This article concerns the study of some properties of the two- component Camassa-Holm system

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu

INITIAL DATA PROBLEMS FOR THE TWO-COMPONENT CAMASSA-HOLM SYSTEM

XIAOHUAN WANG

Abstract. This article concerns the study of some properties of the two- component Camassa-Holm system. By constructing two sequences of solutions of the two-component Camassa-Holm system, we prove that the solution map of the Cauchy problem of the two-component Camassa-Holm system is not uniformly continuous inH^s(R),s >5/2.

1. Introduction

Many authors have studied shallow water equations, of which a typical exam- ple is Camassa-Holm (CH) equation. This equation has been extended to a two- component integrable system (CH2) by combining its integrability property with compressibility, or free-surface elevation dynamics in its shallow-water interpreta- tion [10, 23]:

mt+umx+ 2mux+σρρx= 0, t >0, x∈R,

ρt+ (ρu)x= 0, t >0, x∈R, (1.1) 1.1 where m=u−u_xx and σ=±1. We remark thatσ= 1 is the hydrodynamically relevant choice, see the discussion in [10]. Local well-posedness of (1.1) withσ= 1 was obtained by [10, 11]. The precise blow-up scenarios and blow-up phenomena of strong solution for (1.1) was established by [10, 11, 13, 15, 19, 17]. Guan-Yin obtained the existence of global weak solution to (1.1). Just recently, Gui and Liu [18] studied (1.1) with σ = 1 in Besov space and they obtained the local well- posedness. In this paper, we consider the Cauchy problem of (1.1) and study the some properties of it.

Ifρ≡0, then (1.1) becomes the well-known Camassa-Holm equation [3]. In the past decade, the Camassa-Holm equation has attracted much attention because of its integrability and the existence of multi-peakon solutions, see [1]-[7] and [33]- [35] for the details. The Cauchy problem and initial boundary value problem of the Camassa-Holm equation have been studied extensively [5, 12]. It has been shown that the Camassa-Holm equation is locally well-posedness [5] for initial data u₀∈ H^s(R), s >3/2. Moreover, it has global strong solutions [5] and finite time

2000Mathematics Subject Classification. 35G25, 35B30, 35L05.

Key words and phrases. Non-uniform dependence; Camassa-Holm system; well-posedness;

energy estimates; initial value problem.

c

2014 Texas State University - San Marcos.

Submitted June 4, 2014. Published June 24, 2014.

This work was supported by PRC Grant NSFC 11301146.

1

blow-up solutions [5, 6, 8]. On the other hand, it has global weak solution in H¹(R) [1, 2, 3, 7]. The advantage of the Camassa-Holm equation in comparison with the KdV equation lies in the fact that the Camassa-Holm equation has peaked solutions and models wave breaking (i.e. the solution remains bounded while its slope becomes unbounded in finite time [3, 5, 6, 30]). Here peaked solutions are actually peaked traveling waves, similar to the waves of greatest height encountered in classical hydrodynamics, see the discussion in the papers [4, 9, 31]. Moreover, there is a rich geometric structure underlying the Camassa-Holm equation, see the discussion in the papers [25, 26].

Recently, some properties of solutions to the Camassa-Holm equation have been studied by many authors. Himonas et al. [20] studied the persistence properties and unique continuation of solutions of the Camassa-Holm equation. They showed that a strong solution of the Camassa-Holm equation, initially decaying exponen- tially together with its spacial derivative, must be identically equal to zero if it also decays exponentially at a later time, see [35, 14] for the similar properties of solutions to other shallow water equation. Just recently, Himonas-Kenig [21] and Himonas et al. [22] considered the non-uniform dependence on initial data for the Camassa-Holm equation on the line and on the circle, respectively. Lv et al. [27]

obtained the non-uniform dependence on initial data for µ-b equation. Lv-Wang [28] considered the (1.1) withρ=γ−γxxand obtained the non-uniform dependence on initial data. Wang [32] obtained the non-uniform dependence on initial data of periodic Camassa-Holm system. Tang-Wang [29] obtained the H¨older continuous of Camassa-Holm system.

In this paper, we consider the non-uniform dependence on initial data for (1.1).

We remark that there is significant difference between (1.1) and (1.1) with ρ = γ−γxx. It is easy to see that whenρ=γ−γxx, there are some similar properties between the two equations in (1.1). Thus the proof of non-uniform dependence on initial data to (1.1) with ρ = γ −γxx is similar to the single equation, for example, Camassa-Holm equation. But in (1.1), ρanduhave different properties, see Theorem 2.1. This needs construct different asymptotic solution, see section 3. Besides, the results in this paper are different from those in [27] because of the difference of the two operators 1−∂xxandµ−∂xx.

This article is organized as follows. In section 2, we recall the well-posedness result of Constantin-Ivanov [10] and Escher et al. [11] and use it to prove the basic energy estimate from which we derive a lower bound for the lifespan of the solution as well as an estimate of theH^s(R)×H^s−1(R) norm of the solution (u(t, x), ρ(t, x)) in terms of H^s(R)×H^s−1(R) norm of the initial data (u₀, ρ₀). In section 3, we construct approximate solutions, compute the error and estimate theH¹-norm of this error. In section 4, we estimate the difference between approximate and actual solutions, where the exact solution is a solution to (1.1) with initial data given by the approximate solutions evaluated at time zero. The non-uniform dependence on initial data for (1.1) is established in section 5 by constructing two sequences of solutions to (1.1) in a bounded subset of the Sobolev spaceH^s(R), whose distance at the initial time is converging to zero while at any later time it is bounded below by a positive constant.

