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Well-posed Solutions of the Third Order
Benjamin–Ono Equation in Weighted Sobolev
Spaces
Xueshang Feng∗
Abstract
Here we continue the study of the initial value problem for the third order
T
Benjamin-Ono equation in the weighted Sobolev spaces Hγs = H s L2γ , where
s > 3, γ ≥ 0. The result is the proof of well-posedness of the afore mentioned
problem in Hγs , s > 3, γ ∈ [0, 1]. The proof involves the use of parabolic
regularization, the Riesz-Thorin interpolation theorem and the construction
technique of auxiliary functions.
1 Introduction
In recent years, the initial value problem for Benjamin-Ono(BO) equation of deep
water,
∂tu + ∂x (2Hux + u2 ) = 0
has been investigated by many authors [1, 4, 5, 7, 8]. Iorio[5] established the wellposedness of the above BO equation in Hγ2 , where γ ∈ [0, 1], by using Kato’s theory
of linear evolution equations of “hyperbolic type”. Ponce [7] proved the global wellposedness of the BO equation in H s , s ≥ 3/2. In [3], the authors obtained the
Supported by National Natural Science Foundation of China
Received by the editors May 1996.
Communicated by J. Mawhin.
1991 Mathematics Subject Classification : 35Q20, 35G25, 35D99.
Key words and phrases : The third order BO equation; Initial value problem; Weighted
Sobolev spaces; Global well-posedness.
∗
Bull. Belg. Math. Soc. 4 (1997), 525–537
526
X. Feng
global existence and uniqueness both in the usual Sobolev space H s and weighted
Sobolev spaces Hγs for the third order BO equation
∂tu = −∂x(u3 + 3uHux + 3H(uux ) − 4∂x2u), t > 0, x ∈ R
(1.1)
u(x, 0) = φ(x)
(1.2)
which can be considered as the high order approximation of the BO equation. Here
∂k
u = u(x, t) is real valued, ∂xk = ∂x
k and the subscripts appended to u stand for
partial differentiations, and H denotes the Hilbert transform
(Hf)(x) = P.V.
In this paper, Hγs = H s
T
Z
f(y)
dy
π(y − x)
L2γ with the following norm
||u||2γ,s = ||f||2s + |ωγ f|22 , ωγ (x) = (1 + x2 )γ
where || • ||s and | • |2 are the usual norms of the Sobolev spaces H s (R) and Lebesgue
space L2 (R), respectively. Throughout the paper, | • |p will denote the usual norm
of the Lebesgue spaces Lp (R), p ∈ [1, ∞]. If X and Y are Banach spaces we denote
the set of all bounded operators from X to Y by B(X, Y ). If X = Y we write B(X)
instead of B(X, Y ). As usual [A, B] will indicate the commutator of two linear
operators(not necessarily continuous).
The BO equation arises in the study of unidirectional propagation of nonlinear
dispersive waves or long nonlinear sausage-wave propagation in a magnetic slab in
an incompressible plasma of the solar atmosphere, and presents the interesting fact
that operrators modelling the dispersive effect in the BO equation or the effect of
derivative nonlinearity in (1.1) are nonlocal. In [3, 5], the authors have shown that
u ∈ C([0, T ]; H 43 ) or u ∈ C([0, T ]; H24) solves equation (1.1) or the BO equation
2
if and only if u ≡ 0, and pointed out in [3] that this phenomena are caused by
the nonlocal terms modelling the equations. The papers [1, 7] are devoted to the
proof of the well-posedness of the global solutions to the initial value problem of
the BO equation in H s , for s ≥ 32 . Many physicists such as Ruderman (see [8]
and references therein) have used this equation to study the wave propagation in a
magnetically structured atmosphere and found its solutions in the form of periodic
waves of permanent shape numerically. As the high order approximation of the
BO equation, the third order BO equation (1.1) is interesting in its own rights and
therefore it is of importance to study its well-posedness whether mathematically or
physically considered. I would here like to apologize if I miss citing other papers on
the BO equation.
Our aim here is to investigate the global well-posedness of problem (1.1), (1.2)
in the weighted Sobolev spaces Hγs as done in [5] for the BO equation.
Our main result is as follows:
Theorem 1.1. Let φ ∈ Hγs , s ≥ 4 and γ ∈ [0, 1]. Then for all fixed µ ≥ 0
there exists a unique uµ ∈ C([0, T ]; H s) for any given T > 0 such that ∂tuµ ∈
C([0, T ]; H s−4) and ωγ′ ∂xk uµ ∈ C([0, T ]; L2) for k ≤ s/2 and the following (2.6), (2.7)
are satisfied. Moreover,
|ωγ′ ∂xk u|2 ≤ C(γ)||u||γ,s
Third Order Benjamin–Ono Equation in Weighted Sobolev Spaces
527
Furthermore, the map φ 7→ uµ is continuous in the sense that if φn → φ in Hγs−2
and unγ , uµ are the corresponding solutions then
lim sup[0,T ]||unµ (t) − uµ (t)||γ,s−2 = 0
n→∞
where s ≥ 6.
In fact, we shall show that the map φ 7→ u is globally Lipschitz. In our cases, it
is difficult to apply Kato’s theory of linear evolution equations of “hyperbolic type”
because of the appearance of the derivative nonlinear nonsmooth terms. The proof
in this paper is very technical and long. In the following two sections we shall finish
the proof of Theorem 1.1 by presenting a series of statements.
2 Preliminary results
In this part, a series of lemmas are given for the purpose of proving Theorem 1.1.
The lemmas below are established following the same lines as those of [5].
Lemma 2.1. Suppose
n a(x) isodefined on R and |a(x)−a(y)| ≤ M|x−y|, x, y ∈
R
R. Let Tǫ (f) = |x−y|≥ǫ a(x)−a(y)
f(y)dy. Then
(x−y)2
|Tǫ (f)|p ≤ Ap |f|p , for all f ∈ Lp (R)
with Ap independent of ǫ, 1 < p < ∞.
For details and further results of this kind see Calderon [2] and P.238 of Stein
[9]. The following well-known Gagliardo-Nirenberg’s inequality will be unexplainedly
used throughout the paper.
Lemma 2.2. For f ∈ H m (R) and p ≥ 2, then we have
|∂xd f|p ≤ 2
(p−2)
2p
|∂xk u|λ2 |u|1−λ
(2.1)
where d, m, k ∈ N {0}, λ = (d + 21 − 1p )/k, and 0 ≤ d < k ≤ m. More precisely,
for f ∈ H d+1 (R) we have
S
1
2d+1
|∂xd f|∞ ≤ |f|22(d+1) |∂xd+1f|22(d+1)
(2.2)
Proof. (2.1) has been well-known. To prove (2.2), noting that
2
|f(x)| = 2
Z
2
|f(x)| = −2
x
−∞
Z
ffx dx ≤ 2
Z
∞
x
ffx dx ≤ 2
x
−∞
Z
|ffx |dx
∞
x
|ffx |dx
We get by using the Cauchy-Schwartz inequality
|f|2∞ ≤
Z
|ffx |dx ≤ |f|2 |fx |2
(2.3)
which gives (2.2) for the case d=0. By using (2.1) and (2.3) we get
2d+1
1
|∂xdu|2∞ ≤ |∂xdu|2|∂xd+1 u|2 ≤ |∂xd+1u|2d+1 |u|2d+1
528
X. Feng
which finishes the proof of (2.2).
It should be pointed out that inequality (2.2) can be used to improve the uniqueness result for the smallness case of the initial data in [3].
Lemma 2.3. Let λ ∈ [0, ∞), γ ∈ [0, 1], µ > 0, and Qµ = −µ∂x4 + 4∂x3 . Then
Sµ (t) = exp(tQµ ) ∈ B(Hγs , Hγs+λ ) for all t > 0, s ≥ 0 and satisfies the estimate
||Sµ (t)φ||γ,s ≤ G(t, µ, λ, γ)||φ||γ,s
(2.4)
for all φ ∈ Hγs , where G(t, µ, λ, γ) is locally integrable with respect to t. Moreover
the map t ∈ (0, ∞) 7→ Sµ (t)φ is continuous with respect to the topology of Hγs+λ .
