on the solutions of the dullin–gottwald–holm equation in besov spaces

Nonlinear Analysis: Real World Applications 13 (2012) 2580–2592

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Nonlinear Analysis: Real World Applications

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On the solutions of the Dullin–Gottwald–Holm equation in Besov spacesKai Yan ∗, Zhaoyang YinDepartment of Mathematics, Sun Yat-sen University, 510275 Guangzhou, China

a r t i c l e i n f o

Article history:Received 25 July 2011Accepted 8 March 2012

Keywords:The Dullin–Gottwald–Holm equationLocal well-posednessBesov spacesBlow-upGlobal solutionsSharp estimate from belowLower semicontinuity

a b s t r a c t

This paper is concernedwith the Cauchy problem for the Dullin–Gottwald–Holm equation.First, the local well-posedness for this system in Besov spaces is established. Second,the blow-up criterion for solutions to the equation is derived. Then, the existence anduniqueness of global solutions to the equation are investigated. Finally, the sharp estimatefrom below and lower semicontinuity for the existence time of solutions to this equationare presented.

© 2012 Elsevier Ltd. All rights reserved.

1. Introduction

In this paper, we consider the Cauchy problem of the following Dullin–Gottwald–Holm (DGH) equation:ut − α2utxx + c0ux + 3uux + γ uxxx = α2(2uxuxx + uuxxx), t > 0, x ∈ R,u(0, x) = u0(x), x ∈ R,

(1.1)

where the constants α2 and γ

c0are squares of length scales, and c0 > 0 is the linear wave speed for undisturbed water at

rest at spatial infinity, u(t, x) stands for the fluid velocity.Eq. (1.1) was derived by Dullin et al. [1] as a model to describe unidirectional propagation of surface waves on a shallow

layer of water which is at rest at infinity. This equation is obtained by asymptotic analysis and a near-identity normal formtransform fromwater wave theory. It combines the linear dispersion of the well-known Korteweg–de Vries (KdV) equationwith the nonlinear and nonlocal dispersion of the Camassa–Holm (CH) equation. Moreover, it is completely integrable andits traveling wave solutions include both the KdV solitons and the CH peakons as limiting cases.

Forα = 0, Eq. (1.1) becomes the KdV equationwhich describes the unidirectional propagation ofwaves at the free surfaceof shallow water under the influence of gravity. u(t, x) represents the wave height above a flat bottom, x is proportional todistance in the direction of propagation and t is proportional to elapsed time. It is integrable and its solitary waves aresolitons [2], traveling wave is linearly unstable [3]. The Cauchy problem of the KdV equation has been studied extensively,e.g. [4–6].

For α = 1 and γ = 0, Eq. (1.1) becomes the CH equation, modeling the unidirectional propagation of shallow waterwaves over a flat bottom. Here u(t, x) stands for the fluid velocity at time t in the spatial x direction [1,7–9]. It is also amodel for the propagation axially symmetric waves in hyper-elastic rods [10]. It has a bi-Hamiltonian structure [11,12] andis completely integrable [7,13]. Its solitary waves are peaked [14], capturing thus the shape of solitary wave solutions tothe governing equations for water waves [15,16]. The orbital stability of the peaked solutions is proved in [17]. The explicit

∗ Corresponding author.E-mail addresses: [email protected] (K. Yan), [email protected] (Z. Yin).

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K. Yan, Z. Yin / Nonlinear Analysis: Real World Applications 13 (2012) 2580–2592 2581

interaction of the peaked solutions is given in [18]. The Cauchy problem and initial-boundary value problem for the CHequation have been studied extensively. It has been shown that this equation is locally well-posed [19–28]. Furthermore, ithas not only global strong solutions [19,20,29] but also finite time blow-up solutions [19–21,25,26,29,30]. Apart from these,it has global weak solutions in H1(R) [31–36]. The advantage of the CH equation in comparison with the KdV equation liesin the fact that the CH equation has peaked solitons and models wave breaking that the wave remains bounded while itsslop becomes unbounded in finite time [14,30,37].

Recently, a large amount of literaturewas devoted to the DGH equation [38–42]. Some new global existence and blow-upresults for strong solutions to the DGH equation on the line were presented in [27,43,44] and on the circle in [45,46]. On theother hand, it has not only global weak solutions [47] and global conservative solutions [48] but also analytic solutions [49].However, the goal of this paper is to establish the local well-posedness for Cauchy problem of Eq. (1.1) (concerning withα > 0) in Besov spaces, derive the blow-up criterion of solutions to the equation, and give the unique global solution undersome certain assumption for the initial datum and some properties on the existence time. Most of our results can be carriedout to the periodic case and to homogeneous Besov spaces. It is noticing that an interesting application of the technique inthe Besov spaces for the local well-posedness is on the two-component Camassa–Holm system, which can be seen in [50].

Given α > 0. Using the Green function p(x) , 12α e

−|xα |, x ∈ R and the identity (1 − α2∂2

x )−1f = p ∗ f for all f ∈ L2, we

can rewrite Eq. (1.1) as follows:∂tu +

u −

γ

α2

∂xu = P(D)

u2

+α2

2u2x +

c0 +

γ

α2

u

, t > 0, x ∈ R,

u(0, x) = u0(x), x ∈ R(1.2)

with the operator P(D) , −∂x(1 − α2∂2x )

−1.Our paper is organized as follows. In Section 2, we will recall some facts on the Littlewood–Paley decomposition, Besov

spaces and their some useful properties, and the transport equation theory. In Section 3, we establish the local well-posedness of Eq. (1.2) in Besov spaces. In Section 4, we derive the blow-up criterion of solutions to Eq. (1.2). In Section 5,we investigate the existence and uniqueness of global solutions to Eq. (1.2). Section 6 is devoted to some properties onthe existence time, including the sharp estimate from below and lower semicontinuity for the existence time of solutionsto Eq. (1.2).

2. Preliminaries

In this section, wewill recall some facts on the Littlewood–Paley decomposition, the nonhomogeneous Besov spaces andtheir some useful properties, and the transport equation theory, which will be used in the sequel.