Notation. In the following, we denote by ∗ the spatial convolution. Given a Banach space Z, we denote its norm by k · kZ. Since all space of functions are overR, for simplicity, we dropRin our notations of function spaces if there is no

ambiguity. Let [A, B] = AB−BA denotes the commutator of linear operator A andB. Setkzk²_Hs×H^s−1=kuk²_Hs+kρk²_Hs−1, wherez= (u, ρ).

2. Local well-posedness

In this section we first recall the known results of Constantin-Ivanov [10] and Escher et al. [11] and give a new estimate of the solution to (1.1).

Let Λ = (1−∂_x²)^1/2. Then the operator Λ⁻² acting onL²(R) can be expressed by its associated Green’s functionG(x) =¹₂e^−|x| as

Λ⁻²f(x) = (G∗f)(x) = 1 2

Z ∞

−∞

e^−|x−y|f(y)dy, f ∈L²(R).

Hence (1.1) is equivalent to the system u_t+uu_x=−∂xΛ⁻² u²+1

2u²_x+1 2ρ²

, t >0, x∈R, ρt+uρx=−uxρ, t >0, x∈R,

(2.1) 2.1 with initial data

u(0, x) =u₀(x), ρ(0, x) =ρ₀(x), x∈R. (2.2) 2.1a The following result is given by Constantin-Ivanov [10] and Escher et al. [11].

t2.1 Theorem 2.1. Given z₀ = (u₀, ρ₀) ∈ H^s×H^s−1, s ≥ 2. Then there exists a maximal existence time T =T(kz₀k_Hs×H^s−1)>0 and a unique solutionz= (u, ρ) to (2.1)with (2.2)such that

z=z(·, z0)∈C([0, T);H^s×H^s−1)∩C¹([0, T);H^s−1×H^s−2).

Moreover, the solution depends continuously on the initial data, i.e. the mapping z₀7→z(·, z0) :H^s×H^s−1→C([0, T);H^s×H^s−1)∩C¹([0, T);H^s−1×H^s−2) is continuous.

Next, we will give an explicit estimate for the maximal existence timeT. Also, we will show that at any time t in the time interval [0, T0] the H^s-norm of the solutionz(t, x) is dominated by theH^s-norm of the initial data z₀(x). In order to do this, we need the following lemmas.

l2.3 Lemma 2.2 ([24]). Ifr >0, then

k[Λ^r, f]gk2≤C(kfxk∞kΛ^r−1gk2+kΛ^rfk2kgk∞), whereC is a positive constant depending only on r.

t2.2 Theorem 2.3. Let s >5/2. Ifz= (u, ρ)is a solution of (2.1)with initial dataz0

described in Theorem 2.1, then the maximal existence timeT satisfies T ≥T0:= 1

2C_skz₀k_Hs×H^s−1

, (2.3) 2.2

whereC_s is a constant depending only ons. Also, we have

kz(t)kH^s×H^s−1 ≤2kz0kH^s×H^s−1, 0≤t≤T₀. (2.4) 2.3

Proof. The derivation of the lower bound for the maximal existence time (2.3) and the solution size estimate (2.4) is based on the following differential inequality for the solutionz:

1 2

d

dtkz(t)k²_Hs×H^s−1≤Cskz(t)k³_Hs×H^s−1, 0≤t < T. (2.5) 2.4 Suppose that (2.5) holds. Then, integrating (2.5) from 0 tot, we have

kz(t)k_Hs×H^s−1 ≤ kz0k_Hs×H^s−1

1−Cskz0k_Hs×H^s−1t.

From this inequality it follows thatkz(t)k_Hs×H^s−1 is finite ifCskz0k_Hs×H^s−1t <1.

LetT₀= _2C ¹

skz0k_Hs×Hs−1, then, for 0≤t≤T₀, we have kz(t)k_Hs×H^s−1 ≤ kz0k_Hs×H^s−1

1−C_skz0kH^s×H^s−1T₀ = 2kz₀k_Hs×H^s−1.

Now we prove the inequality (2.5). Note that the productsuuxanduρxare only in H^s−1 if u, ρ ∈ H^s. To deal with this problem, we will consider the following modified system

(Jεu)t+Jε(uux) =−∂xΛ⁻²

Jεu²+1

2Jεu²_x+1 2Jερ²

, t >0, x∈R, (J_ερ)_t+J_ε(uρ_x) =−J_ε(u_xρ), t >0, x∈R,

(2.6) 2.5 where for eachε∈(0,1] the operatorJ_εis the Friedrichs mollifier defined by

J_εf(x) =J_ε(f)(x) =j_ε∗f.