Proof. Due to [3], where the Lemma is established in the case of H0s = H s , s ∈ R,
it suffices to examine Sµ (t) as an operator from Hγs into Hγ0 = L2γ . If γ = 0 we obtain
T
H00 = L2 so that ||Sµ (t)φ||0,0 ≤ ||φ||0,0. Next, if γ = 1 and φ ∈ H1s = H s L21 we
have
(ω1 Sµ (t)φ)∧(ξ) = Eµ (ξ, t)(1 − ∂ξ2)φˆ − (∂ξ2Eµ (ξ, t))φˆ − 2(∂ξ Eµ (ξ, t))∂ξ φˆ
(2.5)
where Eµ (ξ, t) = exp[−(µξ 4 + 4iξ 3 )t]. The result then follows by combining (2.5)
with the formulas
∂ξ Eµ (ξ, t) = −(4µξ 3 + 12iξ 2 )tEµ (ξ, t)
∂ξ2Eµ (ξ, t) = −12t(µξ 2 + 2iξ)Eµ (ξ, t) + 16t2 ξ 4 (µξ + 3i)Eµ (ξ, t)
and the estimate
λ
0 ≤ ξ 2λ exp(−µtξ 4 ) ≤ λλ exp( )(2µt)−λ/2
2
The statement in the Lemma is now an easy consequence of the Riesz-Thorin
interpolation theorem.
We are now in position to establish a local existence for the following parabolic
regularized problem
∂tuµ = −µ∂x4uµ − ∂x (u3µ + 3uµ H∂x uµ + 3H(uµ ∂x uµ ) − 4∂x2uµ )
(2.6)
uµ (x, 0) = φ(x)
(2.7)
in Hγs , γ ∈ [0, 1] and µ > 0. Let T be a positive number and consider the set
n
Xγ,s (T ) = f ∈ C([0, T ]; Hγs); ||f(t) − Sµ (t)φ||γ,s ≤ ||φ||γ,s , t ∈ [0, T ]
o
which becomes a complete metric space when provided with the distance obtained
from the sup norm. Note that if f ∈ Xγ,s (T ), then by Lemma 2.3 it is not difficult
to show that one can choose T > 0 such that the map
(Af)(t) = Sµ (t)φ −
Z
0
t
Sµ (t − τ )∂x f 3 + 3fHfx + 3H(ffx ) dτ
defines a contraction in Xγ,s (T ). In view of the uniqueness and regularity results
established in [3] we have
Corollary 2.4. Let φ ∈ Hγs , s ≥ 4, γ ∈ [0, 1], µ > 0. Then there exist a
T > 0 and a unique uµ ∈ C([0, T ]; Hγs) such that ∂t uµ ∈ C([0, T ]; H s−4) satisfies
problem (2.6), (2.7). Moreover, uµ ∈ C((0, T ]; Hγq ) for all q ∈ R.
Third Order Benjamin–Ono Equation in Weighted Sobolev Spaces
529
Let µ > 0 be fixed. From now on, unless otherwise specifed, we write u = uµ for
simplicity. The next step is to establish global existence in Hγs , s ≥ 4 and γ ∈ [0, 1].
This has already been done in [3] in the cases γ = 0 and γ = 1 (the result for γ = 1
will be re-obtained below from a slightly different point of view). In order to obtain
global estimates for the Hγs norm of the solution it suffices to study what happens
in L2γ since the H s result was proved in [3]. Now, by Corollary 2.4 and integration
by parts, at least formally we have
t
Z
ωγ2 u22x dx
uu2xωγ ωγ′′dx − 4µ
Z
uu2x(ωγ′ )2 dx − 8µ
∂t ||u(t)||2γ,0
= −4µ
−8
Z
Z
ωγ2 u∂x (u3)
+ 2µ
dx − 3
Z
+2 (ωγ2 )′′′u2 dx − 24
Z
Z
Z
0
ωγ2 u2 H(∂x2u)dx
+6
ωγ ωγ′ uu2x dx − 3
Z
Z
Z
ux u2xωγ ωγ′′ dx
ωγ ωγ′ u2 Hux dx
ωγ2 uH(∂x2(u2))dx
(2.8)
In order to extend the local solution to the whole time interval [0, T ], for any fixed
T > 0 we must estimate the right-hand side of (2.8). For this purpose we need the
following Lemmas.
Lemma 2.5. Let u ∈ Hγ4 , γ ∈ [0, 1]. Then ωγ/2 u2x ∈ L2 and satisfies
1
1
|ω γ2 u2x |2 ≤ C||u||42 |ωγ u|22 ≤ C||u||γ,4
(2.9)
where C is a generic constant.
Proof. By definition, for u ∈ S(R)(the Schwartz space) we have
|ω
γ
2
u2x|22
=
Z
ωγ u22xdx
=
Z
ωγ′′ uu2x
dx + 2
Z
ωγ′ uu3xdx
+
Z
ωγ uu4xdx
≤ |ωγ′′u|2|u2x |2 + 2|ωγ′ u|2|u3x |2 + |ωγ u|2|u4x|2 ≤ C||u||4|ωγ u|2
(2.10)
which combines a limiting argument to yield the result.
Lemma 2.6. Let f ∈ Hγs , γ ∈ [0, 1]. Then x(1 + x2 )γ ∂xk f ∈ L2 and
|x(1 + x2)(r−1) ∂xk f|2 ≤ C(r)||f||γ,s , 2k ≤ s
(2.11)
where C(γ) is a constant depending only on γ. The same results are true if x(x2 +
1)γ−1 is replaced by ωγ′ (x).
Proof. A simple limiting argument shows that it suffices to prove (2.11) for
f ∈ S(R). Assume therefore that this is the case and integrate by parts to obtain
for k = 1
|x(1 + x2)γ−1 fx |22 = −
3
X
Ij
j=1
where
Z
I1 = 2 (1 + x2)γ f(x)[x(1 + x2)γ−2 ]fxdx
I2 = 2(2γ − 2)
Z
(1 + x2)γ f(x)[x3(1 + x2)γ−3 ]fx dx
530
X. Feng
I3 =
Z
(1 + x3)γ f(x)[x2(1 + x2)γ−2 ]∂x2f(x)dx
It is easy to verify that the expressions inside the square brackets in all the three
ingegrals are bounded of x. By this remark and the Cauchy-Schwartz inequality we
obtain
|I1| ≤ 2|fx |2|ωγ |2, |I2| ≤ |4γ − 4||fx |2 |ωγ f|2 , |I3| ≤ |∂x2f|2 |ωγ f|2
Now, we have
|x(1 + x2)γ−1 fx |2 ≤ {(2 + |4γ − 4|)|fx |2 + |f2x|2 } |ωγ f|2 ≤ C(γ)||f||2|ωγ f|2
which finishes the proof. For the general case, a similar argument yields
|x(1 + x2)γ−1 ∂xk f|22 =
2k Z
X
(1 + x2 )γ f(x)pj (x)∂xj f(x)dx
j=k
where pj (x) is bounded in x. So the result follows.
Corollary 2.7. Let f ∈ Hγ4 , γ ∈ [0, 1]. Then ω γ2 fx ∈ L2 , ω γ2 fx ∈ L∞ and
satisfy the estimates
1
1
|ω γ2 fx |2 ≤ C(γ)||f||22 |ωγ f|22
(2.12)
1
1
|ω γ2 fx |∞ ≤ C(γ)||f||44 |ωγ f|22
(2.13)
where C(γ) > 0 depends only on γ.
Proof. Similar to the proof of (2.9), it suffices to prove (2.12) for f ∈ S(R). If
this is the case, then we have
|ω γ2 fx |22 =
Z
ωγ fx fx dx = −
Z
ωγ ff2xdx −
Z
ωγ′ ffx dx
≤ |fx |2|ωγ′ f|2 + |f2x|2 |ωγ f|2 ≤ C(γ)||f||2|ωγ f|2
which gives (2.12). By (2.2), (2.9), (2.11) and (2.12) we obtain
|ω γ2 fx |2∞ ≤ |ω γ2 fx |2|ω ′γ fx + ω γ2 f2x|2 ≤ C(γ)||f||4|ωγ f|2
2
This finishes (2.13).
Lemma 2.8. Let f ∈ Hγs , γ ∈ [0, 1]. Then [ω γ2 , H∂x ]f ∈ L2 and
|[ω γ2 , H∂x ]f|2 ≤ C(γ)|f|2
(2.14)
where C(γ) depends only on γ.
Proof. First observe that it suffices to prove (2.14) in case of f ∈ S(R). Now
[ω γ2 , H∂x ]f = [ω γ2 , H]∂x f − H[∂x , ω γ2 ]f
(2.15)
For the second commutator in (2.15) we have
H[∂x, ω γ2 ]f = H(ω ′γ f)
2
(2.16)
Third Order Benjamin–Ono Equation in Weighted Sobolev Spaces
531
For the first commutator in (2.15) we integrate by parts once to obtain
[ω γ2 , H]∂xf = P.V.