Proposition 2.1 ([51], Littlewood–Paley Decomposition). There exists a couple of smooth functions (χ, ϕ) valued in [0, 1], suchthat χ is supported in the ball B , {ξ ∈ Rn

: |ξ | ≤43 }, and ϕ is supported in the ring C , {ξ ∈ Rn

:34 ≤ |ξ | ≤

83 }. Moreover,

∀ξ ∈ Rn, χ(ξ) +

q∈N

ϕ(2−qξ) = 1,

and

supp ϕ(2−q·) ∩ supp ϕ(2−q′

·) = ∅, if |q − q′| ≥ 2,

supp χ(·) ∩ supp ϕ(2−q·) = ∅, if q ≥ 1.

Then for all u ∈ S′, we can define the nonhomogeneous dyadic blocks as follows. Let

∆qu , 0, if q ≤ −2,

∆−1u , χ(D)u = F −1χF u,∆qu , ϕ(2−qD)u = F −1ϕ(2−qξ)F u, if q ≥ 0.

Hence,

u =

q∈Z

∆qu in S′(Rn),

where the right-hand side is called the nonhomogeneous Littlewood–Paley decomposition of u.

Remark 2.1. (1) The low frequency cut-off Sq is defined by

Squ ,

q−1p=−1

∆pu = χ(2−qD)u = F −1χ(2−qξ)F u, ∀q ∈ N.

2582 K. Yan, Z. Yin / Nonlinear Analysis: Real World Applications 13 (2012) 2580–2592

(2) The Littlewood–Paley decomposition is quasi-orthogonal in L2 in the following sense:

∆p∆qu ≡ 0, if |p − q| ≥ 2,∆q(Sp−1u∆pv) ≡ 0, if |p − q| ≥ 5,∆q(∆p−1u∆pv) ≡ 0, if p ≤ q − 1,

for all u, v ∈ S′(Rn).(3) Thanks to Young’s inequality, we get

∥∆qu∥Lp , ∥Squ∥Lp ≤ C∥u∥Lp , ∀1 ≤ p ≤ ∞,

where C is a positive constant independent of q.

Definition 2.1 ([51], Besov Spaces). Let s ∈ R, 1 ≤ p, r ≤ ∞. The nonhomogeneous Besov space Bsp,r(R

n) (Bsp,r for short) is

defined by

Bsp,r(R

n) , {f ∈ S′(Rn) : ∥f ∥Bsp,r < ∞},

where

∥f ∥Bsp,r , ∥2qs∆qf ∥lr (Lp) = ∥(2qs∥∆qf ∥Lp)q≥−1∥lr .

If s = ∞, B∞p,r ,

s∈R Bs

p,r .

Definition 2.2. Let T > 0, s ∈ R and 1 ≤ p ≤ ∞. Set

Esp,r(T ) , C([0, T ]; Bs

p,r) ∩ C1([0, T ]; Bs−1p,r ), if r < ∞,

Esp,∞(T ) , L∞([0, T ]; Bs

p,∞) ∩ Lip ([0, T ]; Bs−1p,∞)

and

Esp,r ,

T>0

Esp,r(T ).

Remark 2.2. By Definition 2.1 and Remark 2.1(3), we can deduce that

∥∆qu∥Bsp,r , ∥Squ∥Bsp,r ≤ C∥u∥Bsp,r ,

where C is a positive constant independent of q.

In the following proposition, we list some important properties of Besov spaces.

Proposition 2.2 ([21,22,51]). Suppose that s ∈ R, 1 ≤ p, r, pi, ri ≤ ∞, i = 1, 2. We have(1) Topological properties: Bs

p,r is a Banach space which is continuously embedded in S′.(2) Density: C∞

c is dense in Bsp,r ⇐⇒ 1 ≤ p, r < ∞.

(3) Embedding: Bsp1,r1 ↩→ B

s−n( 1p1

−1p2

)

p2,r2 , if p1 ≤ p2 and r1 ≤ r2,

Bs2p,r2 ↩→ Bs1

p,r1 locally compact, if s1 < s2.

(4) Algebraic properties: ∀s > 0, Bsp,r ∩ L∞ is an algebra. Moreover, Bs

p,r is an algebra, provided that s > np or s ≥

np and r = 1.

(5) 1-D Morse-type estimates:(i) For any s > 0, we have

∥fg∥Bsp,r (R) ≤ C(∥f ∥Bsp,r (R)∥g∥L∞(R) + ∥g∥Bsp,r (R)∥f ∥L∞(R)).

(ii) ∀s1 ≤1p < s2 (s2 ≥

1p if r = 1) and s1 + s2 > 0, we have

∥fg∥Bs1p,r (R)

≤ C∥f ∥Bs1p,r (R)

∥g∥Bs2p,r (R)

, (2.1)

where C is a positive constant independent of f and g.(6) Complex interpolation:

∥f ∥Bθs1+(1−θ)s2p,r

≤ ∥f ∥θ

Bs1p,r

∥f ∥1−θ

Bs2p,r

, ∀u ∈ Bs1p,r ∩ Bs1

p,r , ∀θ ∈ [0, 1]. (2.2)

(7) A logarithmic interpolation inequality:

∥f ∥Bsp,1≤ C

1 + ε

ε∥f ∥Bsp,∞

1 + ln

∥f ∥Bs+εp,∞

∥f ∥Bsp,∞

, ∀ε > 0.


(8) Fatou lemma: if (un)n∈N is bounded in Bsp,r and un → u in S′, then u ∈ Bs

p,r and

∥u∥Bsp,r ≤ lim infn→∞

∥un∥Bsp,r .

(9) Let m ∈ R and f be a Sm-multiplier (i.e., f : Rn→ R is smooth and satisfies that ∀α ∈ Nn, ∃ a constant Cα, s.t. |∂α f (ξ)| ≤

Cα(1 + |ξ |)m−|α| for all ξ ∈ Rn). Then the operator f (D) is continuous from Bsp,r to Bs−m

p,r .

