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Hölder continuity on μ-b equation
Nonlinear Analysis 102 (2014) 30–35
Contents lists available at ScienceDirect
Nonlinear Analysis journal homepage: www.elsevier.com/locate/na
Hölder continuity on µ-b equation Guangying Lv a,b,∗ , Xiaohuan Wang a a
College of Mathematics and Information Science, Henan University, Kaifeng 475001, PR China
b
Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL 60616, USA
article
info
Article history: Received 30 November 2013 Accepted 5 February 2014 Communicated by Enzo Mitidieri MSC: 35G25 35B30 35L05
abstract This paper is concerned with the Hölder continuous on initial data for the µ-b equation on the circle. It is shown that the data to solution for the µ-b equation is Hölder continuous from bounded set of Sobolev spaces with exponent s > 3/2 measured in a weaker Sobolev norm with index r < s in periodic case. © 2014 Elsevier Ltd. All rights reserved.
Keywords: Hölder continuity µ-b equation Energy estimates
1. Introduction The b-equation [1–3], ut − uxxt + u(ux − uxxx ) + bux (u − uxx ) = 0,
(1.1)
which in fact comprises a family of equations parameterized by the real value b, models wave phenomena in shallow fluid flow, and has been the subject of considerable interest both to mathematicians and physicists. When b = 2 and 3, respectively, (1.1) reduces to the well-known Camassa–Holm (CH) [4] and Degasperis–Procesi (DP) [2] equations, which arise as models for shallow water waves of moderate amplitude [5,6] and have been extensively studied, particularly from the point of view of integrability and geodesic flow on infinite dimensional Lie groups [7–11]. In contrast to classical model equations for water waves, like KdV and BBM, novel features are captured by CH and DP. For example, the development of singularities in finite time in the form of breaking waves (meaning that the solution remains uniformly bounded but its slope becomes unbounded in finite time), see [12] and the reference therein. Of great interest is also the existence of traveling wave solutions that are peaked at their crest. These peakons are similar to the waves of greatest height that arise as solutions to the governing equations for water waves (see the discussion in [13–17]). The Cauchy problem of the b-equation, and in particular its well-posedness and blow-up behavior, has also been well-studied both on the real line and on the circle, see for example [12,18–21]; in particular, [22] looks at the non-uniform dependence on initial data for the b-equation. In [23], Khesin, Lenells and Misiołek studied a generalization of the Hunter–Saxton (HS) [24] equation, which is a high frequency limit of the CH equation, and introduced the µ-HS equation. The µ-HS equation, which is now more commonly referred to as the µ-CH equation, describes the propagation of weakly nonlinear orientation waves in a massive nematic
∗
Corresponding author at: College of Mathematics and Information Science, Henan University, Kaifeng 475001, PR China. Tel.: +86 15137877126. E-mail addresses: [email protected], [email protected], [email protected] (G. Lv), [email protected] (X. Wang).
http://dx.doi.org/10.1016/j.na.2014.02.003 0362-546X/© 2014 Elsevier Ltd. All rights reserved.
G. Lv, X. Wang / Nonlinear Analysis 102 (2014) 30–35
31
liquid crystal with external magnetic field and self-interaction. It was also shown in [23] that the µ-CH equation on the circle is formally integrable, admits a bi-Hamiltonian structure and an infinite hierarchy of conservation laws, and is an Euler equation for geodesic flow on the Lie group of circle diffeomorphisms. Subsequently, in addition to µ-CH, Lenells, Misiołek and Tiğlay [25] also introduced the µ-DP as well as µ-Burgers equations, and the µ-b equation (see also [26,27]). In this paper, we consider the Cauchy problem for the (spatially) periodic µ-b equation:
µ(ut ) − uxxt − uuxxx + bux (µ(u) − uxx ) = 0, u(0, x) = u0 (x),