1
π
Z ω γ (x) − ω γ (y)
2
2
(x − y)2
f(y)dy − H(ω ′γ f)
(2.17)
f(y)dy − 2H(ω ′γ f)
(2.18)
2
From (2.15)-(2.17) we know that
[ω γ2 , H∂x ]f = P.V.
1
π
Z ω γ (x) − ω γ (y)
2
2
(x − y)2
2
Since ω ′γ is bounded and ω γ2 is Lipschitz, by Lemma 2.1 and the fact that H ∈ B(L2 )
2
it follows from (2.18) that
|[ωγ , H∂x2]f|2 ≤ C(γ)|f|2
which finishes the proof.
From the proof above we see that (2.14) holds for all f ∈ L2 .
Lemma 2.9. Let f ∈ Hγ2 , γ ∈ [0, 1]. Then [ωγ , H∂x2 ]f ∈ L2 and
|[ωγ , H∂x2]f|2 ≤ C(γ)||f||γ,2
(2.19)
where C(γ) ≥ 0 depends only on γ.
Proof. First note that it suffices to prove (2.19) for f ∈ S(R). Evidently
[ωγ , H∂x2]f = [ωγ , H]∂x2f − H[∂x2, ωγ ]f
(2.20)
So it suffices to establish (2.19) with [ωγ , H∂x2]f replaced by H[∂x2, ωγ ]f and [ωγ , H]∂x2f.
The result then follows for the first of these functions by combining the formula
H[∂x2, ωγ ]f = H(ωγ′′f + 2ωγ′ ∂xf)
(2.21)
with Lemma 2.6 and the facts that ωγ′′ ∈ L∞ and H ∈ B(L2). In order to handle
[ωγ , H]∂x2f we integrate by parts once to obtain
[ωγ , H]∂x2f
=
H(ωγ′ ∂x f)
1
− P.V.
π
Z
ωγ (x) − ωγ (y)
f(y)dy
(x − y)2
(2.22)
In view of Lemma 2.6 the first term on the right-hand side of (2.22) satisfies
|H(ωγ′ ∂xf)|2 ≤ C(γ)||f||γ,2, so it remains to bound the second. Since ωγ′ is bounded
if and only if γ ∈ [0, 21 ], in this case ωγ is Lipschitz. Hence, by Lemma 2.1 we have
Z
1 ωγ (x) − ωγ (y)
P.V.
f(y)dy
≤ A|f|2
π
(x − y)2
(2.23)
where A > 0 is a generic constant, γ ∈ [0, 12 ]. In the case of γ ∈ ( 21 , 1] we integrate
by parts once in (2.22) to get
[ωγ , H]∂x2f = −P.V.
1
π
Z
K(x, y)g(y)dy
(2.24)
532
X. Feng
2(ωγ (x) − ωγ (y)) − 2ωγ′ (y)(x − y) − ωγ′′(y)(x − y)2
(x − y)3(1 + y 2 )γ
K(x, y) =
(2.25)
where g(y) = (1 + y 2)γ f(y) ∈ L2 . If γ ∈ ( 21 , 1], it is not difficult to verify that
K(x, y) is a Hilbert-Schmidt kernel. Thus
1Z
P.V.
K(x, y)ωγ (y)f(y)dy ≤ C|ωγ f|2
(2.26)
π
where C > 0 is a generic constant, γ ∈ ( 21 , 1]. Now the result of the lemma follows
from (2.20)-(2.26).
Now we are in position to estimate each term in the right-hand side of (2.8) as
follows:
Z
µ
µ
′′
R1 ) 4µ uu2xωγ ωγ dx ≤ |ωγ u2x|22 + Cµ|ωγ′′u|2 ≤ |ωγ u2x |22 + Cµ|u|22
3
3
Z
µ
′ 2
R2 ) 4µ uu2x (ωγ ) dx ≤ |ωγ u2x|22 + Cµ|ωγ u|22
3
Z
µ
R3 ) 8µ ux u2xωγ ωγ′′dx ≤ |ωγ u2x |22 + Cµ|ux |22
3
Z
R4 ) 8 (ωγ )2u∂x (u3)dx ≤ C(||u||2)|ωγ u|22
Z
2 2
2
R5 ) 3 (ωγ ) u H(∂x u)dx ≤ C(||u||3)|ωγ u|22
Z
′
2
R6 ) 6 ωγ ωγ u Hux dx ≤ C(||u||2)|ωγ u|22
Z
′′′ 2
R7 ) 2 (ωγ ) u dx ≤ |ωγ u|22
Z
′
R8 ) 24 ωγ ωγ uu2xdx ≤ 24|ωγ u|2|ωγ′ u2x |2 ≤ C|ωγ u|2|ω γ2 u2x|2 ≤ C(||u||4)|ωγ u|22
Z
Z
Z
(ωγ )2uH∂x2 (u2)dx =
R9 )
=
Z
ωγ u[ωγ , H∂x2]u2 dx +
Z
ωγ u[ωγ , H∂x2]u2dx +
ωγ uH∂x2(ωγ u2 )dx
ωγ uH[ωγ′′u2 + 2ωγ′ uux + 2ωγ u2x + 2ωγ uu2x)dx
n
≤ C|ωγ u|2 |[ωγ , H∂x2 ]u2|2 + |ωγ′′u2 |2|ωγ′ uux|2 + |ωγ u2x |2 + |ωγ uu2x|2
n
o
≤ C(γ)|ωγ u|2 ||u2||γ,2 + |u2|2 + |ωγ′ uux |2 + |ω γ2 ux |∞ |ω γ2 ux |2 + |u2x|∞ |ωγ u|2
o
≤ C(γ, ||u||4)|ωγ u|2
Considering R1 –R9 ) and (2.8) we obtain
Z
d
||u(t)||2γ,0 + µ (ωγ )2 u22xdx ≤ C(µ, γ, ||u||4)|ωγ u|22
(2.8)′
dt
By [3] we know that ||u(t)||s is bounded uniformly for t in any fixed interval [0, T ]
and µ varying in any bounded interval. That is,
||u(t)||s ≤ C(µ, ||φ||s ), t ∈ [0, T ]
(2.27)
533
Third Order Benjamin–Ono Equation in Weighted Sobolev Spaces
where C(µ, ||φ||s) > 0 depends in an nondecreasing way on µ and ||φ||s , respectively.
If µ remains in a bounded set [0, µ0], one can choose C(µ, ||φ||s) depending only on
||φ||s. With this remark in mind, apply Gronwall’s inequality to (2.8)′ to obtain
||u(t)||γ,0 ≤ C(µ, γ, T, ||φ||γ,4), t ∈ [0, T ]
(2.28)
where C(µ, γ, T, ||φ||γ,4) has the same property as that of (2.27).
Now we return to the global existence result. If µ > 0, the global existence result
follows from (2.27) and (2.28).The existence result in the case µ = 0 can be proved
by the standard limiting argument.
Proof of uniqueness. Here the proof is reproduced from [3]. Let u, v be two
solutions to problem (2.6), (2.7) with the initial data φ(x), ψ(x) ∈ H s , respectively.
Putting w = u − v, then w satisfies the following equation
wt =
−µ∂x4w
4∂x3 w
+
− ∂x
3
(u − v ) + 3wHux + 3vHwx + H∂x (u2 − v 2)
2
3
3
w(x, 0) = φ(x) − ψ(x)
(2.29)
(2.30)
For µ > 0 fixed, the uniqueness result follows from the standard energy estimate.
So in the following we only give the proof for µ = 0.
From (2.29), a direct calculation of (2.29) and use of Lemma 2.2 yield
d
dt
Z
w2 dx = 2
1
− w2 (u2 + uv + v 2)x + 3wwx Hux + 3vwx Hwx
2
Z
3
3
− (u + v)wxHwx − (ux + vx )wHwx dx
2
2
Now it is not difficult to find that
|w|22
≤ |w(0)|2 + C(||φ||2, ||ψ||2)
Z
0
t
(2.31)
||w(t)||21dt
(2.32)
Since s ≥ 4, we know that |ut |L∞ ([0,T ]×R), |vt|L∞ ([0,T ]×R), |∂xk u(x, t)|L∞([0,T ]×R) and
|∂xk v(x, t)|L∞([0,T ]×R) are bounded for 0 ≤ k ≤ s − 1. Multiply equation (2.29) by
K4 (u) − K4 (v) to know
Z
wt (K4 (u) − K4 (v)) dx = 0
(2.33)
where K4 (u) = u3 + 3uHux + 3H(uux ) − 4u2x .
The bound for the term in the right-hand side of (2.33) is carried out as follows.