(10) The paraproduct is continuous from B−

1p

p,1 (R) × (B1pp,∞(R) ∩ L∞(R)) to B

−1p

p,∞(R), i.e.,

∥fg∥B−

1p

p,∞(R)

≤ C∥f ∥B−

1p

p,1 (R)

∥g∥B1pp,∞(R)∩L∞(R)

.

Nowwe state some useful results in the transport equation theory, which are crucial to the proofs of our main theoremslater.

Lemma 2.1 ([21,51], A Priori Estimates in Besov Spaces). Let 1 ≤ p, r ≤ ∞ and s > −min( 1p , 1 −

1p ). Assume that f0 ∈ Bs

p,r ,

F ∈ L1(0, T ; Bsp,r), and ∂xv belongs to L1(0, T ; Bs−1

p,r ) if s > 1 +1p or to L1(0, T ; B

1pp,r ∩ L∞) otherwise. If f ∈ L∞(0, T ;

Bsp,r)

C([0, T ]; S′) solves the following 1-D linear transport equation:

(T )

∂t f + v∂xf = F ,f |t=0 = f0.

Then there exists a constant C depending only on s, p and r, and such that the following statements hold:

(1) If r = 1 or s = 1 +1p ,

∥f (t)∥Bsp,r ≤ ∥f0∥Bsp,r +

t

0∥F(τ )∥Bsp,rdτ + C

t

0V ′(τ )∥f (τ )∥Bsp,rdτ

or hence,

∥f (t)∥Bsp,r ≤ eCV (t)(∥f0∥Bsp,r +

t

0e−CV (τ )

∥F(τ )∥Bsp,rdτ)

with V (t) = t0 ∥∂xv(τ)∥

B1pp,r∩L∞

dτ if s < 1 +1p and V (t) =

t0 ∥∂xv(τ)∥Bs−1

p,rdτ else.

(2) If s ≤ 1 +1p , and ∂xf0 ∈ L∞, ∂xf ∈ L∞((0, T ] × R) and ∂xF ∈ L1(0, T ; L∞), then

∥f (t)∥Bsp,r + ∥∂xf (t)∥L∞ ≤ eCV (t)(∥f0∥Bsp,r + ∥∂xf0∥L∞ +

t

0e−CV (τ )(∥F(τ )∥Bsp,r + ∥∂xF(τ )∥L∞) dτ),

with V (t) = t0 ∥∂xv(τ)∥

B1pp,r∩L∞

dτ .

(3) If v = C1f + C2, then for all s > 0, (1) holds true with V (t) = t0 ∥∂xf (τ )∥L∞dτ .

(4) If r < ∞, then f ∈ C([0, T ]; Bsp,r). If r = ∞, then f ∈ C([0, T ]; Bs′

p,1) for all s′ < s.

Remark 2.3. (3) in Lemma2.1 has been proven in the case v = f in [51], but it also holds true in the general case v = C1f +C2following the similar argument.

Lemma 2.2 ([51], Existence and Uniqueness). Let p, r, s, f0 and F be as in the statement of Lemma 2.1. Assume that v ∈

Lρ(0, T ; B−M∞,∞) for some ρ > 1 and M > 0, and ∂xv ∈ L1(0, T ; Bs−1

p,r ) if s > 1 +1p or s = 1 +

1p and r = 1, and

∂xv ∈ L1(0, T ; B1pp,∞ ∩ L∞) if s < 1 +

1p . Then (T ) has a unique solution f ∈ L∞(0, T ; Bs

p,r)

(

s′<s C([0, T ]; Bs′p,1)) and

the inequalities of Lemma 2.1 can hold true. Moreover, if r < ∞, then f ∈ C([0, T ]; Bsp,r).

3. Local well-posedness

In this section, we will establish the local well-posedness of Eq. (1.2) in Besov spaces.Uniqueness and continuity with respect to the initial data in some sense can be obtained by the following a priori

estimates.


Lemma 3.1. Let 1 ≤ p, r ≤ ∞ and s > max(1 +1p ,

32 ). Suppose that we are given u, v ∈ L∞(0, T ; Bs

p,r) ∩ C([0, T ]; S′) twosolutions of Eq. (1.2) with the initial data u0, v0 ∈ Bs

p,r . Then for all t ∈ [0, T ], we have

(1) if s > max(1 +1p ,

32 ) but s = 2 +

1p , then

∥u(t) − v(t)∥Bs−1p,r

≤ ∥u0 − v0∥Bs−1p,r

eC t0 (∥u(τ )∥Bsp,r

+∥v(τ)∥Bsp,r+1)dτ

; (3.1)

(2) if s = 2 +1p , then

∥u(t) − v(t)∥Bs−1p,r

≤ C∥u0 − v0∥θ

Bs−1p,r

eθC t0 (∥u(τ )∥Bsp,r


(∥u(t)∥Bsp,r + ∥v(t)∥Bsp,r )1−θ ,

where θ ∈ (0, 1), C = C(s, p, r, c0, α, γ ).

Proof. Set w , v − u. It is obvious that w ∈ L∞(0, T ; Bsp,r) ∩ C([0, T ]; S′) solves the following Cauchy problem of the

transport equation:∂tw +

u −

γ

α2

∂xw = R(t, x),

w|t=0 = w0 , v0 − u0,(3.2)

where R(t, x) , −w∂xv + P(D)((u + v + c0 +γ

α2 )w +α2

2 ∂x(u + v)∂xw).

Claim. For all s > max(1 +1p ,

32 ) and t ∈ [0, T ], we have

∥R(t)∥Bs−1p,r

≤ C(∥u(t)∥Bsp,r + ∥v(t)∥Bsp,r + 1)∥w(t)∥Bs−1p,r

. (3.3)

where C = C(s, p, r, c0, α, γ ) is a positive constant.Indeed, for s > 1 +

1p , B

s−1p,r is an algebra, by Proposition 2.2 (4), we have

∥w∂xv∥Bs−1p,r

≤ C∥w∥Bs−1p,r

∥∂xv∥Bs−1p,r

≤ C∥v∥Bsp,r ∥w∥Bs−1p,r

.