t > 0, x ∈ S1 , x ∈ S1 ,
(1.2)
where b is a real constant, u = u(t , x) is a time-dependent function on the unit circle S1 = R/Z ≃ [0, 1), and u(t , x)dx denotes the mean value of u. We note that under spatial periodicity, µ(ut ) = 0 = µ(ux ). S1 In the case b = 2 and 3, respectively, the µ-b equation reduces to the µ-CH and µ-DP equations. In addition, if µ(u) = 0, they reduce to the HS and µ-Burgers equations, respectively. The local well-posedness of the µ-CH and µ-DP Cauchy problems have been studied in [23,25]. Recently, Fu et al. [28] described precise blow-up scenarios for µ-CH and µ-DP. Our attention in this paper is on the property on initial data. Himonas et al. [29,30] showed that solutions of the CH equation on the real line as well as on the circle do not depend uniformly on the initial data. They made use of the method of approximate solutions [31–34]. Lv et al. [27] studied the µ-b equation and obtained the non-uniform dependence on initial data. Himonas and Holmes [35] studied the Novikov equation and obtained the Hölder continuity of the solution map. The Hölder continuous of solution map of b-equation was established by Chen et al. [36]. The Hölder continuous of µ-b equation is an open problem and we will give a positive answer in this paper. Our work has been inspired by [36,35]. We show the solution map of (1.2) is Hölder continuous from bounded set of Sobolev spaces with exponent s > 3/2 measured in a weaker Sobolev norm with index r < s in periodic case. We remark that there is significant difference between the b-equation and µ-b equation because of the difference of the two operators µ − ∂x2 and 1 − ∂x2 . This paper is organized as follows: In Section 2, we recall the well-posedness results of [23,25] and some well-known results, and then state our main result. In Section 3, we prove the main result. In this paper, the symbols ., ≈ and & are used to denote inequality/equality up to a positive universal constant. For example, f (x) . g (x) means that f (x) ≤ cg (x) for some positive universal constant c. Since all spaces of functions are over S1 , the reference to S1 will be dropped if no ambiguity arises. [A, B] = AB − BA denotes the commutator of linear operators A and B.
µ(u) =
2. Some well-known results As µ(ux ) = 0 under spatial periodicity, we can re-write (1.2) as follows:
ut + uux = −∂x A−1
u(0, x) = u0 (x),
bµ(u)u +
3−b 2
u2x
,
t > 0, x ∈ S1 ,
(2.1)
x ∈ S1 ,
where A = µ − ∂x2 is an isomorphism between H s (S1 ) and H s−2 (S1 ) with the inverse v = A−1 u given by
v(x) =
x2 2
−
x 2
+
13
12
µ(u) + (x − 1/2)
1
0
y
u(s)dsdy −
0
x
u(s)dsdy + 0
1
0
y 0
s
u(r )drdsdy. 0
Since A−1 and ∂x commute, the following identities hold: A−1 ∂x u(x) = (x − 1/2)
1
u(x)dx − 0
A
∂x u(x) = −u(x) +
−1 2
x
u(y)dy + 0
1
0
x
u(y)dydx,
(2.2)
0
1
u(x)dx.
(2.3)
0
It is easy to show that µ(A−1 ∂x u(x)) = 0. We recall the following local well-posedness result: Proposition 2.1 ([25, Theorem 5.5]). Let u0 ∈ H s , s > 3/2. Then there exist a maximal existence time T = T (∥u0 ∥H s ) > 0 and a unique solution u to (2.1) such that u = u(·, u0 ) ∈ C ([0, T ); H s ) ∩ C 1 ([0, T ); H s−1 ). Moreover, the solution depends continuously on the initial data, i.e., the map u0 → u(·, u0 ) : H s → C ([0, T ); H s ) ∩ C 1 ([0, T ); H s−1 ) is continuous.
32
G. Lv, X. Wang / Nonlinear Analysis 102 (2014) 30–35
Remark 2.1. The maximal existence time T > 0 in Proposition 2.1 is independent of the Sobolev index s > 3/2; this can be proved using the Kato method [37], as in [38]. Next, we will give a result about the lower bound for the maximal existence time T with respect to the H s -norm of the initial data. Proposition 2.2 ([27, Lemma 2.1]). Let s > 3/2, and let u be the solution of (2.1) with initial data u0 described in Proposition 2.1. Then the maximal existence time T satisfies T ≥ T0 :=
1 2Cs ∥u0 ∥H s
,
(2.4)
where Cs is a constant depending only on s. Also, we have
∥u(t )∥H s ≤ 2∥u0 ∥H s ,
0 ≤ t ≤ T0 .