1)
Z
1
=
2
Z
t
0
Z
wt(u3 − v 3)dxdt
t
Z Z
1 t
w2 (u2 + uv + v 2)tdxdt
w (u + uv + v )dx −
2
2
2
2
0
≤ C(|w(t)|22 + |w(0)|22 +
Z
0
t
0
|w(τ )|22dτ )
(2.34)
534
X. Feng
3
=
2
t
Z
2)
0
Z
Z
3
wt [3uHwx + 3wHvx + H(u2 − v 2)x ]dxdt
2
2
w Hvx dx + 3
t
Z
uwHwx dx − 3
Z
0
t
0
Z
ut [wHwx + H(wwx )]dxdt
Z
≤ δ|wx|22 + C(δ)|w|22 + C||w(0)||21 + C
3) − 4
Z
t
0
Z
0
t
||w(τ )||21dτ
(2.35)
wt w2x dxdt = 2|wx |22 − 2|wx (0)|22
(2.36)
Adjusting the value of δ, from (2.32)-(2.36) we derive
||w(t)||21 ≤ C(||φ||1, ||ψ||1)||w(0)||21 + C(||φ||4, ||ψ||4)
Z
0
t
||w(τ )||21dτ
(2.37)
Applying Gronwall’s inequality to (2.37) to yield
||w(t)||21 ≤ C(T, ||φ||4, ||ψ||4)||w(0)||21 , t ∈ [0, T ]
(2.38)
which implies the uniqueness.
Up to now we have proved the following
Corollary 2.10. Let φ ∈ Hγs , s ≥ 4 and γ ∈ [0, 1]. Then for all fixed
µ ≥ 0 there exists a unique uµ ∈ C([0, T ]; Hγs) for any given T > 0 such that
∂tuµ ∈ C([0, T ]; H s−4) and ωγ′ ∂xk uµ ∈ C([0, T ]; L2) for 2k ≤ s and (2.6), (2.7) are
satisfied.
3 Proof of Theorem 1.1 Continued
In this section we continue the proof of Theorem 1.1.
Lemma 3.1. Let φ, ψ ∈ Hγs , s ≥ 4 and γ ∈ [0, 1], µ ≥ 0. Let uµ , vµ
be the solutions(obtained in Corollary 2.10) to (2.6) with φ, ψ as the initial data,
respectively. Suppose w = uµ − vµ . Then we have
||w(t)||2γ,0
≤
||w(0)||2γ,4
+ C(µ, T, ||φ||γ,4, ||ψ||γ,4)
Z
t
0
||w(t)||2γ,4dt, t ∈ [0, T ]
(3.1)
where C depends in an nondecreasing way on its arguments.
Proof. By the assumptions of the Lemma we know that w satisfies problem
(2.29), (2.30). Multiply (2.29) by ωγ2 w and integrate by parts to obtain
Z
d Z 2 2
ωγ w dx + 2µ ωγ2 (∂x2w)2 dx
dt
= −4µ
+ −8
Z
Z
+ −3
Z
ww2x ωγ ωγ′′ dx − 4µ
Z
ww2x (ωγ′ )2dx − 8µ
Z
Z
ωγ2 w∂x (u3 − v 3)dx + 2 (ωγ2 )′′′w2 dx − 24
ωγ2 w2 Hv2xdx + 3
Z
(ωγ2 )′ w2 Hvxdx − 6
Z
wx w2xωγ ωγ′′dx
Z
ωγ ωγ′ ww2xdx
ωγ2 wuHw2xdx
Third Order Benjamin–Ono Equation in Weighted Sobolev Spaces
535
7
X
(3.2)
−6
Z
ωγ2 wux Hwx dx − 3
Z
ωγ2 wH∂x2 (u2 − v 2)dx ≡
Bj
j=1
The estimate for each Bj is as follows. By using (2.9), (2.11), (2.12), (2.13) and
(2.14), we obtain
|B1 | ≤ µ|ωγ w2x|22 + µC|ωγ w|22 , |B2 | ≤ C(||φ||2, ||ψ||2)|ωγ w|22
Z
′
|B3 | ≤ 24 ωγ ωγ ww2x dx ≤ 24|ωγ w|2 |ωγ′ w2x|2 ≤ C|ωγ w|2 |ω γ2 w2x|2 ≤ C||w||2γ,4
|B4| ≤ C(T, ||φ||3, ||ψ||3)|ωγ w|22 , |B5 | ≤ C|ωγ u|2||w||2γ,2
B6 = 6
Z
ωγ2 wux Hwx dx = 6
Z
ωγ wω γ2 ux[ω γ2 , H∂x ]wdx + 6
Z
ωγ wω γ2 ux H∂x (ω γ2 w)dx
≤ 6|ω γ2 ux |∞ |ωγ w|2|[ω γ2 , H∂x ]w|2 + |ωγ w|2|∂x (ω γ2 w)|2 ≤ C(||u||γ,4)||w||2γ,4
B7 = 3
=3
Z
Z
ωγ2 wH∂x2 (u2 − v 2)dx
ωγ w[ωγ , H∂x2 ](u2 − v 2)dx + 3
Z
ωγ wH∂x2(ωγ w(u + v))dx
≤ C|ωγ w|2 ||u2 − v 2||γ,2 + |∂x2(ωγ (u + v)w)|2
≤ C(T, ||φ||γ,4, ||ψ||γ,4)||w||2γ,4
Considering (3.1) and (3.2) we know that
d
|ωγ w|22 + µ (ωγ )2(∂x2 w)2 dx ≤ C(µ, T, ||φ||γ,4, ||ψ||γ,4)||w||2γ,4
dt
Z
(3.3)
Integrate (3.3) with respect to t to yield the result.
Lemma 3.2. Let the assumptions in Lemma 3.1 be satisfied and s ≥ 6. Then
we have
Z
2s − 3
d
s−1
2
s−2
s−2
|∂x w|2 +
(u + v)∂x wH∂x wdx
dt
4
≤ C(µ, T, ||φ||s, ||ψ||s)||w(t)||2s−2, t ∈ [0, T ]
(3.4)
where C depends in an increasing way on its arguments and remains bounded whenever its arguments stay in a bounded set.
Proof. The proof of (3.4) for µ > 0 is simple and is omitted. But the proof for
µ = 0 is technical. For s = 6, a complicated calculation yields
d s−2 2
|∂ w|2 ≤ −27 (u + v)x∂x4wH∂x5 wdx + C(T, ||φ||6, ||ψ||6)||w(t)||24
dt x
Z
and
d
dt
Z
(u + v)∂xs−3wH∂xs−2 wdx
≤ 12
Z
(u + v)x∂x4wH∂x5 wdx + C(T, ||φ||6, ||ψ||6)||w(t)||24
These two inequalities mean that the derivative term in the left hand side equals
the first term in the right hand side plus a term less than the second term in the
536
X. Feng
right hand side. Hence, (3.4) holds for s = 6. For general s we can prove (3.4) by
induction on s. The details are omitted.
The Completion of Proof of Theorem 1.1. By Lemma 2.2 we have
2s − 3
s−3
s−2
s−3
s−2
(u + v)∂x wH∂x wdx ≤ C(|φ|2 , |ψ|2 )|∂x w|∞ |∂x w|2
4
Z
1
≤ |∂xs−2 w|22 + C|w|22
2
From (2.38), (3.4) and (3.5) there appears that
1
2s−5
≤ C|w| 2(s−2) |∂xs−2 w|22(s−2)
+1
|∂xs−2 w|22 ≤ C||w(0)||2s−2 + C
Z
0
t
||w(t)||2s−2 dt
(3.5)
(3.6)
Combining (2.38), (3.1), (3.6) with Gronwall’s inequality we can obtain
||w(t)||2γ,s−2 ≤ C(T, ||φ||γ,s, ||ψ||γ,s)||w(0)||2γ,s−2 , t ∈ [0, T ]
which implies the last statement of Theorem 1.1.
References
[1] Abdelouhab L. et al., Nonlocal models for nonlinear, dispersive waves, Physica
D, 40(1989), 360-392
[2] Calderon A. P., Commutators of singular integral operators, Proc. Nat. Acad.
Sci., 53(1965), 1092-1099
[3] Feng X., Han X., On the Cauchy problem for the third order Benjamin-Ono
equation, J. London Math. Soc., 53(1996), 512-528
[4] Iorio R. J., On the Cauchy problem for the Benjamin-Ono equations, Comm.
PDEs, 11(1986), 1031-1086
[5] Iorio R. J., The Benjamin-Ono equation in weighted Sobolev spaces, J. Math.
Anal. Appl., 157(1991), 577-590
[6] Kato T., Linear evolution equations of “hyperbolic” type, I, J. Math. Soc.
