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非線性薛丁格方程數值計算的離散能量法(2/2)
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博士班研究生-兼任助理：舒宇宸、舒宇宸、李易霖
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中 華 民 國 96 年 12 月 11 日
Schr¨
odinger–Langevin equation
I-Liang Chern
Department of Mathematics and Taidar Institute of Mathematical Sciences, National Taiwan University Taipei 106, Taiwan, e-mail: chern@math.ntu.edu.tw
Hai-Liang Li
Department of Mathematics, Capital Normal University Beijing, P. R. China, email: hailiang−li@mail.cnu.edu.cn
Abstract
We consider the large time behavior for nonlinear Schr¨odinger–Langevin equation in one dimen-sion for WKB-initial data with different density at left/right far fields. We show that the momen-tum damping overwhelms the quanmomen-tum dispersion. Thus, unlike those in scattering theory, the solution tends to an asymptotic state determined by a porous media equation. More precisely, the total density tends pointwise to a nonlinear diffusion wave and the phase tends to a corresponding function.
1
Introduction and Main results
The theory of quantum mechanics was employed to deal with the dissipative system which were ob-served, for example, in heavy ion physics and frictional phenomena in fission, etc [25, 20]. Recently, the nonlinear Schr¨odinger–Langevin equation is taken into granted to describe the dissipative process due to frictional force, for instance, in the motion of a Brownian particle in heat bath by Kostin [15], to characterize directly a class of nonlinear quantum mechanics through nonlinear gauge generaliza-tion by Doebner-Goldin-Nattermann [5], and to study the mogeneraliza-tion of charged (quantum) particles in semiconductor of nano-size [12, 19], and so on. The starting point for the derivation of Schr¨odinger– Langevin equation is the (quantum) Langevin equation. It is well-known that the Langevin equation has been widely used in order to investigate the diffusion of Brownian particles, dissipation and other non-equilibrium phenomena. In classical mechanics, the Langevin equation for a Brownian particle of mass m acted on by an external force F (x) is
˙ x(t) =1 mk, ˙k(t) = − ξ mk + F (x) + Γ(t), (1.1)
where k = m ˙x is the momentum, ξ > 0 is the friction constant, and Γ(t) is the stochastic force due to heat bath. This force a purely random centered Gaussian process characterized by
hΓ(t)i = 0, hΓ(t), Γ(t0)i = 2ξκT δ(t − t0),
where T > 0 is the temperature of heat bath and κ is the Boltzmann constant. Based on this fundamental equation (1.1) one can derive the well-known Fokker-Planck equations [23].
In quantum mechanical analogy, Ford-Kac-Mazur [7, 8] have proposed the quantum Langevin equation which is the Heisenberg equation of motion for the (operator) coordinate of a Brownian particle coupled to a heat bath:
˙ X(t) =1 mK, ˙ K(t) = − ξ mK + F (X) + Γ(t). (1.2)
Here K is the Heisenberg momentum operator, and X is the Heisenberg position operator. Starting with a friction term propositional to the expectation of the Heisenberg momentum operator K in the Ehrenfest equation (the second equation above), Kostin [15] was able to derived the nonlinear Schr¨odinger–Langevin equation for a Brownian particle interacting with a thermal background.
In general, the nonlinear Schr¨odinger–Langevin for the wave function Ψ takes the form iε∂tΨ = − 1 2ε 2∆Ψ + h(|Ψ|2)Ψ +1 τSΨ, in R d × R+, (1.3) S =1 2ε ln(Ψ/Ψ ∗), (1.4)
where d ≥ 1, ε > 0 is the scaled Planck constant, τ > 0 is the scaled frictional constant, and Ψ∗denotes the complex conjugate of the wave function Ψ. The function h(|Ψ|2) represents the self-interaction potential. We shall assume h0 > 0. Physically it means that the interaction of particles is repulsive. There are other derivations of the Schr¨odinger–Langevin equation based on different assumptions, see [14, 27, 4, 26, 9].