(2.5)
Now, we state our main result. Theorem 2.1. Assume s > 23 and 0 ≤ r < s. Then the solution map to (2.1) is Hölder continuous with exponent α = α(b, s, r ) as a map from B(0, h) with H r (R) norm to C ([0, T0 ], H r (R)), where T0 is defined as in Proposition 2.2. More precisely, for initial data u(0), w(0) in a ball B(0, h) := {u ∈ H s : ∥u∥H s ≤ h} of H s , the corresponding µ-b equation u(t ), w(t ) satisfy the inequality
∥u(t ) − w(t )∥C ([0,T0 ];H r ) ≤ c ∥u(0) − w(0)∥αH r ,
(2.6)
where α is given by
α=
1 s−r
if b = 3 and (s, r ) ∈ Ω1 , or (s, r ) ∈ Ω2 if b = 3 and (s, r ) ∈ Ω3 , or (s, r ) ∈ Ω4
(2.7)
and the regions Ω1 , Ω2 and Ω3 are defined by
Ω1 Ω2 Ω3 Ω4
= {(s, r ) : = {(s, r ) : = {(s, r ) : = {(s, r ) :
s > 3/2, 0 ≤ r ≤ s − 1}, s ≥ 2, 0 ≤ r ≤ s − 1}, s > 3/2, s − 1 < r < s}, s ≥ 2, s − 1 < r < s}.
Remark 2.2. It follows from the proof of Theorem 2.1 in Section 3 that α can be taken as 1 when (s, r ) ∈ Ω5 , where Ω5 = {(s, r ) : s > 3/2, 1 ≤ r ≤ s − 1}. It is easy to see that Ω5 ⊂ Ω2 , and thus we will not discuss it in detail. 3. Proof of Theorem 2.1 In this section, we shall use energy estimate and the property of the operator µ−∂x2 to prove Theorem 2.1. Let D = 1 −∂x . In order to prove Theorem 2.1, we need the following Lemmas. Lemma 3.1 ([35, Lemma 1]). If r + 1 ≥ 0, then
∥[Dr ∂x , f ]v∥L2 ≤ c ∥f ∥H s ∥v∥H r provided that s > 3/2 and r + 1 ≤ s. Proof of Theorem 2.1. Let u0 (x), w0 (x) ∈ B(0, h) and let u(x, t ) and w(x, t ) be the two solutions to (2.1) with initial data u0 (x) and w0 (x), respectively. Define v = u − w , then v satisfies that
1 3−b vt + ∂x [v(u + w)] = −∂x A−1 bµ(u)v + µ(v)w + vx (u + w)x , 2 2 v(0, x) = u0 (x) − w0 (x),
t > 0, x ∈ S1 ,
(3.1)
x ∈ S1 ,
where we have used the fact that µ(u) − µ(w) = µ(v). Applying Dr to both sides of (2.1), then multiplying both sides by Dr v and integrating, we get 1 d 2 dt
∥v(t )∥2H r = − −
1
Dr ∂x [v(u + w)] · Dr v dx − b
2 S 3−b 2
Dr ∂x A−1 [µ(u)v + µ(v)w ] · Dr v dx S
D ∂x A r
−1
vx (u + w)x · D v dx r
S
:= E1 + E2 + E3 .
(3.2)
G. Lv, X. Wang / Nonlinear Analysis 102 (2014) 30–35
33
We now estimate E1 , E2 and E3 . Estimate E1 . It is easy to see that
1 |E1 | = − Dr ∂x [v(u + w)] · Dr v dx 2 S 1 r r r r = [D ∂x , u + w]v · D v dx − (u + w)D ∂x v · D v dx 2 S S . [Dr ∂x , u + w]v · Dr v dx + (u + w)Dr ∂x v · Dr v dx . S
(3.3)
S
It follows from Lemma 3.1 that
[Dr ∂x , u + w]v · Dr v dx ≤ ∥[Dr ∂x , u + w]v∥L2 · ∥v∥H r S
≤ ∥u + w∥H s · ∥v∥2H r ,
(3.4)
where we used the fact that ∥Dr v∥L2 = ∥v∥H r . Direct calculation shows that
(u + w)Dr ∂x v · Dr v dx = ∂x (u + w) · (Dr v)2 dx S
S
≤ ∥∂x (u + w)∥L∞ ∥v∥2H r ≤ ∥u + w∥H s ∥v∥2H r , 1
where we used Sobolev embedding theorem H 2 + ↩→ L∞ . Submitting the above inequality and (3.4) into (3.3), we obtain
|E1 | ≤ ∥u + w∥H s ∥v∥2H r .
(3.5)
Estimate E2 . It is easy to show that
|E2 | = −b Dr ∂x A−1 [µ(u)v + µ(v)w] · Dr v dx S
−1
. ∥∂x A
[µ(u)v + µ(v)w ] ∥H r · ∥v(t )∥H r .