Japan, 25(1973), 648-666
[7] Ponce G., On the global well-posedness of the Benjamin-Ono equation, Differential and Integral Equations, 4(1991), 527-542
[8] Ruderman M. S., Nonlinear sausage-wave propagation in a magnetic slab in an
incompressible fluid, J. Plasma Physics, 49(1993), 271-282
[9] Stein E. M., Singular integrals and differentiability properties of functions,
Princeton University Press, Princeton, New Jersey, 1970
Third Order Benjamin–Ono Equation in Weighted Sobolev Spaces
Laboratory of Numerical Study for Helioshperic Physics
CAS, P.O.Box 8701, Beijing 100080
P. R. China
537
Benjamin–Ono Equation in Weighted Sobolev
Spaces
Xueshang Feng∗
Abstract
Here we continue the study of the initial value problem for the third order
T
Benjamin-Ono equation in the weighted Sobolev spaces Hγs = H s L2γ , where
s > 3, γ ≥ 0. The result is the proof of well-posedness of the afore mentioned
problem in Hγs , s > 3, γ ∈ [0, 1]. The proof involves the use of parabolic
regularization, the Riesz-Thorin interpolation theorem and the construction
technique of auxiliary functions.
1 Introduction
In recent years, the initial value problem for Benjamin-Ono(BO) equation of deep
water,
∂tu + ∂x (2Hux + u2 ) = 0
has been investigated by many authors [1, 4, 5, 7, 8]. Iorio[5] established the wellposedness of the above BO equation in Hγ2 , where γ ∈ [0, 1], by using Kato’s theory
of linear evolution equations of “hyperbolic type”. Ponce [7] proved the global wellposedness of the BO equation in H s , s ≥ 3/2. In [3], the authors obtained the
Supported by National Natural Science Foundation of China
Received by the editors May 1996.
Communicated by J. Mawhin.
1991 Mathematics Subject Classification : 35Q20, 35G25, 35D99.
Key words and phrases : The third order BO equation; Initial value problem; Weighted
Sobolev spaces; Global well-posedness.
∗
Bull. Belg. Math. Soc. 4 (1997), 525–537
526
X. Feng
global existence and uniqueness both in the usual Sobolev space H s and weighted
Sobolev spaces Hγs for the third order BO equation
∂tu = −∂x(u3 + 3uHux + 3H(uux ) − 4∂x2u), t > 0, x ∈ R
(1.1)
u(x, 0) = φ(x)
(1.2)
which can be considered as the high order approximation of the BO equation. Here
∂k
u = u(x, t) is real valued, ∂xk = ∂x
k and the subscripts appended to u stand for
partial differentiations, and H denotes the Hilbert transform
(Hf)(x) = P.V.
In this paper, Hγs = H s
T
Z
f(y)
dy
π(y − x)
L2γ with the following norm
||u||2γ,s = ||f||2s + |ωγ f|22 , ωγ (x) = (1 + x2 )γ
where || • ||s and | • |2 are the usual norms of the Sobolev spaces H s (R) and Lebesgue
space L2 (R), respectively. Throughout the paper, | • |p will denote the usual norm
of the Lebesgue spaces Lp (R), p ∈ [1, ∞]. If X and Y are Banach spaces we denote
the set of all bounded operators from X to Y by B(X, Y ). If X = Y we write B(X)
instead of B(X, Y ). As usual [A, B] will indicate the commutator of two linear
operators(not necessarily continuous).
The BO equation arises in the study of unidirectional propagation of nonlinear
dispersive waves or long nonlinear sausage-wave propagation in a magnetic slab in
an incompressible plasma of the solar atmosphere, and presents the interesting fact
that operrators modelling the dispersive effect in the BO equation or the effect of
derivative nonlinearity in (1.1) are nonlocal. In [3, 5], the authors have shown that
u ∈ C([0, T ]; H 43 ) or u ∈ C([0, T ]; H24) solves equation (1.1) or the BO equation
2
if and only if u ≡ 0, and pointed out in [3] that this phenomena are caused by
the nonlocal terms modelling the equations. The papers [1, 7] are devoted to the
proof of the well-posedness of the global solutions to the initial value problem of
the BO equation in H s , for s ≥ 32 . Many physicists such as Ruderman (see [8]
and references therein) have used this equation to study the wave propagation in a
magnetically structured atmosphere and found its solutions in the form of periodic
waves of permanent shape numerically. As the high order approximation of the
BO equation, the third order BO equation (1.1) is interesting in its own rights and
therefore it is of importance to study its well-posedness whether mathematically or
physically considered. I would here like to apologize if I miss citing other papers on
the BO equation.
Our aim here is to investigate the global well-posedness of problem (1.1), (1.2)
in the weighted Sobolev spaces Hγs as done in [5] for the BO equation.
Our main result is as follows:
Theorem 1.1. Let φ ∈ Hγs , s ≥ 4 and γ ∈ [0, 1]. Then for all fixed µ ≥ 0
there exists a unique uµ ∈ C([0, T ]; H s) for any given T > 0 such that ∂tuµ ∈
C([0, T ]; H s−4) and ωγ′ ∂xk uµ ∈ C([0, T ]; L2) for k ≤ s/2 and the following (2.6), (2.7)
are satisfied. Moreover,
|ωγ′ ∂xk u|2 ≤ C(γ)||u||γ,s
Third Order Benjamin–Ono Equation in Weighted Sobolev Spaces
527
Furthermore, the map φ 7→ uµ is continuous in the sense that if φn → φ in Hγs−2
and unγ , uµ are the corresponding solutions then
lim sup[0,T ]||unµ (t) − uµ (t)||γ,s−2 = 0
n→∞
where s ≥ 6.
In fact, we shall show that the map φ 7→ u is globally Lipschitz. In our cases, it
is difficult to apply Kato’s theory of linear evolution equations of “hyperbolic type”
because of the appearance of the derivative nonlinear nonsmooth terms. The proof
in this paper is very technical and long. In the following two sections we shall finish
the proof of Theorem 1.1 by presenting a series of statements.
2 Preliminary results
In this part, a series of lemmas are given for the purpose of proving Theorem 1.1.
The lemmas below are established following the same lines as those of [5].
Lemma 2.1. Suppose
n a(x) isodefined on R and |a(x)−a(y)| ≤ M|x−y|, x, y ∈
R
R. Let Tǫ (f) = |x−y|≥ǫ a(x)−a(y)
f(y)dy. Then
(x−y)2
|Tǫ (f)|p ≤ Ap |f|p , for all f ∈ Lp (R)
with Ap independent of ǫ, 1 < p < ∞.
For details and further results of this kind see Calderon [2] and P.238 of Stein
[9]. The following well-known Gagliardo-Nirenberg’s inequality will be unexplainedly
used throughout the paper.
Lemma 2.2. For f ∈ H m (R) and p ≥ 2, then we have
|∂xd f|p ≤ 2
(p−2)
2p
|∂xk u|λ2 |u|1−λ
(2.1)
where d, m, k ∈ N {0}, λ = (d + 21 − 1p )/k, and 0 ≤ d < k ≤ m. More precisely,
for f ∈ H d+1 (R) we have
S
1
2d+1
|∂xd f|∞ ≤ |f|22(d+1) |∂xd+1f|22(d+1)
(2.2)
Proof. (2.1) has been well-known. To prove (2.2), noting that
2
|f(x)| = 2
Z
2
|f(x)| = −2
x
−∞
Z
ffx dx ≤ 2
Z
∞
x
ffx dx ≤ 2
x
−∞
Z
|ffx |dx
∞
x
|ffx |dx
We get by using the Cauchy-Schwartz inequality
|f|2∞ ≤
Z
|ffx |dx ≤ |f|2 |fx |2
(2.3)
which gives (2.2) for the case d=0. By using (2.1) and (2.3) we get
2d+1
1
|∂xdu|2∞ ≤ |∂xdu|2|∂xd+1 u|2 ≤ |∂xd+1u|2d+1 |u|2d+1
528
X. Feng
which finishes the proof of (2.2).
It should be pointed out that inequality (2.2) can be used to improve the uniqueness result for the smallness case of the initial data in [3].
Lemma 2.3. Let λ ∈ [0, ∞), γ ∈ [0, 1], µ > 0, and Qµ = −µ∂x4 + 4∂x3 . Then
Sµ (t) = exp(tQµ ) ∈ B(Hγs , Hγs+λ ) for all t > 0, s ≥ 0 and satisfies the estimate
||Sµ (t)φ||γ,s ≤ G(t, µ, λ, γ)||φ||γ,s
(2.4)
for all φ ∈ Hγs , where G(t, µ, λ, γ) is locally integrable with respect to t. Moreover
the map t ∈ (0, ∞) 7→ Sµ (t)φ is continuous with respect to the topology of Hγs+λ .