Note that P(D) ∈ Op(S−1). According to Proposition 2.2 (9) and (2.1), we obtain

∥P(D)

u + v + c0 +

γ

α2

w +

α2

2∂x(u + v)∂xw

∥Bs−1

p,r≤ C

u + v + c0 +γ

α2

w +

α2

2∂x(u + v)∂xw

Bs−2p,r

≤ C

u + v + c0 +γ

α2

w

Bs−1p,r

+

α2

2∂x(u + v)∂xw

Bs−2p,r

≤ C(∥u(t)∥Bsp,r + ∥v(t)∥Bsp,r + 1)∥w(t)∥Bs−1

p,r,

if max(1 +1p ,

32 ) < s ≤ 2 +

1p .

Otherwise, the above inequality can also hold true in view of the fact Bs−2p,r is an algebra as s > 2 +

1p . This proves our

Claim.Applying Lemma 2.1 (1) and the fact that ∥∂xw(t)∥

B1pp,r∩L∞

, ∥∂xw(t)∥Bs−2p,r

≤ C∥w(t)∥Bsp,r , if w ∈ Bsp,r with s > max(1 +

1p ,

32 ), one can obtain, for case (1),

∥w(t)∥Bs−1p,r

≤ ∥w0∥Bs−1p,r

+

t

0∥R(τ )∥Bs−1

p,rdτ + C

t

0∥u(τ )∥Bsp,r ∥w(τ)∥Bs−1

p,rdτ ,

which together with the Claim yield

∥w(t)∥Bs−1p,r

≤ ∥w0∥Bs−1p,r

+ C t

0(∥u(τ )∥Bsp,r + ∥v(τ)∥Bsp,r + 1)∥w(τ)∥Bs−1

p,rdτ .

Taking advantage of Gronwall’s inequality, we get (3.1).For the critical case (2) s = 2 +

1p , we here use the interpolation method to deal with it. Indeed, if we choose

s1 ∈ (max(1 +1p ,

32 ) − 1, s − 1), s2 ∈ (s − 1, s) and θ =

s2−(s−1)s2−s1

∈ (0, 1), then s − 1 = θs1 + (1 − θ)s2. According


to (2.2) and the consequence of case (1), we have

∥w(t)∥Bs−1p,r

≤ ∥w(t)∥θ

Bs1p,r

∥w(t)∥1−θ

Bs2p,r

≤ ∥w0∥θ

Bs1p,reθC t0 (∥u(τ )∥

Bs1+1p,r

+∥v(τ)∥Bs1+1p,r

+1) dτ(∥u(t)∥B

s2p,r

+ ∥v(t)∥Bs2p,r

)1−θ

≤ C∥w0∥θ

Bs−1p,r

eθC t0 (∥u(τ )∥Bsp,r


(∥u(t)∥Bsp,r + ∥v(t)∥Bsp,r )1−θ .

Hence, we get the desired result. �

We next construct the approximation solutions to Eq. (1.2) as follows.

Lemma 3.2. Let p, r and s be as in the statement of Lemma 3.1. Assume that u0 ∈ Bsp,r and u0

≡ 0. Then

(1) there exists a sequence of smooth functions (un)n∈N belonging to C(R+; B∞

p,r) and solving the following linear transportequations by induction:

(Tn)

∂tun+1+

un

−γ

α2

∂xun+1

= P(D)

(un)2 +

α2

2(∂xun)2 +

c0 +

γ

α2

un

,

un+1|t=0 , un+1

0 (x) = Sn+1u0.

(2) there exists T > 0 such that the solutions (un)n∈N is uniformly bounded in Esp,r(T ) and a Cauchy sequence in C([0, T ]; Bs−1

p,r ),whence it converges to some limit u ∈ C([0, T ]; Bs−1

p,r ).

Proof. Since all the data Sn+1u0 ∈ B∞p,r , it then follows from Lemma 2.2 and by induction with respect to the index n that (1)

holds.To prove (2), applying Remark 2.2 and simulating the proof of Lemma 3.1 (1), we obtain that for s > max(1 +

1p ,

32 ) and

s = 2 +1p ,

e−CUn(t)∥un+1(t)∥Bsp,r ≤ C∥u0∥Bsp,r +

C2

t

0e−CUn(τ )(∥un(τ )∥Bsp,r + 1)∥un(τ )∥Bsp,rdτ , (3.4)

with Un(t) , t0 ∥un(τ )∥Bsp,rdτ .

Choose 0 < T < min( 14C2∥u0∥Bsp,r

, 12C2 ) and suppose that

∥un(t)∥Bsp,r ≤2C∥u0∥Bsp,r

1 − 4C2∥u0∥Bsp,r t, ∀t ∈ [0, T ]. (3.5)

Noting that eC(Un(t)−Un(τ ))≤

1−4C2∥u0∥Bsp,r

τ

1−4C2∥u0∥Bsp,rtand substituting (3.6) into (3.5) yield, for all t ∈ [0, T ],

∥un+1(t)∥Bsp,r ≤C∥u0∥Bsp,r

1 − 4C2∥u0∥Bsp,r t+

C1 − 4C2∥u0∥Bsp,r t

t

0

2C2∥u0∥

2Bsp,r

(1 − 4C2∥u0∥Bsp,r τ)32dτ

+

t

0

C∥u0∥Bsp,r

(1 − 4C2∥u0∥Bsp,r τ)12dτ

≤

2C∥u0∥Bsp,r

1 − 4C2∥u0∥Bsp,r t,

which implies that (un)n∈N is uniformly bounded in C([0, T ]; Bsp,r). By using the equation (Tn) and the similar proof of (3.3),

one can easily prove that (∂tun+1)n∈N is uniformly bounded in C([0, T ]; Bs−1p,r ). Hence, (un)n∈N is uniformly bounded in Es

p,r(T ).Now it suffices to show that (un)n∈N is a Cauchy sequence in C([0, T ]; Bs−1

p,r ). Indeed, For allm, n ∈ N, from (Tn), we have

∂t(un+m+1− un+1) +

un+m

−γ

α2

∂x(un+m+1

− un+1)

= (un− un+m) ∂xun+1

+ P(D)

(un+m

− un)un+m

+ un+ c0 +

γ

α2

+

α2

2(∂xun+m

− ∂xun)(∂xun+m+ ∂xun)

.