(3.6)
By (2.2) and (2.3), we have −1
∥∂x A u∥H r
1 x 1 x 1 = x− u(x)dx − u(y)dy + u(y)dydx r 2 0 0 0 0 H 1 1 x 1 . |u(x)|dx + ∥u(t )∥H r −1 + |u(y)|dydx. x − 2
Hr
0
0
0
Using the above inequality, we have
∥∂x A
−1
[µ(u)v + µ(v)w ] ∥H r
1 1 x 1 . |µ(u)| x − |v(x)|dx + ∥v(t )∥H r −1 + |v(y)|dydx 2 Hr 0 0 0 1 1 x 1 + |µ(v)| x − |w(x)|dx + ∥w(t )∥H r −1 + |w(y)|dydx 2
Hr
0
. (∥u∥H s + ∥w∥H s )∥v(t )∥H r ,
0
0
(3.7)
where we have used the following inequality
|µ(v)| = v(x, t )dx ≤ |v(x, t )|dx ≤ ∥v(t )∥H r S
S
provided that r ≥ 0. Substituting (3.7) into (3.6), we get
|E2 | . (∥u∥H s + ∥w∥H s )∥v(t )∥2H r .
(3.8)
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G. Lv, X. Wang / Nonlinear Analysis 102 (2014) 30–35
Estimate E3 . Similar to E2 and integrating by part, we have
3−b |E3 | = − Dr ∂x A−1 vx (u + w)x · Dr v dx 2 S
vx (u + w)x ∥H r · ∥v(t )∥H r . ∥v(t )∥L2 (∥u∥H 2 + ∥w∥H 2 ) · ∥v(t )∥H r −1
. ∥∂x A
. (∥u∥H s + ∥w∥H s ) · ∥v(t )∥2H r
(3.9)
provided that s ≥ 2. In the above inequality, we used
vx (x, t )ux (x, t )dx = v(x, t )uxx (x, t )dx ≤ ∥v(t )∥L2 ∥u∥H 2 . S
S
On the other hand, we also have
vx (x, t )ux (x, t )dx ≤ ∥v(t )∥H 1 ∥u∥H 1 . S
Hence we can prove the result in Remark 2.2 as the following step. Lipschitz continuous in Ω2 and Ω1 with b = 3. When b = 3, substituting (3.3) and (3.9) into (3.2), we have 1 d 2 dt
∥v(t )∥2H r . (∥u∥H s + ∥w∥H s )∥v(t )∥2H r .
It follows from Proposition 2.2 that ∥u(t )∥H s , ∥w(t )∥H s ≤ c (∥u0 ∥H s + ∥w0 ∥H s ) ≤ c since u0 , w0 ∈ B(0, h). Consequently, we obtain 1 d 2 dt
∥v(t )∥2H r . c ∥v(t )∥2H r ,
which implies that
∥v(t )∥H r ≤ ecT0 ∥v(0)∥H r .
(3.10)
Or equivalently
∥u(t ) − w(t )∥H r ≤ ecT0 ∥u(0) − w(0)∥H r ,
(3.11)
which is the desired Lipschitz continuity in Ω1 with b = 3. Similarly, when b ̸= 3, substituting (3.3), (3.6) and (3.9) into (3.2), we can obtain (3.10) and thus the Lipschitz continuity in Ω2 is proved. Hölder continuous in Ω4 and Ω3 with b = 3. Since s − 1 < r < s, by interpolating between H s−1 and H s norms, we get
∥v(t )∥H r ≤ ∥v(t )∥sH−s−r 1 ∥v(t )∥rH−s s+1 . Moreover, from Proposition 2.2, we have that
∥v(t )∥H s . ∥u0 ∥H s + ∥w0 ∥H s . h, and thus we have
∥v(t )∥H r . ∥v(t )∥sH−s−r 1 .
(3.12)
We see that (3.11) is valid for r = s − 1, s > 3/2 when b = 3 and s ≥ 2 when b ̸= 3. Therefore, applying (3.11) into (3.12), we obtain
∥v(t )∥H r . ∥v(0)∥sH−s−r 1 , which is the desired Hölder continuity. The proof of Theorem 2.1 is completed.
Acknowledgment The research of the work is supported in part by PRC Grants NSFC 11301146 and 11226168.