Proof. Due to [3], where the Lemma is established in the case of H0s = H s , s ∈ R,
it suffices to examine Sµ (t) as an operator from Hγs into Hγ0 = L2γ . If γ = 0 we obtain
T
H00 = L2 so that ||Sµ (t)φ||0,0 ≤ ||φ||0,0. Next, if γ = 1 and φ ∈ H1s = H s L21 we
have
(ω1 Sµ (t)φ)∧(ξ) = Eµ (ξ, t)(1 − ∂ξ2)φˆ − (∂ξ2Eµ (ξ, t))φˆ − 2(∂ξ Eµ (ξ, t))∂ξ φˆ
(2.5)
where Eµ (ξ, t) = exp[−(µξ 4 + 4iξ 3 )t]. The result then follows by combining (2.5)
with the formulas
∂ξ Eµ (ξ, t) = −(4µξ 3 + 12iξ 2 )tEµ (ξ, t)
∂ξ2Eµ (ξ, t) = −12t(µξ 2 + 2iξ)Eµ (ξ, t) + 16t2 ξ 4 (µξ + 3i)Eµ (ξ, t)
and the estimate
λ
0 ≤ ξ 2λ exp(−µtξ 4 ) ≤ λλ exp( )(2µt)−λ/2
2
The statement in the Lemma is now an easy consequence of the Riesz-Thorin
interpolation theorem.
We are now in position to establish a local existence for the following parabolic
regularized problem
∂tuµ = −µ∂x4uµ − ∂x (u3µ + 3uµ H∂x uµ + 3H(uµ ∂x uµ ) − 4∂x2uµ )
(2.6)
uµ (x, 0) = φ(x)
(2.7)
in Hγs , γ ∈ [0, 1] and µ > 0. Let T be a positive number and consider the set
n
Xγ,s (T ) = f ∈ C([0, T ]; Hγs); ||f(t) − Sµ (t)φ||γ,s ≤ ||φ||γ,s , t ∈ [0, T ]
o
which becomes a complete metric space when provided with the distance obtained
from the sup norm. Note that if f ∈ Xγ,s (T ), then by Lemma 2.3 it is not difficult
to show that one can choose T > 0 such that the map
(Af)(t) = Sµ (t)φ −
Z
0
t
Sµ (t − τ )∂x f 3 + 3fHfx + 3H(ffx ) dτ
defines a contraction in Xγ,s (T ). In view of the uniqueness and regularity results
established in [3] we have
Corollary 2.4. Let φ ∈ Hγs , s ≥ 4, γ ∈ [0, 1], µ > 0. Then there exist a
T > 0 and a unique uµ ∈ C([0, T ]; Hγs) such that ∂t uµ ∈ C([0, T ]; H s−4) satisfies
problem (2.6), (2.7). Moreover, uµ ∈ C((0, T ]; Hγq ) for all q ∈ R.
Third Order Benjamin–Ono Equation in Weighted Sobolev Spaces
529
Let µ > 0 be fixed. From now on, unless otherwise specifed, we write u = uµ for
simplicity. The next step is to establish global existence in Hγs , s ≥ 4 and γ ∈ [0, 1].
This has already been done in [3] in the cases γ = 0 and γ = 1 (the result for γ = 1
will be re-obtained below from a slightly different point of view). In order to obtain
global estimates for the Hγs norm of the solution it suffices to study what happens
in L2γ since the H s result was proved in [3]. Now, by Corollary 2.4 and integration
by parts, at least formally we have
t
Z
ωγ2 u22x dx
uu2xωγ ωγ′′dx − 4µ
Z
uu2x(ωγ′ )2 dx − 8µ
∂t ||u(t)||2γ,0
= −4µ
−8
Z
Z
ωγ2 u∂x (u3)
+ 2µ
dx − 3
Z
+2 (ωγ2 )′′′u2 dx − 24
Z
Z
Z
0
ωγ2 u2 H(∂x2u)dx
+6
ωγ ωγ′ uu2x dx − 3
Z
Z
Z
ux u2xωγ ωγ′′ dx
ωγ ωγ′ u2 Hux dx
ωγ2 uH(∂x2(u2))dx
(2.8)
In order to extend the local solution to the whole time interval [0, T ], for any fixed
T > 0 we must estimate the right-hand side of (2.8). For this purpose we need the
following Lemmas.
Lemma 2.5. Let u ∈ Hγ4 , γ ∈ [0, 1]. Then ωγ/2 u2x ∈ L2 and satisfies
1
1
|ω γ2 u2x |2 ≤ C||u||42 |ωγ u|22 ≤ C||u||γ,4
(2.9)
where C is a generic constant.
Proof. By definition, for u ∈ S(R)(the Schwartz space) we have
|ω
γ
2
u2x|22
=
Z
ωγ u22xdx
=
Z
ωγ′′ uu2x
dx + 2
Z
ωγ′ uu3xdx
+
Z
ωγ uu4xdx
≤ |ωγ′′u|2|u2x |2 + 2|ωγ′ u|2|u3x |2 + |ωγ u|2|u4x|2 ≤ C||u||4|ωγ u|2
(2.10)
which combines a limiting argument to yield the result.
Lemma 2.6. Let f ∈ Hγs , γ ∈ [0, 1]. Then x(1 + x2 )γ ∂xk f ∈ L2 and
|x(1 + x2)(r−1) ∂xk f|2 ≤ C(r)||f||γ,s , 2k ≤ s
(2.11)
where C(γ) is a constant depending only on γ. The same results are true if x(x2 +
1)γ−1 is replaced by ωγ′ (x).
Proof. A simple limiting argument shows that it suffices to prove (2.11) for
f ∈ S(R). Assume therefore that this is the case and integrate by parts to obtain
for k = 1
|x(1 + x2)γ−1 fx |22 = −
3
X
Ij
j=1
where
Z
I1 = 2 (1 + x2)γ f(x)[x(1 + x2)γ−2 ]fxdx
I2 = 2(2γ − 2)
Z
(1 + x2)γ f(x)[x3(1 + x2)γ−3 ]fx dx
530
X. Feng
I3 =
Z
(1 + x3)γ f(x)[x2(1 + x2)γ−2 ]∂x2f(x)dx
It is easy to verify that the expressions inside the square brackets in all the three
ingegrals are bounded of x. By this remark and the Cauchy-Schwartz inequality we
obtain
|I1| ≤ 2|fx |2|ωγ |2, |I2| ≤ |4γ − 4||fx |2 |ωγ f|2 , |I3| ≤ |∂x2f|2 |ωγ f|2
Now, we have
|x(1 + x2)γ−1 fx |2 ≤ {(2 + |4γ − 4|)|fx |2 + |f2x|2 } |ωγ f|2 ≤ C(γ)||f||2|ωγ f|2
which finishes the proof. For the general case, a similar argument yields
|x(1 + x2)γ−1 ∂xk f|22 =
2k Z
X
(1 + x2 )γ f(x)pj (x)∂xj f(x)dx
j=k
where pj (x) is bounded in x. So the result follows.
Corollary 2.7. Let f ∈ Hγ4 , γ ∈ [0, 1]. Then ω γ2 fx ∈ L2 , ω γ2 fx ∈ L∞ and
satisfy the estimates
1
1
|ω γ2 fx |2 ≤ C(γ)||f||22 |ωγ f|22
(2.12)
1
1
|ω γ2 fx |∞ ≤ C(γ)||f||44 |ωγ f|22
(2.13)
where C(γ) > 0 depends only on γ.
Proof. Similar to the proof of (2.9), it suffices to prove (2.12) for f ∈ S(R). If
this is the case, then we have
|ω γ2 fx |22 =
Z
ωγ fx fx dx = −
Z
ωγ ff2xdx −
Z
ωγ′ ffx dx
≤ |fx |2|ωγ′ f|2 + |f2x|2 |ωγ f|2 ≤ C(γ)||f||2|ωγ f|2
which gives (2.12). By (2.2), (2.9), (2.11) and (2.12) we obtain
|ω γ2 fx |2∞ ≤ |ω γ2 fx |2|ω ′γ fx + ω γ2 f2x|2 ≤ C(γ)||f||4|ωγ f|2
2
This finishes (2.13).
Lemma 2.8. Let f ∈ Hγs , γ ∈ [0, 1]. Then [ω γ2 , H∂x ]f ∈ L2 and
|[ω γ2 , H∂x ]f|2 ≤ C(γ)|f|2
(2.14)
where C(γ) depends only on γ.