Similar to the proof of Lemma 3.1(1), for s > max(1 +1p ,

32 ) and s = 2 +

1p , we can obtain that

amn+1(t) ≤ CeCUn+m(t)

amn+1(0) +

t

0e−CUn+m(τ )amn (τ )bmn (τ ) dτ

,

where amn (t) , ∥(un+m−un)(t)∥Bs−1

p,r,Un+m(t) ,

t0 ∥un+m(τ )∥Bsp,r dτ , and bmn (t) , ∥un(t)∥Bsp,r +∥un+1(t)∥Bsp,r +∥un+m(t)∥Bsp,r

+ 1.Thanks to Remark 2.1(3), we have

amn+1(0) =

n+mq=n+1

∆qu0

Bs−1p,r

=

k≥−1

2k(s−1)r

∆k

n+m

q=n+1

∆qu0

r

Lp

1r

≤ C

n+m+1k=n

2−kr2ksr∥∆ku0∥

rLp

1r

≤ C2−n∥u0∥Bsp,r .

Then, according to the fact that (un)n∈N is uniformly bounded in Esp,r(T ), we can find a positive constant CT independent of

n,m such that

amn+1(t) ≤ CT

2−n

+

t

0amn (τ ) dτ

, ∀t ∈ [0, T ].

Arguing by induction with respect to the index n, we can obtain

amn+1(t) ≤ CT

2−n

nk=0

(2TCT )k

k!+ Cn+1

T

t

0

(t − τ)n

n!dτ

≤

CT

nk=0

(2TCT )k

k!

2−n

+ CT(TCT )

n+1

(n + 1)!,

which implies the desired result.On the other hand, for the critical points s = 2 +

1p , we can apply the interpolation method which has been used in the

proof of Lemma 3.1 to show that (un)n∈N is also a Cauchy sequence in C([0, T ]; Bs−1p,r ) for this critical case. Therefore, we have

completed the proof of Lemma 3.2. �

Now we are in the position to prove the main theorem of this section.

Theorem 3.1. Assume that 1 ≤ p, r ≤ ∞ and s > max(1 +1p ,

32 ) (or s = 1 +

1p with r = 1 and 1 ≤ p < ∞). Let u0 ∈ Bs

p,rand u be the obtained limit in Lemma 3.2. Then there exists a time T > 0 such that u ∈ Es

p,r(T ) is the unique solution to Eq. (1.2),and the mapping u0 → u is continuous from Bs

p,r into C([0, T ]; Bs′p,r) ∩ C1([0, T ]; Bs′−1

p,r ) for all s′ < s if r = ∞ and s′ = sotherwise.

Proof. Case 1. s > max(1 +1p ,

32 ) and 1 ≤ p, r ≤ ∞.

We first claim that u ∈ Esp,r(T ) solves Eq. (1.2).

In fact, according to Lemma 3.2(2) and Proposition 2.2(8), one can get u ∈ L∞([0, T ]; Bsp,r). For all s

′ < s, Lemma 3.2(2)applied again, together with an interpolation argument yields un tends to u in C([0, T ]; Bs′

p,r). Taking limit in (Tn), we can seethat u solves Eq. (1.2) in the sense of C([0, T ]; Bs′−1

p,r ) for all s′ < s. Then making use of Eq. (1.2) itself twice and the similarproof of (3.3), together with Lemma 2.1(4) and Lemma 2.2 yield u ∈ Es

p,r(T ).

On the other hand, for any s′ < s, the continuity with respect to the initial data in C([0, T ]; Bs′p,r) ∩ C1([0, T ]; Bs′−1

p,r ) canbe obtained by Lemma 3.1 and a simple interpolation argument. While the continuity in C([0, T ]; Bs

p,r) ∩ C1([0, T ]; Bs−1p,r )

when r < ∞ can be proved through the use of a sequence of viscosity approximation solutions (uε)ε>0 for Eq. (1.2) whichconverges uniformly in C([0, T ]; Bs

p,r) ∩ C1([0, T ]; Bs−1p,r ).


Case 2. s = 1 +1p , r = 1 and 1 ≤ p < ∞.

One can prove the theorem in this critical case following the similar arguments in [22] where Proposition 2.2(7), (9), (10)and Lemma 2.1(1) play a significant role. We shall omit the details here for the sake of conciseness. This completes the proofof Theorem 3.1. �

Remark 3.1. (1) Note that for every s ∈ R, Bs2,2 = Hs. Theorem 3.1 holds true in Sobolev spacesHs with s > 3

2 , which coversthe corresponding result in [27] proved by using Kato’s semigroup theory.

(2) In particular, we have obtained the local well-posedness of Eq. (1.2) in the case B322,1. However, this is not true in the case

B322,∞ in view of the proof of Proposition 4 in [22]. Noting that B

322,1 ↩→ H

32 ↩→ B

322,∞, one can see that s =

32 is the critical

index.(3) Theorem 3.1 also holds true for the more general b-family equation [52] which includes the Camassa–Holm equation

and the Degasperis–Procesi equation.

4. Blow-up criterion

In this section, we will derive the blow-up criterion of the solutions to Eq. (1.2). To this end, we first state the followinguseful estimates.

Lemma 4.1. Assume that 1 ≤ p, r ≤ ∞ and s > 1. Let u ∈ L∞(0, T ; Bsp,r ∩ Lip) solving Eq. (1.2) with the initial datum

u0 ∈ Bsp,r ∩ Lip. Then for all t ∈ [0, T ), we have

∥u(t)∥Bsp,r ≤ ∥u0∥Bsp,r eC t0 (∥u(τ )∥Lip+1)dτ (4.1)

and

∥u(t)∥Lip + 1 ≤ (∥u0∥Lip + 1)eC t0 ∥∂xu(τ )∥L∞ dτ . (4.2)

Proof. Making use of Eq. (1.2) and Lemma 2.1(3), one can obtain that

e−C t0 ∥∂xu(τ )∥L∞ dτ

∥u(t)∥Bsp,r ≤ ∥u0∥Bsp,r + C t

0

P(D)

u2

+α2

2u2x +

c0 +

γ

α2u

(τ )

Bsp,r

e−C τ0 ∥∂xu(τ ′)∥L∞dτ ′

dτ .