Here jε(x) = ¹_εj(^x_ε), and j(x) is a C^∞ function supported in the interval [−1,1]

such thatj(x)≥0,R

Rj(x)dx= 1. Applying the operator Λ^sand Λ^s−1to the first and second equations of (2.6) respectively, then multiplying the resulting equations by Λ^sJ_εuand Λ^s−1J_ερ, respectively, and integrating them with respect tox∈R, we obtain

1 2

d

dtkJεuk²_Hs =− Z

R

Λ^sJε(uux)Λ^sJεudx

− Z

R

∂_xΛ^s−2∂_xΛ⁻²

J_εu²+1

2J_εu²_x+1 2J_ερ²

Λ^sJ_εudx,

(2.7) 2.6 1

2 d

dtkJερk²_Hs−1 =− Z

R

Λ^s−1Jε(uρx)Λ^s−1Jερdx− Z

R

Λ^s−1Jε(uxρ)Λ^s−1Jερdx.

(2.8) 2.7 Similar to [32], we can estimate the right-hand sides of (2.7) and (2.8). We obtain

1 2

d

dtkJεuk²_Hs≤Cs(kuk_∞+kρk_∞+kuxk_∞+kρxk_∞)(kuk²_Hs+kρk²_Hs−1), 1

2 d

dtkJερk²_Hs−1≤Cs(kuk∞+kρk∞+kuxk∞+kρxk∞)(kuk²_Hs+kρk²_Hs−1).

Consequently, 1 2

d

dt kJ_εuk²_Hs+kJ_ερk²_Hs−1

≤C_s(kuk_∞+kρk_∞+ku_xk_∞+kρ_xk_∞)(kuk²_Hs+kρk²_Hs−1).

Then, lettingεaproach 0, we have 1

2 d

dt kuk²_Hs+kρk²_Hs−1

≤Cs(kuk_∞+kρk_∞+kuxk_∞+kρxk_∞)(kuk²_Hs+kρk²_Hs−1),

or 1

2 d

dtkz(t)k²_Hs×H^s−1 ≤Cs(ku(t)k_C1+kρk_C1)kz(t)k²_Hs×H^s−1. (2.9) 2.19 Sinces >5/2, using Sobolev’s inequality we have that

ku(t)k_C1 ≤Csku(t)kH^s, kρ(t)k_C1 ≤Cskρ(t)k_H^s−1.

From (2.9) we obtain the desired inequality (2.5). This completes the proof of

Theorem 2.3.

Recall that kz(t)k²_Hs×H^s−1 =ku(t)k²_Hs+kρ(t)k²_Hs−1, where z(t) = (u(t), ρ(t)).

It follows from Theorem 2.3 that

ku(t)kH^s,kρ(t)k_Hs−1 ≤ kz(t)k_Hs×H^s−1 ≤2kz0k_Hs×H^s−1, 0≤t≤T0. (2.10) 2.20 r2.1 Remark 2.4. Comparing Theorem 2.3 with that in [28], we will see that there

exists a significant different between (1.1) and (1.1) withρ=γ−γxx. In the other words, we require s > 5/2 because of the Sobolev embedding Theorem. But in paper [28], sinceuandγhave the same property, we assume thats >3/2.

3. Approximate solutions

In this section we first construct a two-parameter family of approximate solutions by using a similar method to [21], then compute the error and last estimate the H¹-norm of the error.

Following [21], our approximate solutionsu^ω,λ=u^ω,λ(t, x) andρ^ω,λ=ρ^ω,λ(t, x) to (2.1) will consist of a low frequency and a high frequency part, i.e.

u^ω,λ=ul+u^h, ρ^ω,λ=ρl+ρ^h,

whereω is in a bounded set ofRandλ >0. The high frequency part is given by u^h=u^h,ω,λ(t, x) =λ^−12δ−sφ x

λ^δ

cos(λx−ωt), ρ^h=ρ^h,ω,λ(t, x) =λ^{−12δ−s+1}ψ x

λ^δ

cos(λx−ωt),

(3.1) 3.1 whereφandψareC^∞ cut-off functions such that

φ(x) =

(1 if|x|<1,

0 if|x| ≥2, ψ(x) =

(1 if|x|<1, 0 if|x| ≥2.

The low frequency part (ul, ρl) = (ul,ω,λ(t, x), ρl,ω,λ(t, x)) is the solution to (2.1) with initial data

ul(0, x) =ωλ⁻¹φ˜ x λ^δ

, ρl(0, x) =ωλ⁻¹ψ˜ x λ^δ

, x∈R, (3.2) 3.2 where ˜φand ˜ψareC₀^∞(R) functions such that

φ(x) = 1˜ ifx∈suppφ∪suppψ.

We first study the properties of (ul, ρl) and (u^h, ρ^h). The high frequency part (u^h, ρ^h) defined by (3.1) satisfies

ku^h(t)kH^s ≈O(1), kρ^h(t)k_Hs−1 ≈O(1) forλ1 because of the following result.

l3.1 Lemma 3.1 ([21]). Let ψ∈ S(R), 1< δ <2 and α∈R. Then for any s≥0 we have that

λ→∞lim λ^−12δ−skψ x λ^δ

cos(λx−α)kH^s= 1

√

2kψk2. (3.3) 3.3 Relation (3.3)is also true ifcos is replaced bysin.