Proof. First observe that it suffices to prove (2.14) in case of f ∈ S(R). Now
[ω γ2 , H∂x ]f = [ω γ2 , H]∂x f − H[∂x , ω γ2 ]f
(2.15)
For the second commutator in (2.15) we have
H[∂x, ω γ2 ]f = H(ω ′γ f)
2
(2.16)
Third Order Benjamin–Ono Equation in Weighted Sobolev Spaces
531
For the first commutator in (2.15) we integrate by parts once to obtain
[ω γ2 , H]∂xf = P.V.
1
π
Z ω γ (x) − ω γ (y)
2
2
(x − y)2
f(y)dy − H(ω ′γ f)
(2.17)
f(y)dy − 2H(ω ′γ f)
(2.18)
2
From (2.15)-(2.17) we know that
[ω γ2 , H∂x ]f = P.V.
1
π
Z ω γ (x) − ω γ (y)
2
2
(x − y)2
2
Since ω ′γ is bounded and ω γ2 is Lipschitz, by Lemma 2.1 and the fact that H ∈ B(L2 )
2
it follows from (2.18) that
|[ωγ , H∂x2]f|2 ≤ C(γ)|f|2
which finishes the proof.
From the proof above we see that (2.14) holds for all f ∈ L2 .
Lemma 2.9. Let f ∈ Hγ2 , γ ∈ [0, 1]. Then [ωγ , H∂x2 ]f ∈ L2 and
|[ωγ , H∂x2]f|2 ≤ C(γ)||f||γ,2
(2.19)
where C(γ) ≥ 0 depends only on γ.
Proof. First note that it suffices to prove (2.19) for f ∈ S(R). Evidently
[ωγ , H∂x2]f = [ωγ , H]∂x2f − H[∂x2, ωγ ]f
(2.20)
So it suffices to establish (2.19) with [ωγ , H∂x2]f replaced by H[∂x2, ωγ ]f and [ωγ , H]∂x2f.
The result then follows for the first of these functions by combining the formula
H[∂x2, ωγ ]f = H(ωγ′′f + 2ωγ′ ∂xf)
(2.21)
with Lemma 2.6 and the facts that ωγ′′ ∈ L∞ and H ∈ B(L2). In order to handle
[ωγ , H]∂x2f we integrate by parts once to obtain
[ωγ , H]∂x2f
=
H(ωγ′ ∂x f)
1
− P.V.
π
Z
ωγ (x) − ωγ (y)
f(y)dy
(x − y)2
(2.22)
In view of Lemma 2.6 the first term on the right-hand side of (2.22) satisfies
|H(ωγ′ ∂xf)|2 ≤ C(γ)||f||γ,2, so it remains to bound the second. Since ωγ′ is bounded
if and only if γ ∈ [0, 21 ], in this case ωγ is Lipschitz. Hence, by Lemma 2.1 we have
Z
1 ωγ (x) − ωγ (y)
P.V.
f(y)dy
≤ A|f|2
π
(x − y)2
(2.23)
where A > 0 is a generic constant, γ ∈ [0, 12 ]. In the case of γ ∈ ( 21 , 1] we integrate
by parts once in (2.22) to get
[ωγ , H]∂x2f = −P.V.
1
π
Z
K(x, y)g(y)dy
(2.24)
532
X. Feng
2(ωγ (x) − ωγ (y)) − 2ωγ′ (y)(x − y) − ωγ′′(y)(x − y)2
(x − y)3(1 + y 2 )γ
K(x, y) =
(2.25)
where g(y) = (1 + y 2)γ f(y) ∈ L2 . If γ ∈ ( 21 , 1], it is not difficult to verify that
K(x, y) is a Hilbert-Schmidt kernel. Thus
1Z
P.V.
K(x, y)ωγ (y)f(y)dy ≤ C|ωγ f|2
(2.26)
π
where C > 0 is a generic constant, γ ∈ ( 21 , 1]. Now the result of the lemma follows
from (2.20)-(2.26).
Now we are in position to estimate each term in the right-hand side of (2.8) as
follows:
Z
µ
µ
′′
R1 ) 4µ uu2xωγ ωγ dx ≤ |ωγ u2x|22 + Cµ|ωγ′′u|2 ≤ |ωγ u2x |22 + Cµ|u|22
3
3
Z
µ
′ 2
R2 ) 4µ uu2x (ωγ ) dx ≤ |ωγ u2x|22 + Cµ|ωγ u|22
3
Z
µ
R3 ) 8µ ux u2xωγ ωγ′′dx ≤ |ωγ u2x |22 + Cµ|ux |22
3
Z
R4 ) 8 (ωγ )2u∂x (u3)dx ≤ C(||u||2)|ωγ u|22
Z
2 2
2
R5 ) 3 (ωγ ) u H(∂x u)dx ≤ C(||u||3)|ωγ u|22
Z
′
2
R6 ) 6 ωγ ωγ u Hux dx ≤ C(||u||2)|ωγ u|22
Z
′′′ 2
R7 ) 2 (ωγ ) u dx ≤ |ωγ u|22
Z
′
R8 ) 24 ωγ ωγ uu2xdx ≤ 24|ωγ u|2|ωγ′ u2x |2 ≤ C|ωγ u|2|ω γ2 u2x|2 ≤ C(||u||4)|ωγ u|22
Z
Z
Z
(ωγ )2uH∂x2 (u2)dx =
R9 )
=
Z
ωγ u[ωγ , H∂x2]u2 dx +
Z
ωγ u[ωγ , H∂x2]u2dx +
ωγ uH∂x2(ωγ u2 )dx
ωγ uH[ωγ′′u2 + 2ωγ′ uux + 2ωγ u2x + 2ωγ uu2x)dx
n
≤ C|ωγ u|2 |[ωγ , H∂x2 ]u2|2 + |ωγ′′u2 |2|ωγ′ uux|2 + |ωγ u2x |2 + |ωγ uu2x|2
n
o
≤ C(γ)|ωγ u|2 ||u2||γ,2 + |u2|2 + |ωγ′ uux |2 + |ω γ2 ux |∞ |ω γ2 ux |2 + |u2x|∞ |ωγ u|2
o
≤ C(γ, ||u||4)|ωγ u|2
Considering R1 –R9 ) and (2.8) we obtain
Z
d
||u(t)||2γ,0 + µ (ωγ )2 u22xdx ≤ C(µ, γ, ||u||4)|ωγ u|22
(2.8)′
dt
By [3] we know that ||u(t)||s is bounded uniformly for t in any fixed interval [0, T ]
and µ varying in any bounded interval. That is,
||u(t)||s ≤ C(µ, ||φ||s ), t ∈ [0, T ]
(2.27)
533
Third Order Benjamin–Ono Equation in Weighted Sobolev Spaces
where C(µ, ||φ||s) > 0 depends in an nondecreasing way on µ and ||φ||s , respectively.
If µ remains in a bounded set [0, µ0], one can choose C(µ, ||φ||s) depending only on
||φ||s. With this remark in mind, apply Gronwall’s inequality to (2.8)′ to obtain
||u(t)||γ,0 ≤ C(µ, γ, T, ||φ||γ,4), t ∈ [0, T ]
(2.28)
where C(µ, γ, T, ||φ||γ,4) has the same property as that of (2.27).
Now we return to the global existence result. If µ > 0, the global existence result
follows from (2.27) and (2.28).The existence result in the case µ = 0 can be proved
by the standard limiting argument.
Proof of uniqueness. Here the proof is reproduced from [3]. Let u, v be two
solutions to problem (2.6), (2.7) with the initial data φ(x), ψ(x) ∈ H s , respectively.
Putting w = u − v, then w satisfies the following equation
wt =
−µ∂x4w
4∂x3 w
+
− ∂x
3
(u − v ) + 3wHux + 3vHwx + H∂x (u2 − v 2)
2
3
3
w(x, 0) = φ(x) − ψ(x)
(2.29)
(2.30)
For µ > 0 fixed, the uniqueness result follows from the standard energy estimate.
So in the following we only give the proof for µ = 0.
From (2.29), a direct calculation of (2.29) and use of Lemma 2.2 yield
d
dt
Z
w2 dx = 2
1
− w2 (u2 + uv + v 2)x + 3wwx Hux + 3vwx Hwx
2
Z
3
3
− (u + v)wxHwx − (ux + vx )wHwx dx
2
2
Now it is not difficult to find that
|w|22
≤ |w(0)|2 + C(||φ||2, ||ψ||2)
Z
0
t
(2.31)
||w(t)||21dt
(2.32)
Since s ≥ 4, we know that |ut |L∞ ([0,T ]×R), |vt|L∞ ([0,T ]×R), |∂xk u(x, t)|L∞([0,T ]×R) and
|∂xk v(x, t)|L∞([0,T ]×R) are bounded for 0 ≤ k ≤ s − 1. Multiply equation (2.29) by
K4 (u) − K4 (v) to know
Z
wt (K4 (u) − K4 (v)) dx = 0
(2.33)
where K4 (u) = u3 + 3uHux + 3H(uux ) − 4u2x .