Note that as s > 1,P(D)

u2

+α2

2u2x +

c0 +

γ

α2u

Bsp,r

≤ C(∥u∥Lip + 1)∥u∥Bsp,r ,

with C = C(s, p, r, c0, α, γ ).Hence,

e−C t0 ∥∂xu(τ )∥L∞ dτ

∥u(t)∥Bsp,r ≤ ∥u0∥Bsp,r + C t

0(∥u(τ )∥Lip + 1) e−C

τ0 ∥∂xu(τ ′)∥L∞ dτ ′

∥u(τ )∥Bsp,r dτ ,

which together with Gronwall’s inequality yield (4.1).Next we prove (4.2). First we have the following L∞-estimate for Eq. (1.2):

∥u(t)∥L∞ ≤ ∥u0∥L∞ +

t

0

P(D)

u2

+α2

2u2x +

c0 +

γ

α2u

(τ )

L∞

dτ . (4.3)

Differentiating Eq. (1.2) with respect to x, we get

∂t(∂xu) +

u −

γ

α2

∂x(∂xu) = ∂xP(D)

u2

+α2

2u2x +

c0 +

γ

α2u

− u2x ,

which together with the L∞-estimate for transport equation again imply that

∥∂xu(t)∥L∞ ≤ ∥∂xu0∥L∞ +

t

0

∂x

P(D)

u2

+α2

2u2x +

c0 +

γ

α2u

(τ )

L∞

+ ∥∂xu(τ )∥2L∞

dτ . (4.4)


Thus, in view of (4.3) and (4.4), we obtain

∥u(t)∥Lip ≤ ∥u0∥Lip +

t

0

P(D)

u2

+α2

2u2x +

c0 +

γ

α2u

(τ )

Lip

+ ∥∂xu(τ )∥2L∞

dτ

≤ ∥u0∥Lip +

t

0(C(c0, α, γ )(∥u∥Lip∥∂xu∥L∞ + ∥∂xu∥L∞) + ∥∂xu∥2

L∞)(τ ) dτ

≤ ∥u0∥Lip + C(c0, α, γ )

t

0(∥u(τ )∥Lip + 1)∥∂xu(τ )∥L∞ dτ .

Thanks to Gronwall’s inequality, we reach (4.2). This completes the proof of the lemma. �

Definition 4.1 ([21]). Let u0 be in Bsp,r . We define the lifespan T ⋆

u0 of the solutions to Eq. (1.2) with the initial datum u0 as

T ⋆u0 , sup{T > 0 : Eq. (1.2) has a solution u ∈ Es

p,r(T )}.

Now let us prove the following main theorem of this section.

Theorem 4.1. Assume that 1 ≤ p, r ≤ ∞ and s > max(1 +1p ,

32 ) (or s = 1 +

1p with r = 1 and 1 ≤ p < ∞). Let u be the

corresponding solution to Eq. (1.2) with the initial datum u0 ∈ Bsp,r , which is guaranteed by Theorem 3.1. Then T ⋆

u0 < ∞ implies T⋆u0

0 ∥∂xu(τ )∥L∞dτ = +∞.

Proof. Suppose that T ⋆u0 < ∞ satisfies

T⋆u0

0 ∥∂xu(τ )∥L∞ dτ < +∞. Thanks to (4.2), we have T⋆

u00 (∥u(τ )∥Lip +1) dτ < +∞.

Hence, (4.1) implies that

∥u(t)∥Bsp,r ≤ MT⋆u0

, ∥u0∥Bsp,r eC T⋆

u00 (∥u(τ )∥Lip+1) dτ < +∞, ∀t ∈ [0, T ⋆

u0).

Let ε be positive such that ε < min( 14C2MT⋆

u0

, 12C2 ), where C is the same constant used in (3.5). Then we have a solution

u(t) ∈ Esp,r(ε) to Eq. (1.2) with the initial datum u(T ⋆

u0 −ε2 ). In view of uniqueness, u(t) = u(t + T ⋆

u0 −ε2 ) on t ∈ [0, ε

2 ) sothat u extends the solution u beyond T ⋆

u0 . This contradiction completes the proof of the theorem. �

Remark 4.1. (1) A more precise blow-up criterion holds as follows:

T ⋆u0 < ∞ ⇒

T⋆u0

0∥∂xu(τ )∥B0∞,∞

dτ = +∞. (4.5)

The fact that ∥∂xu∥L∞ can be replaced with the weaker norm ∥∂xu∥B0∞,∞can be easily obtained by using the following

logarithmic interpolation inequality:

∥∂xu∥L∞ ≤ C∥∂xu∥B0∞,∞ln

e +

∥u∥Bsp,r

∥∂xu∥B0∞,∞

,

for all s > max(1 +1p ,

32 ) and 1 ≤ p, r ≤ ∞ (or s = 1 +

1p with r = 1 and 1 ≤ p < ∞) which is a simple consequence

of Proposition 2.2(7) and the fact that B0∞,1 ↩→ L∞.

(2) Since the proofs of Theorem 4.1 and (4.5) are independent of the conservation laws, it then follows from that theyare also true for the general b-family equation [52], whence the above conclusions cover the corresponding resultsin [21,53].

5. Global solutions

In this section, we will show that there exists a unique global solution to Eq. (1.2) under a sign assumption for

u0 − α2∂2x u0 +

c0+γ

α22 .

Theorem 5.1. Assume that 1 ≤ p, r ≤ ∞, s > max(1+1p ,

32 ) (or s = 1+

1p with r = 1 and 1 ≤ p < ∞) and u0 ∈ Bs

p,r ∩H1.

If u0 − α2∂2x u0 +

c0+γ

α22 has the same sign as that of

c0+γ

α22 , then Eq. (1.2) has a unique global solution u ∈ Es

p,r ∩ L∞(R+;H1).