For the low frequency part (ul, ρl), we have the following result.

l3.2 Lemma 3.2. Let ω belong to a bounded set of R, 1 < δ < 2 and λ 1. Then the initial-value problem (2.1)-(3.2)has a unique solution(ul, ρl)∈C([0, T);H^s)× C([0, T);H^s−1), for alls >5/2, satisfying the estimates

kul(t)kH^s ≤Csλ^−1+12δ, kρl(t)k_Hs−1 ≤C_s−1λ^−1+12δ.

Proof. The existence and uniqueness of local a solution can be derived from Theo- rem 2.1 fors >5/2.

It follows from [21, Lemma 5] that kψ x

λ^δ

kH^s ≤λ^δ/2kψkH^s,

where s ≥0 and ψ∈ S(R). Using the above inequality, we have that the initial data (ul(0, x), ρl(0, x)) satisfies the estimate

kul(0)kH^s≤ |ω|λ^−1+12δkφk˜ H^s, kρl(0)kH^s−1 ≤ |ω|λ^−1+12δkψk˜ H^s−1,

which decay ifδ <2 andω is in a bounded set ofR. Recall thatkzl(t)k²_Hs×H^s−1= kul(t)k²_Hs+kρl(t)k²_Hs−1, we obtain

kz_l(0)k_Hs×H^s−1 = (ku_l(0)k²_Hs+kρ_l(0)k²_Hs−1)^1/2≤ |ω|λ^−1+12δ(kφk˜ ²_Hs+kψk˜ ²_Hs−1)^1/2. It follows from (3.2) thatz_l(0)∈H^s×H^s−1 for all s >5/2. If s >5/2, then from estimate (2.3) of Theorem 2.3, we have

kul(t)kH^s ≤Cskul(0)kH^s ≤Csλ^−1+12δ, kρl(t)k_H^s−1 ≤Cskρl(0)k_H^s−1 ≤C_s−1λ^−1+12δ.

The proof is complete.

Now we compute the error. Substituting the approximate solution (u^ω,λ, ρ^ω,λ) into the first and second equation of (2.1), we obtain the error

E=u^h_t +ulu^h_x+u^hulx+u^hu^h_x+∂xΛ⁻²

(u^h)²+k1ulu^h +1

2(u^h_x)²+ulxu^h_x+1

2(ρ^h)²+ρlρ^h ,

F =ρ^h_t +u_lρ^h_x+u^hρ_lx+u^hρ^h_x+ρ^hu_lx+ρ_lu^h_x+ρ^hu^h_x, where we have used that (u_l, ρ_l) solves (3.2).

Direct calculation shows that

u^h_t(t, x) =ωλ^−12δ−sφ x λ^δ

sin(λx−ωt), ρ^h_t(t, x) =ωλ^{−12δ−s+1}ψ x

λ^δ

sin(λx−ωt).

Since ˜φ= 1 ifx∈suppφ∪suppψ, we can writeu^h_t andρ^h_t in the form u^h_t(t, x) =ωφ˜ x

λ^δ

λ^−12δ−sφ x λ^δ

sin(λx−ωt)

=λul(0, x)λ^−12δ−sφ x λ^δ

sin(λx−ωt), ρ^h_t(t, x) =ωφ˜ x

λ^δ

λ^{−12δ−s+1}ψ x λ^δ

sin(λx−ωt)

=λu_l(0, x)λ^{−12δ−s+1}ψ x λ^δ

sin(λx−ωt).

(3.4) 3.4

Computing the spacial derivatives ofu^h andρ^h, we have u^h_x(t, x) =−λλ^−12δ−sφ x

λ^δ

sin(λx−ωt) +λ^−32δ−sφ⁰ x λ^δ

cos(λx−ωt), ρ^h_x(t, x) =−λλ^{−12δ−s+1}ψ x

λ^δ

sin(λx−ωt) +λ^{−32δ−s+1}ψ⁰ x λ^δ

cos(λx−ωt).

(3.5) 3.5 Combining (3.4) with (3.5), we obtain

u^h_t(t, x) +u_lu^h_x(t, x) =λ[u_l(0, x)−u_l(t, x)]λ^−12δ−sφx λ^δ

sin(λx−ωt) +u_l(t, x)λ^−32δ−sφ⁰ x

λ^δ

cos(λx−ωt), ρ^h_t(t, x) +ulρ^h_x(t, x) =λ[ul(0, x)−ul(t, x)]λ^{−12δ−s+1}ψ x

λ^δ

sin(λx−ωt) +u_l(t, x)λ^{−32δ−s+1}ψ⁰x

λ^δ

cos(λx−ωt).

Therefore, we can rewrite the errorE andF as

E=E1+E2+· · ·+E8, F =F1+F2+· · ·+F6, where

E1=−λ[ul(0, x)−ul(t, x)]λ^−12δ−sφ x λ^δ

sin(λx+ωt), E2=ul(t, x)λ^−32δ−sφ⁰ x

λ^δ

cos(λx+ωt), E₃=−u^hu_lx, E₄=−u^hu^h_x, E5=−∂xΛ⁻²k1

2 (u^h)²+k2

2 (ρ^h)²

, E6=−∂xΛ⁻² k1ulu^h+k2ρlρ^h , E7=−(3−k1)∂xΛ⁻²(ulxu^h_x), E8= 3−k₁

2 ∂xΛ⁻² (u^h_x)² , F₁=−k₃λ[u_l(0, x)−u_l(t, x)]λ^{−12δ−s+1}ψ x

λ^δ

sin(λx+ωt), F2=k3ul(t, x)λ^{−32δ−s+1}ψ⁰ x

λ^δ

cos(λx+ωt), F3=−k3u^hρlx, F4=−k3u^hρ^h_x, F₅=−k₃ ρ^hu_lx+ρ_lu^h_x+ρ^hu^h_x

.