With the frictional force (1.4) acting up, the dynamics of the wave function Ψ of Eq. (1.3) is completely different from the classical one for nonlinear Schr¨odinger equation. In fact, it was proven that Schr¨odinger-Langevin equation ususally can have no solitary type solutions in the damped free-particle case in energy sapce [1], and that the coherent quantum-oscillation trajectories are damped due to the nonlinear friction force in the Shcr¨odinger-Langevin equation where the coherent oscillations decay exponentially with time [24].
We are interested in the mathematical analysis on the large time behavior of the macroscopic observable–the mass and the momentum of the nonlinear Schr¨odinger–Langevin equation caused by the nonlinear frictional effect. Roughly speaking, the new frictional term SΨ on the right hand side of (1.3) caused by the purely random force through Langevin equation is dissipative. Thus, we may expect a different asymptotic profile of the wave function in large time. To have an intuition, we apply Madelung’s idea [18] to describe quantum systems in terms of a fluid-dynamical description of the macroscopic observables such as mass, momentum, and energy. We look for the solution of the WKB-form Ψ = √ρ exp(iS/ε) of Eq. (1.3)–(1.4), substitute it into equations, and separate the real part and image part respectively, we can obtain the Madelung fluid-type equations for the particle density ρ and the momentum J = ρ∇S for irrotational flow
∂tρ + div(ρ∇S) = 0, (1.5) ∂t(ρ∇S) + div (ρ∇S ⊗ ∇S) + ∇p(ρ) = ε2 2ρ∇ ∆√ρ √ ρ −1 τρ∇S, (1.6) where the pressure p = p(ρ) satisfies p0(ρ) = ρh0(ρ), and the i-th component of the convective term div(ρu ⊗ u) equalsPd
k=1∂xk(ρuiuk). Let us introduce the re-scaling: t → τ t, ρτ= ρ(t τ, x), S τ = 1 τS( t τ, x), (1.7) to transform (1.5)–(1.6) into ∂tρ + div(ρ∇S) = 0, (1.8)
τ2∂t(ρ∇S) + τ2div (ρ∇S ⊗ ∇S) + ∇p(ρ) = ε2 2 ρ∇ ∆√ρ √ ρ − ρ∇S, (1.9)
Performing the formal limits ε → 0 and τ → 0, we obtain the following nonlinear parabolic equation for density
∂tρ = ∆p(ρ). (1.10) Thus, instead of convergence to that of free Schr¨odinger equation, we expect the density may tend to the self-similar solutions of the parabolic equation (1.10). In the present paper, we justify above expected long-time behavior for nonlinear Schr¨odinger–Langevin equation (1.3)–(1.4) in one-dimension for the following WKB initial data:
Ψ(x, t = 0) = Ψ0(x) = p
ρ0(x)eiS0(x)/ε, (1.11) ρ0(±∞) = ρ±> 0, S0(±∞) = −h(ρ±). (1.12) As shown in [6], Eq. (1.10) in one-dimension admits a unique self-similar solution up to a position shift, the nonlinear diffusion wave. It has the form ρ(x, t) = W (ξ), (ξ = √x
1+t) with the boundary conditions:
W (±∞) = ρ±. (1.13) Note that the mass ρ satisfies the conservation law (1.5). When the initial density ρ0is a perturbation of the nonlinear diffusion wave, it causes a shift of the nonlinear diffusion wave in the following sense[10]:
Z ∞ −∞
[(ρ0(x) − W (x + x0, t = 0)]dx = τ (J+− J−), (1.14) where the constant x0 ∈ R is the shift, and J± = ρ±u±. As it was shown in [10], the momentum (J−, J+) can be set to be zero at infinity. In fact, if not, due to the damping of the momentum equation (at infinity), we can define Je(x, t) and ρe(x, t) as the follows:
Je(x, t) = J−e− 1 τt+ (J+− J−)e− 1 τt Z ∞ −∞ ˜ ρ(x)dx, (1.15) ρe(x, t) = J+− J− τ ρ(x)e˜ −1 τt, (1.16) where ˜ρ(x) ≥ 0 belongs to C0∞(R) and satisfies
Z ∞ −∞
˜
ρ(x)dx = 1.