Proof. By a standard density argument, we only need to prove the theorem for u0 ∈ Bsp,r ∩ H3. According to Theorem 3.1,

Eq. (1.2) has a unique solution u ∈

T<T⋆u0

(Esp,r(T ) ∩ E3

2,2(T )).If we denote by q the flow of u, that is, q is the solution of the following ODE:

qt(t, x) = u(t, q(t, x)) −γ

α2,

q(0, x) = x.

Then we have (cf. [52]) for all t ∈ [0, T ⋆u0), x ∈ R,

y0 +c0 +

γ

α2

2=

y(t, q(t, x)) +

c0 +γ

α2

2

(qx(t, x))2,

with y , u − α2∂2x u and y0 , u0 − α2∂2

x u0.Hence, for all t ∈ [0, T ⋆

u0), we have

sign

y(t) +

c0 +γ

α2

2

= sign

y0 +

c0 +γ

α2

2

. (5.1)

Note that E2(u) ,

R(u2+ α2u2

x) dx is a conservation law which is equivalent to the norm ∥u∥2H1(R)

. Then using the explicitexpression of u = p ∗ y and ∂xu = ∂xp ∗ y where p is the Green function mentioned in the Introduction, one can easily getthat for all t ∈ [0, T ⋆

u0),

∥∂xu(t)∥L∞ ≤1α

∥u(t)∥L∞ +

c0 +γ

α2

2α

≤

1√2α

∥u(t)∥H1 +

c0 +γ

α2

2α

≤ C(α)E(u) +

c0 +γ

α2

2α

,which together with Theorem 4.1 yield T ⋆

u0 = +∞. This completes the proof of the theorem. �

Remark 5.1. Theorem 5.1 covers the corresponding results in [21,27].

6. Some properties on the existence time

In this section, we first improve the blow-up result in Theorem 4.1 and obtain a lower bound Tu0 for the lifespan T ⋆u0 .

Besides, Tu0 is sharp in the following sense: For any ε > 0, there exists a smooth function u0 such that T ⋆u0 < (1 + ε)Tu0 .

Moreover, we can get the lower semicontinuity of the existence time of solutions to Eq. (1.2) with respect to sufficientlysmooth initial data.

To this end, we first need the following useful lemma.

Lemma 6.1 ([30]). Let T > 0 and v ∈ C1([0, T );H2(R)). Then for every t ∈ [0, T ), there exists at least one point ξ(t)(respectively ζ (t)) in R such that m(t) , infx∈R ∂xv(t, x) = ∂xv(t, ξ(t)) (respectively M(t) , supx∈R ∂xv(t, x) = ∂xv(t, ζ (t)))and the function m(t) (respectively M(t)) is differentiable a.e. on (0, T ) with m′(t) = ∂x∂tv(t, ξ(t)) (respectively M ′(t) =

∂x∂tv(t, ζ (t))).

Theorem 6.1. Assume that 1 ≤ p, r ≤ ∞ and s > max(1+1p ,

32 ) (or s = 1+

1p with r = 1 and 1 ≤ p < ∞). Let u be the unique

solution to Eq. (1.2)with the initial datum u0 ∈ Bsp,r ∩H1, which is guaranteed by Theorem 3.1. Set M(t) , supx∈R ∂xu(t, x) and

m(t) , infx∈R ∂xu(t, x). Then we have for all t ∈ [0, T ⋆u0),

M(t) ≤ max(2M(0), K) (6.1)

with K , ( 1α2 {max(1, 1

α2 )E2(u) + 2√2 |c0 +

γ

α2 |

max(1, 1

α2 )E(u)})12 and E2(u) ,

R(u2

+ α2u2x)dx.

Besides, we can get the following blow-up criterion:

T ⋆u0 < ∞ ⇒

T⋆u0

0m(t) dt = −∞. (6.2)


Moreover, the lifespan T ⋆u0 satisfies

T ⋆u0 ≥ Tu0 ,

2garctan

g

−m(0)

(6.3)

with g , ( 2α2 {

1αE2(u) +

1√2|c0 +

γ

α2 |

max(1, 1

α2 )E(u)})12 , and the lower bound Tu0 is sharp.

Proof. By a standard density argument, we only need to prove the theorem for u0 ∈ Bsp,r ∩ H3. According to Theorem 3.1,

Eq. (1.2) has a unique solution u ∈ C([0, T ⋆u0); B

sp,r ∩ H3) ∩ C1([0, T ⋆

u0); Bs−1p,r ∩ H2). Applying Lemma 6.1 to u, one can infer

that for a.e. t ∈ (0, T ⋆u0),

dMdt

+M2

2=

1α2

u2(t, ζ (t)) +

c0 +

γ

α2

u(t, ζ (t)) −

p ∗

u2

+α2

2u2x +

c0 +

γ

α2

u

(t, ζ (t))

(6.4)

and

dmdt

+m2

2=

1α2

u2(t, ξ(t)) +

c0 +

γ

α2

u(t, ξ(t)) −

p ∗

u2

+α2

2u2x +

c0 +

γ

α2

u

(t, ξ(t))

. (6.5)

Using (6.4), and noting that E2(u) is a conservation law, one can gather that for a.e. t ∈ (0, T ⋆u0),

dMdt

+M2

2≤

1α2

∥u∥2

L∞ +

c0 +γ

α2

(∥u∥L∞ + ∥p ∗ u∥L∞)

≤1α2

12∥u∥2

H1 +

2c0 +

γ

α2

√2

∥u∥H1

≤

12K 2. (6.6)

Case 1. IfM(0) ≤ K , then M(t) ≤ K on [0, T ⋆u0).

Case 2. If M(0) > K , there exists a maximal T ⋆≤ T ⋆

u0 such that M(t) > K on [0, T ⋆). Then M(t) is nonincreasing on [0, T ⋆).Hence, we getdM(t)

dt≤ −

M2(t)2

+12K 2 < 0, t ∈ (0, T ⋆),

M(0) > K .

A direct calculation yieldsM(t) ≤ K M(0)+K tanh( 12 Kt)

K+M(0) tanh( 12 Kt)

≤ 2M(0), on [0, T ⋆). Note that if T ⋆ < T ⋆u0 , then M(t) ≤ K on [T ⋆, T ⋆

u0).