Now we are ready to estimate theH¹-norm of each errorEi and theL²-norm of each errorFj (i= 1, . . . ,8, j= 1, . . . ,6). LetC be a generic positive constant. For any positive quantities P and Q, we write P .Q(P &Q) means that P ≤CQ (P ≥CQ) in the following.

Estimates of kE1kH¹ and kF1kL². Note that kf gkH¹≤√

2kfkC¹kgkH¹, ∀f ∈C¹, g∈H¹, andkφ _λ^xδ

sin(λx−ωt)kC¹=λkφk∞, we have kE1kH¹=λ^1−12δ−skφx

λ^δ

sin(λx−ωt)[ul(0, x)−ul(t, x)]kH¹

.λ^1−12δ−skφx λ^δ

sin(λx−ωt)k_C1kul(0, x)−ul(t, x)k_H1

.λ^2−12δ−skul(0, x)−u_l(t, x)kH¹.

(3.6) 3.6

To estimate the H¹-norm of the difference u_l(0, x)−u_l(t, x), we apply the funda- mental theorem of calculus in time variable to obtain

kul(0, x)−ul(t, x)k_H1 = Z t

0

kult(τ)k_H1dτ.

It follows from the first equation of (3.2) that kult(t)k_H1≤ kululxk_H1+k∂xΛ⁻² u²_l +1

2u²_lx+1 2ρ²_l

k_H1

≤ kulk_H1kulk_H2+ku²_l +1 2u²_lx+1

2ρ²_lk2

.kulk²_H2+kulk_∞kulk2+kulxk_∞kulk_H1+kρlk_∞kρlk2

.kulk²_H2+kulk²_H1+kρlk²_H2

.kulk²_H3+kρlk²_H3

.λ^−2+δ, λ1,

(3.7) 3.7

where we have used Lemma 3.2 and the Sobolev embedding Theorem H^s,→L^∞ fors >3/2.

Combining (3.6) and (3.7), we obtain

kE1k_H1.λ^−s+12δ, λ1.

Similarly,

kF1kL² .λ^−s+12δ, λ1.

Estimates of kEikH¹ and kFjkH¹, i = 2, . . . ,8, j = 2,3. In [28], the authors obtained the following estimates

kE2kH¹.λ^−s−δ,

kE3kH¹,kE6kH¹,kE7kH¹ .λ^{−12δ−s+1}λ^−1+12δ, kE4kH¹,kE5kH¹,kE8kH¹.λ^{−12δ−2s+2} Similar to the estimate ofkE2kH¹, we have

kF2kL² .λ^−s−δ, λ1.

Direct calculation shows that

kF3k_L2 =ku^hρlxk_L2 .ku^hkL^∞kρlxk_H1 .λ^−12δ−sλ^−1+12δ, λ1.

Estimates of kF4k_L2. It follows from (3.1) that

ku^h_x(t)k∞.λ^{−12δ−s+1}, kρ^h_x(t)k∞.λ^{−12δ−s+2}, λ1. (3.8) 3.8

By using Lemma 3.1, we have

ku^h(t)k_Hk=λ^−12δ−skφ x λ^δ

cos(λx−ωt)k_Hk

=λ^−s+kλ^−12δ−kkφ x λ^δ

cos(λx−ωt)k_Hk

.λ^−s+k, λ1.

(3.9) 3.9

The above inequality also holds for ρ^h(t). Combining (3.8) and (3.9), we obtain that, forλ1,

kF4kL² =ku^hρ^h_xkL² .ku^hk∞kρ^hkH¹ .λ^−12δ−sλ^−s+2.λ^{−12δ−2s+2}. Estimate of kF5k_L2. It follows from (3.8) and (3.9) that

kF5kL² =k ρ^hulx+ρlu^h_x+ρ^hu^h_x kL²

≤ kρ^hk_∞ku_lxk_H1+ku^h_xk_∞kρ_lk_H1+kρ^hk_∞ku^h_xk_L2 .kρ^hk_∞kulk_H2+ku^h_xk_∞kρlk_H2+kρ^hk_∞ku^h_xk_H1

.λ^−12δ−sλ^−1+12δ+λ^{−12δ−s+1}λ^−1+12δ+λ^{−12δ−s+1}λ^−s+1, which giveskF5k_H1 .λ^{−12δ−2s+2},λ1.

Collecting all error estimates together, we have the following theorem.

t3.1 Theorem 3.3. Let s >5/2 and1< δ <2. Whenω is in a bounded set ofRand λ1, we have that

kEk_H1 .λ^−rs, kFk_L2.λ^−rs, forλ1, 0< t < T, (3.10) 3.10 wherers=s−¹₂δ >0.

