x O
Neglectingthehigherordertermsyieldstheapproximationu (x) = w+(x,0)+
1 − x
w−(x,0). Finally, assuming that u = (v ) we get
x 1 0 x
1 1
w+ = (u v ), w− = (u +v ). (8)
0 0 0 0 0 0
2 − 2
Our aim is to compare the solution u of (1) and (7) with the sum w+ +w−.
Obviously, the error function defined by r = u (w+ +w−) satisfies the
−
initial condition r(x,0) = 0. In order to express r (x,0) in terms of the initial
t
values (u ,v ) of the IB equation, we substitute the CH equations (2) and
0 0
5
(3) into r (x,0):
t
r (x,0) =u (x) w+(x,0)+w−(x,0)
t 1 − t t
(cid:16) 5 (cid:17)
= (w+) +(w−) (1 δ2D2)−1 D (w+ w−)
− 0 x 0 x − − 4 x − x 0 − 0
(cid:26)
ǫ 3
D (w+)2 (w−)2 + δ2D3(w+ w−)
− 2 x 0 − 0 4 x 0 − 0
+ 3ǫδ2D(cid:16) 1 (w+) 2(cid:17) (w−) 2 +w+(w+) w−(w−)
x 0 x 0 x 0 0 xx 0 0 xx
4 2 − −
(cid:18) (cid:19)(cid:27)
(cid:16) (cid:17)
5 (cid:0) (cid:1)1 (cid:0) (cid:1)1
=D (1 δ2D2)−1 δ2(v ) ǫu v
x − 4 x − 2 0 xx − 2 0 0
(cid:26)
3
2
ǫδ (u ) (v ) (u v ) . (9)
0 x 0 x 0 0 xx
− 8 −
(cid:27)
(cid:16) (cid:17)
Substituting u = r+w++w− into (1), we observe that the function r satisfies
1 δ2D2 r r ǫ r2 +2(w+ +w−)r = F , (10)
− x tt − xx − xx − x
(cid:16) (cid:17)
(cid:0) (cid:1)
where F is the residual term given by
x e
F = F+ +F− 2ǫ w+w− , (11)
e x x x − xx
with
(cid:0) (cid:1)
F∓ =ew∓ w∓ δ2w∓ ǫ (w∓)2 . (12)
x tt − xx − xxtt − xx
Our main problem is now reduced to finding an(cid:16)upper(cid:17)bound for r in terms
of ǫ and δ.
We note that r (x,0) is of the form q(x) by (9). Since r(x,0) = 0 and
t x
the nonhomogeneous term in (10) is of the form F , one can show that
(cid:0) (cid:1) x
−
r = ρ for some appropriate function ρ(x,t) (see [9] for the homogeneous
x
case). To further simplify the calculations, in what feollows we will express
(10) in terms of both ρ and r as
1 δ2D2 ρ r ǫ r2 +2(w+ +w−)r = F. (13)
− x tt − x − x −
(cid:16) (cid:17)
with the init(cid:0)ial data (cid:1)
e
r(x,0) = 0, (14)
5 δ2 ǫ 3
ρ (x,0) = (1 δ2D2)−1 (v ) u v ǫδ2 (u ) (v ) (u v ) .
t − 4 x − 2 0 xx − 2 0 0 − 8 0 x 0 x − 0 0 xx
(cid:26) (cid:27)
(cid:16) (cid:17)
(15)
6
3 Some Estimates for the CH Equation and
the Nonhomogeneous IB Equation
In this section, we state some previous estimates from [11] concerning solu-
tions of the CH equation and the nonhomogeneous IB-type equation. For
the convenience of the reader we provide short versions of the proofs in the
Appendix.
The following proposition is a direct consequence of the estimates proved
by Constantin and Lannes in [5] for a more general class of equations, con-
taining the CH equation as a special case. We refer the reader to Section
2 of [11] for a more detailed discussion. As a result we have the following
proposition:
Proposition 1 (Corollary 2.1 of [11]). Let w Hs+k+1(R), s > 1/2, k 1.