The functions Je carries the initial momentum at infinity, whereas ρe contains the mass induced by Je at far fields. Then the shift x0 is determined by
Z ∞ −∞
[ρ0(x) − W (x + x0, t = 0) − ρe(x, t = 0)]dx = 0. (1.17)
By removing Jefrom J and ρefrom ρ, we may assume
J±= 0. (1.18)
It is convenient to investigate the large time behavior of the IVP for NLS (1.3)–(1.4) and (1.11)– (1.12) in terms of the physical quantities, the amplitude n =√ρ and the momentum J = n2S
x. The macroscopic equations take
Jt+ J2 ρ + P (n) x = 1 2ε 2n2nxx n x −J τ, (1.20) where P (n) = p(n2). (1.21) The initial and boundary conditions are given by
n(x, 0) = n0(x) > 0, J (x, 0) = J0(x), (1.22) n(±∞, t) = n±:= √ ρ±, J (±∞, t) = J±= 0. (1.23) Set z0(x) = Z x −∞ (n20(y) − W (y + x0, 0))dy, w0(x) =n0(x) − p W (x + x0, 0), η0(x) = J0(x) + p(W (x + x0, 0))x. The main theorem on the large time behavior of IVP (1.19)–(1.23) is
Theorem 1.1 Let p0(ρ) > 0 for ρ > 0, and |n+− n−| 1. Assume that z0 ∈ L2(R), w0 ∈ H5(R), η0 ∈ H4(R) with kz0kL2(R)+ kw0kH5(R)+ kη0kH4(R) sufficiently small, but independent of ε. Then, there is a global classical solution (n, J ) of IVP (1.19)–(1.23) such that
kn(·, t) −pW (· + x0, t) kH5 + kJ (·, t) + τ p (W (· + x0, t))xkH4 → 0, as t → ∞. Moreover, it holds
kn(·, t) −pW (· + x0, t) kL∞≤ C(1 + t)−3/4, kJ (·, t) + τ p (W (· + x0, t))x kL∞ ≤ C(1 + t)−5/4.
From the solution (ρ, u) of IVP (1.19)–(1.23), we can construct the solution of IVP for NLS (1.3)– (1.4) and (1.11)–(1.12). In fact, from (1.20), the equation for velocity u = Sxis
ut+ 1 2(u 2) x+ h(n2)x= 1 2ε 2nxx n x −1 τu, (1.24) from which we reckon the total velocity satisfies
Z ∞ −∞ u(x, t)dx = e−t/τ Z ∞ −∞ u0(x)dx − τ [h(n2+) − h(n 2 −)](1 − e−t/τ) < ∞. Thus, the wave function Ψ(x, t)
Ψ(x, t) = n(x, t)eiS(x,t)/ε with S(x, t) = −τ h(W−) + Z x −∞ u(y, t)dx (1.25) is well-defined and satisfies IVP (1.3)–(1.4) and (1.11)–(1.12).
Set
φ0= S0(x) + τ h (W (x + x0, t = 0)) . (1.26) The large time behavior for the NLS (1.3)–(1.4) and (1.11)–(1.12) is then obtained as the follows:
Theorem 1.2 Let h0(ρ) > 0 for ρ > 0, and |n+− n−| 1. Assume that (z0, φ0) ∈ L2(R), w0 ∈ H5(R), η0∈ H4(R) with kz0kL2(R)+ kw0kH5(R)+ kη0kH4(R)sufficiently small. Then, there is a global classical solution Ψ = neiS/ε of IVP (1.3)–(1.4) and (1.11)–(1.12) such that
k(Ψ − ˜Ψ)(·, t)kH4(R)→ 0, as t → ∞, (1.27) where ˜Ψ =pW (ξ)e−iτ h(W (ξ))/ε, ξ = (x + x0)/
√
1 + t. Moreover, it holds
k(n(·, t) −pW (· + x0, t), S(·, t) + τ h (W (· + x0, t)))kL∞(R)≤ C(1 + t)−3/4. (1.28)
2
Nonlinear diffusion waves
We list some known results concerning the self-similar solution of the nonlinear parabolic equation (1.10) in this section.