4. Difference between approximate and actual solutions In this section, we estimate the difference between the approximate and ac- tual solutions. Let (uω,λ(t, x), ρω,λ(t, x)) be the solution to (2.1) with initial data the value of the approximate solution (u^ω,λ(t, x), ρ^ω,λ(t, x)) at time zero, that is, (u_ω,λ(t, x), ρ_ω,λ(t, x)) satisfies

∂tuω,λ−uω,λ∂xuω,λ−∂xΛ⁻²(u²_ω,λ+1

2(∂xuω,λ)²+1

2ρ²_ω,λ) = 0, t >0, x∈R,

∂tρω,λ−uω,λ∂xρω,λ−(∂xuω,λρω,λ+∂xρω,λuω,λ) = 0, t >0, x∈R, u_ω,λ(0, x) =u^ω,λ(0, x) =ωλ⁻¹φ˜ x

λ^δ

+λ^−12δ−sφ x λ^δ

cos(λx), x∈R, ρω,λ(0, x) =ρ^ω,λ(0, x) =ωλ⁻¹ψ˜ x

λ^δ

+λ^{−12δ−s+1}ψ x λ^δ

cos(λx), x∈R.

(4.1) 4.1 Note that (uω,λ(0, x), ρω,λ(0, x))∈H^s×H^s−1, s≥2, it follows from Lemma 3.2 and (3.9) that

ku_ω,λ(0, x)k_Hs ≤ ku_l(0)k_Hs+ku^h(0)k_Hs .λ^−1+12δ+ 1, λ1, kρω,λ(0, x)kH^s−1≤ kρl(0)kH^s−1+kρ^h(0)kH^s−1 .λ^−1+12δ+ 1, λ1.

Therefore, if s >5/2, by using Theorem 2.1 and 2.3, we have that for anyω in a bounded set andλ1, problem (4.1) has a unique solutionz_ω,λ∈C([0, T];H^s)×

C([0, T];H^s−1) with

T & 1

kzω,λ(0)k_Hs×H^s−1 & 1

1 +λ^−1+12δ &1. (4.2) a.1 To estimate the difference between the approximate and actual solutions, we let

v=u^ω,λ−uω,λ, σ=ρ^ω,λ−ρω,λ. Then (v, σ) satisfies

vt−vvx+u^ω,λvx+vu^ω,λ_x −∂xΛ⁻²h v²+1

2v²_x +1

2σ²−2u^ω,λv−u^ω,λ_x vx−ρ^ω,λσi

= ˜E, t >0, x∈R, σt−vσx+u^ω,λσx+vρ^ω,λ_x − σvx−u^ω,λσ−ρ^ω,λvx

= ˜F , t >0, x∈R, v(0, x) =σ(0, x) = 0, x∈R,

(4.3) 4.2

where

E˜ =u^ω,λ_t +u^ω,λu^ω,λ_x +∂_xΛ⁻²

(u^ω,λ)²+1

2(u^ω,λ_x )²+1

2(ρ^ω,λ)² , F˜ =ρ^ω,λ_t +u^ω,λρ^ω,λ_x + +ρ^ω,λu^ω,λ_x ,

Similar to the prove of Theorem 3.3, ˜E and ˜F satisfy the H¹-norm estimation (3.10). Now we prove that theH¹-norm of difference decays.

t4.1 Theorem 4.1. Let 1< δ <2ands >5/2, then

kv(t)kH¹ .λ^−rs, kσ(t)kL² .λ^−rs, 0≤t≤T, λ1, wherers=s−¹₂δ >0.

Proof. Note that

1 2

d

dtkv(t)k²_H1 = Z

R

(vvt+vxvxt)dx, (4.4) 4.3 1

2 d

dtkσ(t)k²_L2 = Z

R

σσtdx. (4.5) 4.4

Applying the operator 1−∂_x²= Λ² to both sides of the first equations of (4.3), we have

vt= Λ²E˜−Λ²(u^ω,λvx−vu^ω,λ_x )−(2u^ω,λv+u^ω,λ_x vx+ρ^ω,λσ)x

+1

2(σ²)_x+ 3vv_x−2v_xv_xx−vv_xxx+v_xxt,

(4.6) 4.5 σ_t= ˜F−(u^ω,λσ_x+vρ^ω,λ_x )−(u^ω,λ_x σ+ρ^ω,λv_x) + (vσ)_x. (4.7) 4.6 Substituting (4.6) and (4.7) into (4.4) and (4.5), respectively, we obtain

1 2

d

dtkv(t)k²_H1 = Z

R

vΛ²Edx˜ − Z

R

vΛ²(u^ω,λvx+vu^ω,λ_x )dx

− Z

R

v(2u^ω,λv+u^ω,λ_x vx+ρ^ω,λσ)xdx+1 2

Z

R

v(σ²)xdx +

Z

R

(v(3vvx−2vxvxx−vvxxx+vxxt) +vxvxt)dx,

(4.8) 4.7

1 2

d

dtkσ(t)k²_L2 = Z

R

σF˜dx− Z

R

σ(u^ω,λσ_x+vρ^ω,λ_x )dx

− Z

R

σ(ρ^ω,λvx+σu^ω,λ_x )dx+ Z

R

σ(vσ)xdx.