0
∈ ≥
Then, there exist T > 0, C > 0 and a unique family of solutions
T T
wǫ,δ C [0, ],Hs+k(R) C1 [0, ],Hs+k−1(R)
∈ ǫ ∩ ǫ
(cid:18) (cid:19) (cid:18) (cid:19)
to the CH equation
3 5 3
2 2 2
w +w +ǫww δ w δ w ǫδ (2w w +ww ) = 0. (16)
t x x xxx xxt x xx xxx
− 4 − 4 − 4
with initial value w(x,0) = w (x), satisfying
0
wǫ,δ(t) + wǫ,δ(t) C,
Hs+k t Hs+k−1 ≤
(cid:13) (cid:13)
(cid:13) (cid:13)
(cid:13) (cid:13)
for all 0 < ǫ δ 1(cid:13)and t (cid:13) [0, T]. (cid:13) (cid:13)
≤ ≤ ∈ ǫ
Plugging w of (16) in the IB equation we get a residual term f,
2 2
f = w w δ w ǫ(w ) . (17)
tt xx xxtt xx
− − −
7
Calculation in [11] shows that f is of the form f = F where
x
w3 1
2 2 2 2 2 2 2
F =ǫ ( ) ǫ δ 3(w +2ww ) 3w(w ) +2w (w ) +w (w )
3 x − 8 x xx x − xxx xx x x xx
1 (cid:16) (cid:17)
4 2
+ δ (D D 3D )(3w +5w )
16 x t − x xxx xxt
1 (cid:16) (cid:17)
4 2 3 2
+ ǫδ 3(D D 3D )(w +2ww )
32 x t − x x xx
(cid:16)
2
+2( 3wD +2w +w D )(3w +5w )
− x xx x x xxx xxt
1 (cid:17)
2 4 3 2 2
+ ǫ δ ( 9wD +6w D +3w D )(w +2ww ) . (18)
32 − x xx x x x x xx
(cid:16) (cid:17)
Furthermore, using the uniform bounds in Proposition 1, the following esti-
mate for F was proved in [11]:
Lemma 3.1 (Lemma 3.1 of [11]). Let w Hs+6(R), s > 1/2 and let
0
∈
wǫ,δ be the family of solutions to the CH equation (16) with initial value
w(x,0) = w (x). Then, there is some C > 0 so that the family of residual
0
terms F = Fǫ,δ in (18) satisfies
2 4
F (t) C ǫ +δ ,
k kHs ≤
for all 0 < ǫ δ 1 and t [0, T]. (cid:0) (cid:1)
≤ ≤ ∈ ǫ
We next consider the solution r,ρ of the IB-type equation
2 2 2
1 δ D ρ r ǫ r +2wr = F, (19)
− x tt − x − x −
(cid:0) (cid:1) (cid:0) (cid:1)
where r = ρx. We assume that w and F are giveen functioens depending on ǫ
and δ, with
e e
T
w C [0, ],Hs+1(R) , (20)
∈ ǫ
(cid:18) (cid:19)
T
ew(t) C for t [0, ], (21)
k kHs+1 ≤ ∈ ǫ
T
Fe C [0, ],Hs(R) . (22)
∈ ǫ
(cid:18) (cid:19)
e
8
Ourpurposeistofindaboundforsolutionsof(19). Inthatrespect, following
the approach in [12] and [11], we define the ”energy” as
1
E2(t) = ρ (t) 2 +δ2 r (t) 2 + r(t) 2 +ǫ Λs w(t)r(t) ,Λsr(t)
s 2 k t kHs k t kHs k kHs
(cid:16)ǫ (cid:17)
+ Λsr2(t),Λsr(t) , (cid:10) (cid:0) (cid:1) (2(cid:11)3)
2 e
(cid:10) (cid:11)
where Λs = (1 D2)s/2. Taking the energy in the usual form without the ǫ
− x
terms will yield a loss of δ in the final estimate. This is due to the coefficient
δ2 of the term r (t) 2 (see Remark 2 of [11] for further details).
k t kHs
Since w+ and w− exist for all times t T/ǫ, r(x,t) will exist over the
≤
same time interval unless r or equivalently uǫ,δ blows up in a shorter time.