The bound for the term in the right-hand side of (2.33) is carried out as follows.
1)
Z
1
=
2
Z
t
0
Z
wt(u3 − v 3)dxdt
t
Z Z
1 t
w2 (u2 + uv + v 2)tdxdt
w (u + uv + v )dx −
2
2
2
2
0
≤ C(|w(t)|22 + |w(0)|22 +
Z
0
t
0
|w(τ )|22dτ )
(2.34)
534
X. Feng
3
=
2
t
Z
2)
0
Z
Z
3
wt [3uHwx + 3wHvx + H(u2 − v 2)x ]dxdt
2
2
w Hvx dx + 3
t
Z
uwHwx dx − 3
Z
0
t
0
Z
ut [wHwx + H(wwx )]dxdt
Z
≤ δ|wx|22 + C(δ)|w|22 + C||w(0)||21 + C
3) − 4
Z
t
0
Z
0
t
||w(τ )||21dτ
(2.35)
wt w2x dxdt = 2|wx |22 − 2|wx (0)|22
(2.36)
Adjusting the value of δ, from (2.32)-(2.36) we derive
||w(t)||21 ≤ C(||φ||1, ||ψ||1)||w(0)||21 + C(||φ||4, ||ψ||4)
Z
0
t
||w(τ )||21dτ
(2.37)
Applying Gronwall’s inequality to (2.37) to yield
||w(t)||21 ≤ C(T, ||φ||4, ||ψ||4)||w(0)||21 , t ∈ [0, T ]
(2.38)
which implies the uniqueness.
Up to now we have proved the following
Corollary 2.10. Let φ ∈ Hγs , s ≥ 4 and γ ∈ [0, 1]. Then for all fixed
µ ≥ 0 there exists a unique uµ ∈ C([0, T ]; Hγs) for any given T > 0 such that
∂tuµ ∈ C([0, T ]; H s−4) and ωγ′ ∂xk uµ ∈ C([0, T ]; L2) for 2k ≤ s and (2.6), (2.7) are
satisfied.
3 Proof of Theorem 1.1 Continued
In this section we continue the proof of Theorem 1.1.
Lemma 3.1. Let φ, ψ ∈ Hγs , s ≥ 4 and γ ∈ [0, 1], µ ≥ 0. Let uµ , vµ
be the solutions(obtained in Corollary 2.10) to (2.6) with φ, ψ as the initial data,
respectively. Suppose w = uµ − vµ . Then we have
||w(t)||2γ,0
≤
||w(0)||2γ,4
+ C(µ, T, ||φ||γ,4, ||ψ||γ,4)
Z
t
0
||w(t)||2γ,4dt, t ∈ [0, T ]
(3.1)
where C depends in an nondecreasing way on its arguments.
Proof. By the assumptions of the Lemma we know that w satisfies problem
(2.29), (2.30). Multiply (2.29) by ωγ2 w and integrate by parts to obtain
Z
d Z 2 2
ωγ w dx + 2µ ωγ2 (∂x2w)2 dx
dt
= −4µ
+ −8
Z
Z
+ −3
Z
ww2x ωγ ωγ′′ dx − 4µ
Z
ww2x (ωγ′ )2dx − 8µ
Z
Z
ωγ2 w∂x (u3 − v 3)dx + 2 (ωγ2 )′′′w2 dx − 24
ωγ2 w2 Hv2xdx + 3
Z
(ωγ2 )′ w2 Hvxdx − 6
Z
wx w2xωγ ωγ′′dx
Z
ωγ ωγ′ ww2xdx
ωγ2 wuHw2xdx
Third Order Benjamin–Ono Equation in Weighted Sobolev Spaces
535
7
X
(3.2)
−6
Z
ωγ2 wux Hwx dx − 3
Z
ωγ2 wH∂x2 (u2 − v 2)dx ≡
Bj
j=1
The estimate for each Bj is as follows. By using (2.9), (2.11), (2.12), (2.13) and
(2.14), we obtain
|B1 | ≤ µ|ωγ w2x|22 + µC|ωγ w|22 , |B2 | ≤ C(||φ||2, ||ψ||2)|ωγ w|22
Z
′
|B3 | ≤ 24 ωγ ωγ ww2x dx ≤ 24|ωγ w|2 |ωγ′ w2x|2 ≤ C|ωγ w|2 |ω γ2 w2x|2 ≤ C||w||2γ,4
|B4| ≤ C(T, ||φ||3, ||ψ||3)|ωγ w|22 , |B5 | ≤ C|ωγ u|2||w||2γ,2
B6 = 6
Z
ωγ2 wux Hwx dx = 6
Z
ωγ wω γ2 ux[ω γ2 , H∂x ]wdx + 6
Z
ωγ wω γ2 ux H∂x (ω γ2 w)dx
≤ 6|ω γ2 ux |∞ |ωγ w|2|[ω γ2 , H∂x ]w|2 + |ωγ w|2|∂x (ω γ2 w)|2 ≤ C(||u||γ,4)||w||2γ,4
B7 = 3
=3
Z
Z
ωγ2 wH∂x2 (u2 − v 2)dx
ωγ w[ωγ , H∂x2 ](u2 − v 2)dx + 3
Z
ωγ wH∂x2(ωγ w(u + v))dx
≤ C|ωγ w|2 ||u2 − v 2||γ,2 + |∂x2(ωγ (u + v)w)|2
≤ C(T, ||φ||γ,4, ||ψ||γ,4)||w||2γ,4
Considering (3.1) and (3.2) we know that
d
|ωγ w|22 + µ (ωγ )2(∂x2 w)2 dx ≤ C(µ, T, ||φ||γ,4, ||ψ||γ,4)||w||2γ,4
dt
Z
(3.3)
Integrate (3.3) with respect to t to yield the result.
Lemma 3.2. Let the assumptions in Lemma 3.1 be satisfied and s ≥ 6. Then
we have
Z
2s − 3
d
s−1
2
s−2
s−2
|∂x w|2 +
(u + v)∂x wH∂x wdx
dt
4
≤ C(µ, T, ||φ||s, ||ψ||s)||w(t)||2s−2, t ∈ [0, T ]
(3.4)
where C depends in an increasing way on its arguments and remains bounded whenever its arguments stay in a bounded set.
Proof. The proof of (3.4) for µ > 0 is simple and is omitted. But the proof for
µ = 0 is technical. For s = 6, a complicated calculation yields
d s−2 2
|∂ w|2 ≤ −27 (u + v)x∂x4wH∂x5 wdx + C(T, ||φ||6, ||ψ||6)||w(t)||24
dt x
Z
and
d
dt
Z
(u + v)∂xs−3wH∂xs−2 wdx
≤ 12
Z
(u + v)x∂x4wH∂x5 wdx + C(T, ||φ||6, ||ψ||6)||w(t)||24
These two inequalities mean that the derivative term in the left hand side equals
the first term in the right hand side plus a term less than the second term in the
536
X. Feng
right hand side. Hence, (3.4) holds for s = 6. For general s we can prove (3.4) by
induction on s. The details are omitted.
The Completion of Proof of Theorem 1.1. By Lemma 2.2 we have
2s − 3
s−3
s−2
s−3
s−2
(u + v)∂x wH∂x wdx ≤ C(|φ|2 , |ψ|2 )|∂x w|∞ |∂x w|2
4
Z
1
≤ |∂xs−2 w|22 + C|w|22
2
From (2.38), (3.4) and (3.5) there appears that
1
2s−5
≤ C|w| 2(s−2) |∂xs−2 w|22(s−2)
+1
|∂xs−2 w|22 ≤ C||w(0)||2s−2 + C
Z
0
t
||w(t)||2s−2 dt
(3.5)
(3.6)
Combining (2.38), (3.1), (3.6) with Gronwall’s inequality we can obtain
||w(t)||2γ,s−2 ≤ C(T, ||φ||γ,s, ||ψ||γ,s)||w(0)||2γ,s−2 , t ∈ [0, T ]
which implies the last statement of Theorem 1.1.
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Third Order Benjamin–Ono Equation in Weighted Sobolev Spaces
Laboratory of Numerical Study for Helioshperic Physics
CAS, P.O.Box 8701, Beijing 100080
P. R. China
537