(4.9) 4.8

A direct calculation yields Z

R

(v(3vvx−2vxvxx−vvxxx+vxxt) +vxvxt)dx

= Z

R

[(v³)x−(v²vxx)x+ (vvxt)x]dx= 0.

Substituting the above equalities in (4.8), and adding the resulting equations, we obtain

1 2

d

dt kv(t)k²_H1+kσ(t)k²_L2

= Z

R

vΛ²Edx˜ + Z

R

σF˜dx− Z

R

vΛ²(u^ω,λvx+vu^ω,λ_x )dx

− Z

R

σ(u^ω,λσx+vρ^ω,λ_x )dx− Z

R

v(2u^ω,λv+u^ω,λ_x vx+ρ^ω,λσ)xdx

− Z

R

σ(ρ^ω,λvx+σu^ω,λ_x )dx+ Z

R

1

2v(σ²)x+σ(vσ)x

dx :=I1+I2+· · ·+I7.

We first look at the last termI₇. Integrating by parts gives I7=

Z

R

1

2v(σ²)x+σ(vσ)x

dx= 0.

Estimates of integrals I1 and I2. Integrating by parts and applying the Cauchy-Schwarz inequality, we have

Z

R

vΛ²Edx˜ =

Z

R

(vE˜−vxE˜x)dx

≤ kEk˜ _H1kv(t)k_H1,

Z

R

σF˜dx

≤ kF˜k_L2kσ(t)k_L2.

Estimates of integralsI3-I6. Similar to that in [28], we obtain

6

X

i=3

Ii.(ku^ω,λ(t)k_∞+ku^ω,λ_x (t)k_∞+ku^ω,λ_xx (t)k_∞+kρ^ω,λ(t)k_∞)

×(kv(t)k²_H1+kσ(t)k²_L2).

Combining the estimations forI₁–I₇, we have 1

2 d

dt(kv(t)k²_H1+kσ(t)k²_L2)

.(kEk˜ _H1+kF˜k_H1)(kv(t)k_H1+kσ(t)k_L2)

+ (ku^ω,λ(t)k∞+ku^ω,λ_x (t)k∞+ku^ω,λ_xx (t)k∞+kρ^ω,λ(t)k∞+kρ^ω,λ_x (t)k∞)

×(kv(t)k²_H1+kσ(t)k²_H1).

(4.10) 4.9

It follows from (3.1) that u^h_x=−λ^−32δ−sφ⁰x

λ^δ

cos(λx−ωt)−λ^−δ2−s+1φx λ^δ

sin(λx−ωt),

u^h_xx=λ^−52δ−sφ⁰⁰ x λ^δ

cos(λx−ωt)−2λ^{−32δ−s+1}φ⁰ x λ^δ

sin(λx−ωt)

−2λ^{−12δ−s+2}φ x λ^δ

cos(λx−ωt).

Hence

ku^h(t)k_∞+ku^h_x(t)k_∞+ku^h_xx(t)k_∞.λ^{−(12δ+s−2)}, λ1.

By using Lemma 3.2, we have

kul(t)k_∞+kulx(t)k_∞+kulxx(t)k_∞.λ^{−(1−12δ)}, λ1.

Therefore,

ku^ω,λ(t)k∞+ku^ω,λ_x (t)k∞+ku^ω,λ_xx(t)k∞.λ^−ρs, λ1, (4.11) 4.10 whereρs= min{¹₂δ+s−2,1−¹₂δ}>0 for anys >1 if δis chosen appropriately in the interval (1,2). Similarly, we can prove that

kρ^ω,λ(t)k∞.λ^−s, kρ^ω,λ_x (t)k∞.λ^−ρs λ1. (4.12) 4.11 Let ˜z(t, x) = (v(t, x), σ(t, x)) and kz(t)k˜ ²_H1×L² = kv(t)k²_H1 +kσ(t)k²_L2, then by (4.10)-(4.12), we obtain that

1 2

d

dtk˜z(t)k²_H1×L² .(kEk˜ H¹+kF˜kL²)k˜z(t)kH¹×L²+λ^−ρsk˜z(t)k²_H1

.λ^−rsk˜z(t)k_H1×L²+λ^−ρsk˜z(t)k²_H1×L², λ1, where we have used Theorem 3.3. Consequently,

d

dtkz(t)k˜ _H1×L².λ^−ρsk˜z(t)k_H1×L²+λ^−rs, λ1. (4.13) 4.12 Since k˜z(0)kH¹×L² = (kv(0)k²_H1+kσ(0)k²_L2)^1/2= 0 and for s >1, we can choose δ∈(1,2) such that ρs≥0, by (4.13) and Gronwall’s inequality, we obtain

kz(t)k˜ _H1×L².λ^−rs, 0≤t≤T, λ1.

Note that

kv(t)k_H1, kσ(t)k_L2≤ k˜z(t)k_H1×L², we see that

kv(t)k_H1, kσ(t)k_L2 .λ^−rs, 0≤t≤T, λ1.

This completes the proof.