G. Lv, X. Wang / Nonlinear Analysis 102 (2014) 30–35
35
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Contents lists available at ScienceDirect
Nonlinear Analysis journal homepage: www.elsevier.com/locate/na
Hölder continuity on µ-b equation Guangying Lv a,b,∗ , Xiaohuan Wang a a
College of Mathematics and Information Science, Henan University, Kaifeng 475001, PR China
b
Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL 60616, USA
article
info
Article history: Received 30 November 2013 Accepted 5 February 2014 Communicated by Enzo Mitidieri MSC: 35G25 35B30 35L05
abstract This paper is concerned with the Hölder continuous on initial data for the µ-b equation on the circle. It is shown that the data to solution for the µ-b equation is Hölder continuous from bounded set of Sobolev spaces with exponent s > 3/2 measured in a weaker Sobolev norm with index r < s in periodic case. © 2014 Elsevier Ltd. All rights reserved.
Keywords: Hölder continuity µ-b equation Energy estimates
1. Introduction The b-equation [1–3], ut − uxxt + u(ux − uxxx ) + bux (u − uxx ) = 0,
(1.1)
which in fact comprises a family of equations parameterized by the real value b, models wave phenomena in shallow fluid flow, and has been the subject of considerable interest both to mathematicians and physicists. When b = 2 and 3, respectively, (1.1) reduces to the well-known Camassa–Holm (CH) [4] and Degasperis–Procesi (DP) [2] equations, which arise as models for shallow water waves of moderate amplitude [5,6] and have been extensively studied, particularly from the point of view of integrability and geodesic flow on infinite dimensional Lie groups [7–11]. In contrast to classical model equations for water waves, like KdV and BBM, novel features are captured by CH and DP. For example, the development of singularities in finite time in the form of breaking waves (meaning that the solution remains uniformly bounded but its slope becomes unbounded in finite time), see [12] and the reference therein. Of great interest is also the existence of traveling wave solutions that are peaked at their crest. These peakons are similar to the waves of greatest height that arise as solutions to the governing equations for water waves (see the discussion in [13–17]). The Cauchy problem of the b-equation, and in particular its well-posedness and blow-up behavior, has also been well-studied both on the real line and on the circle, see for example [12,18–21]; in particular, [22] looks at the non-uniform dependence on initial data for the b-equation. In [23], Khesin, Lenells and Misiołek studied a generalization of the Hunter–Saxton (HS) [24] equation, which is a high frequency limit of the CH equation, and introduced the µ-HS equation. The µ-HS equation, which is now more commonly referred to as the µ-CH equation, describes the propagation of weakly nonlinear orientation waves in a massive nematic
∗
Corresponding author at: College of Mathematics and Information Science, Henan University, Kaifeng 475001, PR China. Tel.: +86 15137877126. E-mail addresses: [email protected], [email protected], [email protected] (G. Lv), [email protected] (X. Wang).
http://dx.doi.org/10.1016/j.na.2014.02.003 0362-546X/© 2014 Elsevier Ltd. All rights reserved.
G. Lv, X. Wang / Nonlinear Analysis 102 (2014) 30–35
31
liquid crystal with external magnetic field and self-interaction. It was also shown in [23] that the µ-CH equation on the circle is formally integrable, admits a bi-Hamiltonian structure and an infinite hierarchy of conservation laws, and is an Euler equation for geodesic flow on the Lie group of circle diffeomorphisms. Subsequently, in addition to µ-CH, Lenells, Misiołek and Tiğlay [25] also introduced the µ-DP as well as µ-Burgers equations, and the µ-b equation (see also [26,27]). In this paper, we consider the Cauchy problem for the (spatially) periodic µ-b equation:
µ(ut ) − uxxt − uuxxx + bux (µ(u) − uxx ) = 0, u(0, x) = u0 (x),