Assume that the pressure-density functions satisfy p0(ρ) > 0 and τ , ε are set to be one. Then the nonlinear parabolic equation (1.10) reads:
ρt= p(ρ)xx, p0(ρ) > 0, (2.1) which possesses a unique self-similar solution w(x, t) (see [6])
ρ(x, t)= W (ζ),∆ ζ = √ x t + 1, satisfying W00(ζ) +p 00(W (ζ))W0(ζ) −1 2ζ p0(W (ζ)) W 0(ζ) = 0, W (±∞) = ρ±, (ρ+, ρ−> 0). This solution is increasing if ρ−< ρ+ and decreasing if ρ− > ρ+, and satisfies
6 X k=1 | d k dζkΦ(ζ)| + |W (ζ) − ρ+|ζ>0+ |W (ζ) − ρ−|ζ<0 ≤ Cδe −cζ2 , |Wt(x, t)| ≤ Cδ(1 + t)−1, |Wx(x, t)| ≤ Cδ(1 + t)− 1 2, where and throughout δ = |ρ+− ρ−|.
We introduce a new variable ˜ n(x, t) =pW (x + x0, t). From (2.1), ˜n satisfies ˜ nt= 1 2˜np(˜n 2) xx.
We have the following Lp−estimates of the derivatives of W and ˜n as ([17]):
Lemma 2.1 Let W be the self-similar solution of (1.10) and (1.13) and let ˜n =√W . Then it holds that k∂k t∂ j xW (., t)kLp ≤ Cδ(1 + t) −k−j 2+2p1, (2.2) k∂k t∂ j xn(., t)k˜ Lp≤ Cδ(1 + t) −k−j2+1 2p, (2.3)
In the following section, we will often use the Moser-type calculus inequalities [13]: Lemma 2.2 Let f, g ∈ L∞∩ Hs. Then, it holds
k∂α
x(f g)k ≤ CkgkL∞k∂xαf k + Ckf kL∞k∂αxgk, (2.4) k∂α
x(f g) − f ∂xαgk ≤ CkgkL∞k∂xαf k + Ckf kL∞k∂xα−1gk, (2.5) for 1 ≤ α ≤ s. Here, k · k denotes for L2 norm.
3
The perturbed equations
To obtain energy and decay estimates, we shall work on two sets of perturbed equations. One is an equation for the integral of the perturbed mass and the perturbed momentum:
z(x, t) = Z x
−∞
(ρ(x, t) − W (y + x0, t))dy, η = J + p(W )x. (3.1) The other is the equation for the perturbed amplitude and the perturbed momentum:
w = n − ˜n, η = J + P (˜n)x. (3.2) We derive them and explain why we need to use both equations for energy estimates at the end of this section.
From (1.19)–(1.23), the corresponding IVP for (z, η) becomes
zt+ η = 0, (3.3) ηt+ (η − p(W )x)2 W + zx + p(W + zx) − p(W ) x = 1 2ε 2(W + z x) √ W + zx xx √ W + zx ! x − η + p(W )xt, (3.4) z(x, 0) = z0(x), η(x, 0) = η0(x), x ∈ R. (3.5) From (3.3)–(3.5), follows the IVP for the damped “wave equation” for z
ztt+ zt− (p0(W )zx)x+ 1 4ε
2z
xxxx= (f1+ f2+ f3)x. (3.6) The corresponding initial data are
z(x, 0) = z0(x), zt(x, 0) = −η0(x). (3.7) Here, f1= 1 4ε 2(Wx+ zxx)2 W + zx − p(W )t− 1 4ε 2W xx, (3.8) f2= J2 ρ = (p(W )x+ zt)2 W + zx , (3.9) f3= p(W + zx) − p(W ) − p0(W )zx. (3.10) and we have used
ρ ( √ ρ)xx √ ρ x = 1 2ρxxx− 1 2 ρ2 x ρ x .