Therefore, (6.1) holds. (6.2) is a simple consequence of Theorem 4.1 and (6.1).Next, we prove (6.3). Since E(u) is a conservation law, ∥p∗ f ∥L∞ ≤ ∥p∥L∞∥f ∥L1 ≤

12α ∥f ∥L1 and ∥p∗ f ∥L∞ ≤ ∥p∥L1∥f ∥L∞ ≤

∥f ∥L∞ , it follows from (6.5) that

dm(t)dt

≥ −m2(t)

2−

1α2

p ∗

u2

+α2

2u2x +

c0 +

γ

α2

u

L∞

≥ −12(m2(t) + g2). (6.7)

By solving (6.7), one can deduce that, for all t < min(Tu0 , T⋆u0),

−m(t) ≤g tan

12gt− m(0)

g + m(0) tan 12gtg,

which together with (6.2) yield T ⋆u0 ≥ Tu0 . Following the similar proof of Theorem 5.1 in [27], one can get Tu0 is sharp.

Therefore, we have completed the proof of the theorem. �

Remark 6.1. Theorem 6.1 covers the corresponding results in [21,27].

We conclude this sectionwith lower semicontinuity of the existence timewith respect to sufficiently smooth initial data.


Theorem 6.2. Let 1 ≤ p, r ≤ ∞, s > max(1 +1p ,

32 ) (or s = 1 +

1p with r = 1 and 1 ≤ p < ∞) and v0 ∈ Bs

p,r . Assume that

u0 ∈ B2+ 1

pp,1 and T ⋆

u0 > T . If there exists a constant C = C(p, c0, α, γ ) > 0 such that

∥v0 − u0∥B1+ 1

pp,1

<1

C T0 exp

C τ

0 (∥u(τ ′)∥B2+ 1

pp,1

+ 1) dτ ′

dτ

, (6.8)

then Eq. (1.2) has a unique solution v ∈ Esp,r(T ) with the initial datum v0. In other words, T ⋆

v0> T .

Proof. According to Theorem3.1, there exists a uniquemaximal solution v to Eq. (1.2)with the initial datum v0. Setw , v−uagain. Then w solves the following Cauchy problem of the transport equation:

∂tw +

u + w −

γ

α2

∂xw = F(t, x),

w|t=0 = w0 , v0 − u0,

where F(t, x) , −wux + P(D)(w2+ (2u + c0 +

γ

α2 )w + α2uxwx +α2

2 w2x ).

Denote T ⋆ , min(T ⋆u0 , T

⋆v0

). Noting that B1pp,1 is an algebra, one can readily get for all t ∈ [0, T ⋆),

∥F(t)∥B1+ 1

pp,1

≤ C(∥w(t)∥B1+ 1

pp,1

+ ∥u(t)∥B2+ 1

pp,1

+ 1) ∥w(t)∥B1+ 1

pp,1

,

together with Lemma 2.1(1) yield that

∥w(t)∥B1+ 1

pp,1

≤ ∥w0∥B1+ 1

pp,1

+

t

0∥F(τ )∥

B1+ 1

pp,1

dτ + C t

0∥∂x(u + w)(τ)∥

B1pp,1

∥w(τ)∥B1+ 1

pp,1

dτ

≤ ∥w0∥B1+ 1

pp,1

+ C t

0(∥w(τ)∥

B1+ 1

pp,1

+ ∥u(τ )∥B2+ 1

pp,1

+ 1) ∥w(τ)∥B1+ 1

pp,1

dτ .

Taking advantage of Gronwall’s inequality, one can get

∥w(t)∥B1+ 1

pp,1

≤ ∥w0∥B1+ 1

pp,1

exp

C t

0

∥w(τ)∥

B1+ 1

pp,1

+ ∥u(τ )∥B2+ 1

pp,1

+ 1

dτ

. (6.9)

Let T ⋆⋆ , min{t ∈ [0, T ⋆u0 ] : C∥w0∥

B1+ 1

pp,1

t0 exp(C

τ

0 (∥u(τ ′)∥B2+ 1

pp,1

+ 1) dτ ′) dτ }.

By solving (6.9), one can deduce that for all t < min(T ⋆, T ⋆⋆),

∥w(t)∥B1+ 1

pp,1

≤

∥w0∥B1+ 1

pp,1

exp

C t0 (∥u(τ )∥

B2+ 1

pp,1

+ 1) dτ

1 − C∥w0∥B1+ 1

pp,1

t0 exp

C τ

0 (∥u(τ ′)∥B2+ 1

pp,1

+ 1) dτ ′

dτ

. (6.10)

By the assumption (6.8), we have T < T ⋆⋆.Nowweclaim that T ⋆

v0> T . Indeed, ifwe suppose that T ⋆

v0≤ T , thendue to (6.10),we infer that for all t < T ⋆

v0(≤ T < T ⋆⋆),

∥w(t)∥B1+ 1

pp,1

≤

∥w0∥B1+ 1

pp,1

exp

C T0 (∥u(τ )∥

B2+ 1

pp,1

+ 1) dτ

1 − C∥w0∥B1+ 1

pp,1

T0 exp

C τ

0 (∥u(τ ′)∥B2+ 1

pp,1

+ 1) dτ ′

dτ

< +∞.

This implies that ∥w(t)∥B1+ 1

pp,1

is uniformly bounded on [0, T ⋆v0

).

Since B1+ 1

pp,1 ↩→ Lip and u ∈ C([0, T ]; B

1+ 1p

p,1 ) with T ⋆v0

≤ T , it then follows from Theorem 4.1 that v can be extendedbeyond T ⋆

v0. This contradiction completes the proof of the theorem. �

Remark 6.2. Since the proof of Theorem 6.2 is independent of the conservation laws, it then follows from that it is also truefor the b-family equation [52], whence Theorem 6.2 covers the corresponding results in [21,27,53].


Acknowledgments

This work was partially supported by NNSFC (NO. 10971235) and the key project of Sun Yat-sen University. The authorsthank the referees for their valuable comments and suggestions.

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on the solutions of the dullin–gottwald–holm equation in besov spaces

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