t > 0, x ∈ S1 , x ∈ S1 ,
(1.2)
where b is a real constant, u = u(t , x) is a time-dependent function on the unit circle S1 = R/Z ≃ [0, 1), and u(t , x)dx denotes the mean value of u. We note that under spatial periodicity, µ(ut ) = 0 = µ(ux ). S1 In the case b = 2 and 3, respectively, the µ-b equation reduces to the µ-CH and µ-DP equations. In addition, if µ(u) = 0, they reduce to the HS and µ-Burgers equations, respectively. The local well-posedness of the µ-CH and µ-DP Cauchy problems have been studied in [23,25]. Recently, Fu et al. [28] described precise blow-up scenarios for µ-CH and µ-DP. Our attention in this paper is on the property on initial data. Himonas et al. [29,30] showed that solutions of the CH equation on the real line as well as on the circle do not depend uniformly on the initial data. They made use of the method of approximate solutions [31–34]. Lv et al. [27] studied the µ-b equation and obtained the non-uniform dependence on initial data. Himonas and Holmes [35] studied the Novikov equation and obtained the Hölder continuity of the solution map. The Hölder continuous of solution map of b-equation was established by Chen et al. [36]. The Hölder continuous of µ-b equation is an open problem and we will give a positive answer in this paper. Our work has been inspired by [36,35]. We show the solution map of (1.2) is Hölder continuous from bounded set of Sobolev spaces with exponent s > 3/2 measured in a weaker Sobolev norm with index r < s in periodic case. We remark that there is significant difference between the b-equation and µ-b equation because of the difference of the two operators µ − ∂x2 and 1 − ∂x2 . This paper is organized as follows: In Section 2, we recall the well-posedness results of [23,25] and some well-known results, and then state our main result. In Section 3, we prove the main result. In this paper, the symbols ., ≈ and & are used to denote inequality/equality up to a positive universal constant. For example, f (x) . g (x) means that f (x) ≤ cg (x) for some positive universal constant c. Since all spaces of functions are over S1 , the reference to S1 will be dropped if no ambiguity arises. [A, B] = AB − BA denotes the commutator of linear operators A and B.
µ(u) =
2. Some well-known results As µ(ux ) = 0 under spatial periodicity, we can re-write (1.2) as follows:
ut + uux = −∂x A−1
u(0, x) = u0 (x),
bµ(u)u +
3−b 2
u2x
,
t > 0, x ∈ S1 ,
(2.1)
x ∈ S1 ,
where A = µ − ∂x2 is an isomorphism between H s (S1 ) and H s−2 (S1 ) with the inverse v = A−1 u given by
v(x) =
x2 2
−
x 2
+
13
12
µ(u) + (x − 1/2)
1
0
y
u(s)dsdy −
0
x
u(s)dsdy + 0
1
0
y 0
s
u(r )drdsdy. 0
Since A−1 and ∂x commute, the following identities hold: A−1 ∂x u(x) = (x − 1/2)
1
u(x)dx − 0
A
∂x u(x) = −u(x) +
−1 2
x
u(y)dy + 0
1
0
x
u(y)dydx,
(2.2)
0
1
u(x)dx.
(2.3)
0
It is easy to show that µ(A−1 ∂x u(x)) = 0. We recall the following local well-posedness result: Proposition 2.1 ([25, Theorem 5.5]). Let u0 ∈ H s , s > 3/2. Then there exist a maximal existence time T = T (∥u0 ∥H s ) > 0 and a unique solution u to (2.1) such that u = u(·, u0 ) ∈ C ([0, T ); H s ) ∩ C 1 ([0, T ); H s−1 ). Moreover, the solution depends continuously on the initial data, i.e., the map u0 → u(·, u0 ) : H s → C ([0, T ); H s ) ∩ C 1 ([0, T ); H s−1 ) is continuous.
32
G. Lv, X. Wang / Nonlinear Analysis 102 (2014) 30–35
Remark 2.1. The maximal existence time T > 0 in Proposition 2.1 is independent of the Sobolev index s > 3/2; this can be proved using the Kato method [37], as in [38]. Next, we will give a result about the lower bound for the maximal existence time T with respect to the H s -norm of the initial data. Proposition 2.2 ([27, Lemma 2.1]). Let s > 3/2, and let u be the solution of (2.1) with initial data u0 described in Proposition 2.1. Then the maximal existence time T satisfies T ≥ T0 :=
1 2Cs ∥u0 ∥H s
,
(2.4)
where Cs is a constant depending only on s. Also, we have
∥u(t )∥H s ≤ 2∥u0 ∥H s ,
0 ≤ t ≤ T0 .