We recall that we have assumed p0> 0. The term (p0(W )zx)x is a diffusion term. We denote
min p0(W ) = ν > 0. (3.11)
From (1.19)–(1.20), we derive the “wave equation” for n =√ρ as
ntt+ nt+ 1 nn 2 t− 1 2n P (n) +J 2 n2 xx +1 4ε 2n xxxx− 1 4ε 2n 2 xx n = 0, where we recall P (n) = p(n2), and we have used the relation
h n2nxx n x i x = n nxxxx− n2xx n .
Recalling w = n − ˜n and η = J + P (˜n)x, then we obtain the equations for (w, η) as
2(˜n + w)wt+ 2˜ntw + ηx= 0, (3.12) wtt+ wt− (p0(W )wx)x+ 1 4ε 2w xxxx= g1+ g2+ g3, (3.13) imposed with the initial values
η(x, 0) = η0(x), (3.14) w(x, 0) = w0= n0− ˜n, wt(x, 0) = ˙w0(x) =: − η0x+ 2˜nw0 2(˜n + w0) . (3.15) Here, g1(x, t) = (˜nt+ wt)2 ˜ n + w + ε2 4 (˜nxx+ wxx)2 ˜ n + w − ε2 4n˜xxxx− ˜ntt, (3.16) g2(x, t) = 1 2√ρ J2 ρ xx = 1 2(˜n + w) (P (˜n)x− η)2 (˜n + w)2 xx (3.17) g3(x, t) =p0((˜n + w)2)(˜nx+ wx) x−p 0(˜n2)˜n x x−p 0(˜n2)w x x =(p0((˜n + w)2) − p0(˜n2))(˜n x+ wx)x (3.18) with ηx defined by (3.12).
There is a relation equation between w and zx:
(2˜n + w)w = zx or w = 1
2˜n + wzx, (3.19) which follows from ρ = n2.
Below we shall use both equations for the energy estimates. Roughly speaking, the left-hand sides of both z-equation (3.6) and w-equation (3.13) produce two good terms in the energy estimates: the dissipation energies Z t 0 kzx(s)k2ds, Z t 0 kwx(s)k2ds
and the damping energies Z t 0 kzt(s)k2ds, Z t 0 kwt(s)k2ds. The right-hand side of the w-equation (3.13) produces a term Rt
0kwk
2ds which cannot be con-trolled in the energy estimate for the w-equation, but it can be concon-trolled by the dissipation en-ergy of the z-equation, because kwk ∼ kzxk. On the other hand, the bad term on the right-hand side of the z-equation (3.6) is zxxx, which produces R
t 0 kzxtk
2+ kz
xxk2 ds in the energy es-timate. Thus, the energy estimate cannot be closed by itself. Fortunately, this term is bounded by Rt
0 kwtk 2+ kw
xk2+ kwk2 ds from (3.19) and it can be controlled by the dissipation and damping en-ergies of the w-equation and the z-equation. Notice that the termRt
0kz(s)k
2ds does not appear in the energy estimate for the z-equation because its right-hand side is a derivative. Thus, the combination of the energy estimates for z and w can close both energy estimates.
4
A priori estimates
4.1
A priori assumption
In order to perform the a priori energy estimate, let us assume that it holds for local in time solutions that for T ≥ 0, δT = max 0≤t≤T 1 X k=0 k∂tkz(t)k + k∂ k tw(t)kH5−2k 1 (4.1) Here, k · kHs is the Sobolev norm and k · k is the L2 norm. Under the smallness assumption of δT+ δ, our goal is to show that δT is bounded by δ0+ δ, where
δ0= 1 X k=0 k∂k tz(0)k + k∂ k tw(0)kH5−2k (4.2)
involves only the initial data.