5. Non-uniform dependence

In this section, we prove non-uniform dependence for (2.1) by taking advantage of the information provided by Theorem 2.1-2.3, Theorem 3.3 and Theorem 4.1.

Our main result is the following.

t5.1 Theorem 5.1. If s >5/2, then the data-to-solution z(0)→ z(t) for (2.1) is not uniformly continuous from any bounded subset ofH^s×H^s−1 intoC([−T, T];H^s)×

C([−T, T];H^s−1), where z(0) = (u₀(x), ρ₀(x)) and z(t) = (u(t, x), ρ(t, x)). More precisely, there exist two sequences of solutions (u_λ(t), ρ_λ(t)) and(˜u_λ(t),ρ˜_λ(t))to the differential equations of (2.1)inC([−T, T];H^s)×C([−T, T];H^s−1)such that

kuλ(t)kH^s+ku˜_λ(t)kH^s+kρλ(t)kH^s−1+kρ˜_λ(t)kH^s−1 .1, (5.1)

λ→∞lim kuλ(0)−u˜λ(0)kH^s = lim

λ→∞kρλ(0)−ρ˜λ(0)k_H^s−1 = 0, (5.2) 5.2 lim inf

λ→∞ (ku_λ(t)−u˜_λ(t)k_Hs+kρ_λ(t)−ρ˜_λ(t)k_Hs−1)&sint, |t|< T ≤1. (5.3) 5.3

Proof. Let (uλ(t), ρλ(t)) = (u1,λ(t, x), ρ1,λ(t, x)) and let (˜uλ(t),ρ˜λ(t)) =

(u−1,λ(t, x), ρ−1,λ(t, x)), where (u1,λ(t, x), ρ1,λ(t, x)) and (u−1,λ(t, x), ρ−1,λ(t, x)) be the unique solution to problem (4.1) with initial data (u^1,λ(0, x), ρ^1,λ(0, x)) and (u^−1,λ(0, x), ρ^−1,λ(0, x)), respectively. From Theorem 2.1 these solutions belong to C([0, T];H^s)×C([0, T];H^s−1). By (4.2) and the assumptions after Theorem 2.1, we see that T is independent of λ 1. Letting k = [s] + 2 and using estimate (2.10), we have

ku±1,λ(t)kH^k,kρ±1,λ(t)kH^k−1 .kz^±1,λ(0)kH^k×H^k−1, (5.4) 5.4 where z^±1,λ(0) = (u^±1,λ(0), ρ^±1,λ(0)) and kz^±1,λ(0)k²_Hk×H^k−1 = ku^±1,λ(0)k²_Hk+ kρ^±1,λ(0)k²_Hk−1. Ifλis large enough, then from Lemma 3.1 we have

ku^±1,λ(t)k_Hk≤ ku±1,λ(t)k_Hk+λ^−12δ−skφ x λ^δ

cos(λx−ωt)k_Hk

.λ^−1+12δ+λ^k−skφk2, which gives

ku^±1,λ(t)k_Hk.λ^k−s. (5.5) 5.5

Combining (5.4) with (5.5), we obtain

ku±1,λ(t)kH^k .λ^k−s, λ1. (5.6) 5.6 Estimates (5.5) and (5.6) yield

ku^±1,λ(t)−u_±1,λ(t)k_Hk .λ^k−s, λ1. (5.7) 5.7 Theorem 4.1 implies

ku^±1,λ(t)−u_±1,λ(t)kH¹ .λ^−rs, λ1. (5.8) 5.8 Now, applying the interpolation inequality

kϕk_Hs ≤ kϕk^(s_H²s1^{−s)/(s2−s1)}kϕk^(s−s_Hs2^1)/(s2−s1)

withs1= 1 ands2= [s] + 2 =k, and using estimates (5.7) and (5.8), we obtain ku^±1,λ(t)−u±1,λ(t)kH^s

≤ ku^±1,λ(t)−u_±1,λ(t)k(k−s)/(k−1)

H¹ ku^±1,λ(t)−u_±1,λ(t)k(s−1)/(k−1) H^k

.λ^−rs(k−s)/(k−1)λ(k−s)(s−1)/(k−1)

.λ^−(rs−s+1)(k−s)/(k−1), λ1.

Hence

ku^±1,λ(t)−u_±1,λ(t)kH^s .λ^−εs, λ1, (5.9) 5.9 whereεs= (1−¹₂δ)/(s+ 2).

Next, we prove (5.2) and (5.3). Note that 0< δ <2, we have ku1,λ(0)−u_−1,λ(0)kH^s= 2λ⁻¹kφ˜ x

λ^δ

kH^s ≤2λ^−1+12δkφk˜ H^s →0, kρ1,λ(0)−ρ−1,λ(0)kH^s−1 = 2λ⁻¹kψ˜ x

λ^δ

kH^s−1 ≤2λ^−1+12δkψk˜ H^s−1 →0 as λ→ ∞, which implies that (5.2) holds. Now, we prove (5.3). It is easy to see that

lim inf

λ→∞ (ku_λ(t)−˜u_λ(t)k_Hs+kρ_λ(t)−ρ˜_λ(t)k_Hs−1)≥lim inf

λ→∞ ku_λ(t)−u˜_λ(t)k_Hs.