(2.5)
Now, we state our main result. Theorem 2.1. Assume s > 23 and 0 ≤ r < s. Then the solution map to (2.1) is Hölder continuous with exponent α = α(b, s, r ) as a map from B(0, h) with H r (R) norm to C ([0, T0 ], H r (R)), where T0 is defined as in Proposition 2.2. More precisely, for initial data u(0), w(0) in a ball B(0, h) := {u ∈ H s : ∥u∥H s ≤ h} of H s , the corresponding µ-b equation u(t ), w(t ) satisfy the inequality
∥u(t ) − w(t )∥C ([0,T0 ];H r ) ≤ c ∥u(0) − w(0)∥αH r ,
(2.6)
where α is given by
α=
1 s−r
if b = 3 and (s, r ) ∈ Ω1 , or (s, r ) ∈ Ω2 if b = 3 and (s, r ) ∈ Ω3 , or (s, r ) ∈ Ω4
(2.7)
and the regions Ω1 , Ω2 and Ω3 are defined by
Ω1 Ω2 Ω3 Ω4
= {(s, r ) : = {(s, r ) : = {(s, r ) : = {(s, r ) :
s > 3/2, 0 ≤ r ≤ s − 1}, s ≥ 2, 0 ≤ r ≤ s − 1}, s > 3/2, s − 1 < r < s}, s ≥ 2, s − 1 < r < s}.
Remark 2.2. It follows from the proof of Theorem 2.1 in Section 3 that α can be taken as 1 when (s, r ) ∈ Ω5 , where Ω5 = {(s, r ) : s > 3/2, 1 ≤ r ≤ s − 1}. It is easy to see that Ω5 ⊂ Ω2 , and thus we will not discuss it in detail. 3. Proof of Theorem 2.1 In this section, we shall use energy estimate and the property of the operator µ−∂x2 to prove Theorem 2.1. Let D = 1 −∂x . In order to prove Theorem 2.1, we need the following Lemmas. Lemma 3.1 ([35, Lemma 1]). If r + 1 ≥ 0, then
∥[Dr ∂x , f ]v∥L2 ≤ c ∥f ∥H s ∥v∥H r provided that s > 3/2 and r + 1 ≤ s. Proof of Theorem 2.1. Let u0 (x), w0 (x) ∈ B(0, h) and let u(x, t ) and w(x, t ) be the two solutions to (2.1) with initial data u0 (x) and w0 (x), respectively. Define v = u − w , then v satisfies that
1 3−b vt + ∂x [v(u + w)] = −∂x A−1 bµ(u)v + µ(v)w + vx (u + w)x , 2 2 v(0, x) = u0 (x) − w0 (x),
t > 0, x ∈ S1 ,
(3.1)
x ∈ S1 ,
where we have used the fact that µ(u) − µ(w) = µ(v). Applying Dr to both sides of (2.1), then multiplying both sides by Dr v and integrating, we get 1 d 2 dt
∥v(t )∥2H r = − −
1
Dr ∂x [v(u + w)] · Dr v dx − b
2 S 3−b 2
Dr ∂x A−1 [µ(u)v + µ(v)w ] · Dr v dx S
D ∂x A r
−1
vx (u + w)x · D v dx r
S
:= E1 + E2 + E3 .
(3.2)
G. Lv, X. Wang / Nonlinear Analysis 102 (2014) 30–35
33
We now estimate E1 , E2 and E3 . Estimate E1 . It is easy to see that
1 |E1 | = − Dr ∂x [v(u + w)] · Dr v dx 2 S 1 r r r r = [D ∂x , u + w]v · D v dx − (u + w)D ∂x v · D v dx 2 S S . [Dr ∂x , u + w]v · Dr v dx + (u + w)Dr ∂x v · Dr v dx . S
(3.3)
S
It follows from Lemma 3.1 that
[Dr ∂x , u + w]v · Dr v dx ≤ ∥[Dr ∂x , u + w]v∥L2 · ∥v∥H r S
≤ ∥u + w∥H s · ∥v∥2H r ,
(3.4)
where we used the fact that ∥Dr v∥L2 = ∥v∥H r . Direct calculation shows that
(u + w)Dr ∂x v · Dr v dx = ∂x (u + w) · (Dr v)2 dx S
S
≤ ∥∂x (u + w)∥L∞ ∥v∥2H r ≤ ∥u + w∥H s ∥v∥2H r , 1
where we used Sobolev embedding theorem H 2 + ↩→ L∞ . Submitting the above inequality and (3.4) into (3.3), we obtain
|E1 | ≤ ∥u + w∥H s ∥v∥2H r .
(3.5)
Estimate E2 . It is easy to show that
|E2 | = −b Dr ∂x A−1 [µ(u)v + µ(v)w] · Dr v dx S
−1
. ∥∂x A
[µ(u)v + µ(v)w ] ∥H r · ∥v(t )∥H r .