Lemma 4.1 Under the assumption (4.1), we have 1 2 √ ρ−≤ ˜n + w ≤ 3 2 √ ρ+, 1 2ρ−≤ ρ ≤ 3 2ρ+, (4.3) Proof: From (3.19), kzxk ≤ O(kwk). The lemma follows easily from the smallness of kzkH1, kwkH1 and Sobolev embedding.
We have the following relations between zxand w. Lemma 4.2 It holds that
kwk ∼ kzxk (4.4) and k∂k t∂ j xzxk = k X l=0 j X i=0 O(δT + δ)k∂tl∂ i xwk, 0 ≤ 2k + j ≤ 5, (4.5) provided δT 1.
Proof: The proof follows easily from the relation: zx= (2˜n + w)w, assumption (4.1), (2.3) and Lemma 4.1.
Lemma 4.3 Under the assumption (4.1), it holds that for 0 ≤ t ≤ T 2 X k=0 k∂k tw(t)kH5−2k, 2 X k=0 k∂k tz(t)kH6−2k, 1 X k=0 k∂k tη(t)kH4−2k, 1 X k=0 k∂k tJ (t)kH4−2k = O(δT + δ). (4.6)
Proof: From w-equation (3.13), wtt can be expressed in terms of ∂tk∂x4−2kw with k ≤ 1. From assumption (4.1), we get kwttkH1 = O(δT). The estimates for kzxkH5, kzxtkH3 follows from Lemma 4.2 and assumption (4.1). The estimate for kzttkH2 follows from the z-equation (3.6). The estimates for η comes from (3.3), (3.12), (4.1). From J = η − P (˜n)xand the estimates of η and Lemma 4.2, we get the estimates for J .
From the relations: η = −zt, zx= (2˜n + w)w, 2(˜n + w)wt+ 2˜ntw = −ηx, we can get the following equivalent relations.
Lemma 4.4 Under the assumption (4.1), the following norms are equivalent whenever one of them is small:
kzkH6+ kztkH4 ∼ kzkH6+ kηkH4
∼ kzk + kwkH5+ kηkH4 ∼ kzk + kwkH5+ kztk + kwtkH3. (4.7) We recall that the nonlinear terms have the following expression:
f1= ε2 4 (Wx+ zxx)2 W + zx − p(W )t− ε2 4 Wxx, f2= (p(W )x+ zt)2 W + zx , f3= p(W + zx) − p(W ) − p0(W )zx. and g1= (˜nt+ wt)2 ˜ n + w + ε2 4 (˜nxx+ wxx)2 ˜ n + w − ε2 4n˜xxxx− ˜ntt, g2= 1 2(˜n + w) (P (˜n)x− η)2 (˜n + w)2 xx g3=(p0((˜n + w)2) − p0(˜n2))(˜nx+ wx)x. From Lemmas 2.1, 4.1, 4.3, we have the following a priori estimates.
Lemma 4.5 Under the assumption (4.1), the nonlinear terms have the following a-priori estimates: f1= O(δT + δ)zxx+ O(δ)r2,
f2= O(δT + δ)zt+ O(δ)r2, f3= O(δT)zx,
where the function rk(x, t) is related to the kth x−derivative of W . It is defined such that
Lemma 4.6 Under the assumption (4.1), the nonlinear terms have the following a-priori estimates: g1= (a1wx)x+ O(δ + δT)wt+ O(δ)r4 g2= (a2wx)x+ (b2wt)x+ O(δ + δT)(w + wx+ wt) + O(δ)r4 g3= (a3wx)x+ (δT + δ)(w + wx) where a1= 2 4 2˜nxx+ wxx ˜ n + w , a2= − J2 ρ2, b2= 2J ρ , a3= p0((˜n + w)2) − p0(˜n2) = O(w).