By Theorem 2.1 the blow-up of uǫ,δ is controlled by the L∞-norm. Thus the
blow-up of r is also determined by its L∞-norm or equivalently by r(t) .
k kHs
Since r(x,0) = 0 we define
T
Tǫ,δ = sup t : r(τ) 1 for all τ [0,t] . (24)
0 ≤ ǫ k kHs ≤ ∈
(cid:26) (cid:27)
Note that
Λs(wr),Λsr C r(t) 2 , and Λsr2,Λsr r(t) 3 r(t) 2 ,
≤ k kHs ≤ k kHs ≤ k kHs
(cid:12)(cid:10) (cid:11)(cid:12) (cid:12)(cid:10) (cid:11)(cid:12)
w(cid:12) hereewehaveu(cid:12)sed(24)andtheuniform(cid:12)estimatefor(cid:12)w. Thus, forsufficiently
small values of ǫ and t Tǫ,δ, we have
0
≤
e
1
2 2 2 2 2
E (t) ρ (t) +δ r (t) + r(t) ,
s ≥ 4 k t kHs k t kHs k kHs
(cid:16) (cid:17)
which shows that E2(t) is positive definite. The above result also shows
s
that an estimate obtained for E2(t) gives an estimate for r(t) 2 . After a
s k kHs
series of calculations and estimates we obtain the differential inequality for
the energy:
d
E (t) C ǫE (t)+ sup F(t) (25)
s s
dt ≤ t≤T/ǫ Hs!
(cid:13) (cid:13)
(cid:13) (cid:13)
The proofs of Lemma 3.1 and this inequality(cid:13)weere(cid:13)given in [11]. We will
summarize those proofs in the Appendix for the convenience of the reader.
9
4 Convergence Proof for the Decoupled
Approximation
In this section we prove our main result, Theorem 4.1 given below. Recall
from Section 2 that we started with the family of solutions u = uǫ,δ of the IB
equation and chose appropriate solutions w∓ of the uncoupled CH equations.
Our aim is to show that the sum w+ + w− is a good approximation for u.
In other words, we want to find a good estimate for the error, namely the
solution of the problem defined by (13)-(15). This is achieved by the proof
of Theorem 4.1, where we take advantage of the results in Section 3.
Theorem 4.1. Let u Hs+6(R) and v Hs+7(R), s > 1/2. Suppose uǫ,δ
0 0
∈ ∈
is the solution of the IB equation (1) with initial data
u(x,0) = u (x), u (x,0) = (v (x)) .
0 t 0 x
Let
1 1
w+ = (u v ), w− = (u +v ).
0 0 0 0 0 0
2 − 2
Then, for any given t∗ > 0 there exists δ∗ 1 so that the solutions (w∓)ǫ,δ of
≤
the uncoupled CH equations (2) and (3) with initial values w∓(x,0) = w∓(x)
0
satisfy
uǫ,δ(t) (w+)ǫ,δ(t) (w−)ǫ,δ(t) C (ǫ+δ2)+(ǫ+δ4)t
Hs
k − − k ≤
(cid:16) (cid:17)
for all t [0,t∗] and all 0 < ǫ δ δ∗.
∈ ≤ ≤
Proof. We first note that (13) is exactly (19) with w = w+ + w− and F =
F++F− 2ǫ(w+w−) . The explicit form of F+ is obtained by substituting
− x
w+ in place of w in (18). Similarly, the explicit forme of F− is obtainedeby
substituting w− for w and t for t in (18). Since w+ and w− are solutions
−
of the CH equations (2) and (3), by Proposition 1 and Lemma 3.1 we have
the estimates
w∓(t) C, F∓(t) C ǫ2 +δ4 ,
Hs+1 ≤ Hs ≤
for all 0 < ǫ (cid:13)δ 1(cid:13)and t [0, T]. T(cid:13)herefor(cid:13)e, (cid:0) (cid:1)
≤(cid:13) ≤ (cid:13) ∈ ǫ (cid:13) (cid:13)
w(t) w+(t) + w−(t) C, (26)
k kHs+1 ≤ Hs+1 Hs+1 ≤
F (t) F(cid:13)+(t) (cid:13) + F−(cid:13)(t) (cid:13)+2ǫ (w+w−) (t) C ǫ+δ4 .
e Hs ≤ (cid:13) H(cid:13)s (cid:13) Hs(cid:13) x Hs ≤
(cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:0) (2(cid:1)7)
(cid:13) (cid:13)
e (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13)
(cid:13) (cid:13)
10