(3.6)
By (2.2) and (2.3), we have −1
∥∂x A u∥H r
1 x 1 x 1 = x− u(x)dx − u(y)dy + u(y)dydx r 2 0 0 0 0 H 1 1 x 1 . |u(x)|dx + ∥u(t )∥H r −1 + |u(y)|dydx. x − 2
Hr
0
0
0
Using the above inequality, we have
∥∂x A
−1
[µ(u)v + µ(v)w ] ∥H r
1 1 x 1 . |µ(u)| x − |v(x)|dx + ∥v(t )∥H r −1 + |v(y)|dydx 2 Hr 0 0 0 1 1 x 1 + |µ(v)| x − |w(x)|dx + ∥w(t )∥H r −1 + |w(y)|dydx 2
Hr
0
. (∥u∥H s + ∥w∥H s )∥v(t )∥H r ,
0
0
(3.7)
where we have used the following inequality
|µ(v)| = v(x, t )dx ≤ |v(x, t )|dx ≤ ∥v(t )∥H r S
S
provided that r ≥ 0. Substituting (3.7) into (3.6), we get
|E2 | . (∥u∥H s + ∥w∥H s )∥v(t )∥2H r .
(3.8)
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G. Lv, X. Wang / Nonlinear Analysis 102 (2014) 30–35
Estimate E3 . Similar to E2 and integrating by part, we have
3−b |E3 | = − Dr ∂x A−1 vx (u + w)x · Dr v dx 2 S
vx (u + w)x ∥H r · ∥v(t )∥H r . ∥v(t )∥L2 (∥u∥H 2 + ∥w∥H 2 ) · ∥v(t )∥H r −1
. ∥∂x A
. (∥u∥H s + ∥w∥H s ) · ∥v(t )∥2H r
(3.9)
provided that s ≥ 2. In the above inequality, we used
vx (x, t )ux (x, t )dx = v(x, t )uxx (x, t )dx ≤ ∥v(t )∥L2 ∥u∥H 2 . S
S
On the other hand, we also have
vx (x, t )ux (x, t )dx ≤ ∥v(t )∥H 1 ∥u∥H 1 . S
Hence we can prove the result in Remark 2.2 as the following step. Lipschitz continuous in Ω2 and Ω1 with b = 3. When b = 3, substituting (3.3) and (3.9) into (3.2), we have 1 d 2 dt
∥v(t )∥2H r . (∥u∥H s + ∥w∥H s )∥v(t )∥2H r .
It follows from Proposition 2.2 that ∥u(t )∥H s , ∥w(t )∥H s ≤ c (∥u0 ∥H s + ∥w0 ∥H s ) ≤ c since u0 , w0 ∈ B(0, h). Consequently, we obtain 1 d 2 dt
∥v(t )∥2H r . c ∥v(t )∥2H r ,
which implies that
∥v(t )∥H r ≤ ecT0 ∥v(0)∥H r .
(3.10)
Or equivalently
∥u(t ) − w(t )∥H r ≤ ecT0 ∥u(0) − w(0)∥H r ,
(3.11)
which is the desired Lipschitz continuity in Ω1 with b = 3. Similarly, when b ̸= 3, substituting (3.3), (3.6) and (3.9) into (3.2), we can obtain (3.10) and thus the Lipschitz continuity in Ω2 is proved. Hölder continuous in Ω4 and Ω3 with b = 3. Since s − 1 < r < s, by interpolating between H s−1 and H s norms, we get
∥v(t )∥H r ≤ ∥v(t )∥sH−s−r 1 ∥v(t )∥rH−s s+1 . Moreover, from Proposition 2.2, we have that
∥v(t )∥H s . ∥u0 ∥H s + ∥w0 ∥H s . h, and thus we have
∥v(t )∥H r . ∥v(t )∥sH−s−r 1 .
(3.12)
We see that (3.11) is valid for r = s − 1, s > 3/2 when b = 3 and s ≥ 2 when b ̸= 3. Therefore, applying (3.11) into (3.12), we obtain
∥v(t )∥H r . ∥v(0)∥sH−s−r 1 , which is the desired Hölder continuity. The proof of Theorem 2.1 is completed.
Acknowledgment The research of the work is supported in part by PRC Grants NSFC 11301146 and 11226168.
G. Lv, X. Wang / Nonlinear Analysis 102 (2014) 30–35
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