Lemma 4.7 Under the assumption (4.1), the higher order derivatives of the nonlinear terms have the following a-priori estimates: for 0 ≤ j ≤ 3,
∂xjg1=(a1∂xj+1w)x+ O(δT+ δ) j+1 X i=1 ∂xiw + j X i=1 ∂ixwt ! + O(δ)rj+4, ∂xjg2=(a2∂xj+1w)x+ (b2∂xjwt)x+ O(δT+ δ) j+1 X i=1 ∂xiw + j X i=1 ∂ixwt ! + O(δ)rj+4 ∂xjg3=(a3∂xj+1w)x+ O(δT+ δ) j+1 X i=1 ∂xiw.
4.2
Estimates for (z, z
t)
Lemma 4.8 For the local in time solutions z(t), it holds for 0 ≤ t ≤ T that 1 2kzt(t)k 2+1 4kz(t)k 2+Z ∞ −∞ p0(W )zx2dx + ε 2 4kzxx(t)k 2 + Z t 0 ε2 4 kzxxk 2+1 2kztk 2+1 2 Z ∞ −∞ p0(W )zx2dx ds ≤ O(δ0+ δ)2+ (α + O(δT+ δ)) Z t 0 kzxx(s)k2+ kzxt(s)k2 ds, (4.9)
where α is a constant such that
α + O(δ + δT) ≤ 1
10min(1, ν). (4.10) Proof: Multiplying (3.6) with (z + 2zt) and integrating over R, we get after integration by parts
d dt kztk2+ 1 2kzk 2+ε 2 4kzxxk 2+ Z ∞ −∞ p0(W )zx2dx + Z ∞ −∞ ztz dx +ε 2 4kzxxk 2+ kz tk2+ Z ∞ −∞ p0(W )zx· (z + 2zt)xdx = − Z ∞ −∞ (z + 2zt)x(f1+ f2+ f3)dx.
The diffusion term on the left-hand side has the following estimate: Z ∞ −∞ p0(W )zx· (z + 2zt)xdx ≥ Z ∞ −∞ (p0(W ) − O(δ + δT))zx2dx + d dt Z ∞ −∞ p0(W )zx2dx
Using Lemma 4.5 for fmand Cauchy’s inequality, we get Z ∞ −∞ (f1+ f2+ f3) · (z + 2zt)xdx ≤ α(kzxk2+ kzxtk2) + O(1)(kf1k2+ kf2k2+ kf3k2) ≤ α(kzxk2+ kzxtk2) + O(δT+ δ)(kzxxk2+ kzxk2+ kztk2) + O(δ2)(1 + t)−3/2.
Combining these estimates, we get d dt kztk2+ 1 2kzk 2+ε2 4 kzxxk 2+Z ∞ −∞ p0(W )zx2dx + Z ∞ −∞ ztz dx +ε 2 4kzxxk 2+ (1 − α − O(δ + δ T))kztk2+ Z ∞ −∞ (p0(W ) − α − O(δ + δT))zx2dx ≤ (α + O(δT + δ)) kzxxk2+ kzxtk2 + O(δ2)(1 + t)−3/2.
Integrating this in time from 0 to t, applying Cauchy’s inequality forR ztz dx, we get (4.9), provided α and δ + δT satisfy (4.10).
4.3
Estimates for (w, w
t, w
tt)
4.3.1 Basic estimates
Lemma 4.9 For the local in time solutions w, it holds 1 2kwt(t)k 2+1 4kw(t)k 2+1 2 Z ∞ −∞ p0(W )w2xdx +ε 2 4 kwxx(t)k 2 + Z t 0 ε2 4 kwxx(s)k 2+1 2kwt(s)k 2+1 2 Z ∞ −∞ p0(W )wx2dx ds ≤ O(δ0+ δ)2+ (α + O(δT+ δ)) Z t 0 kw(s)k2ds, (4.11)
for 0 ≤ t ≤ T , provided that δT+ δ is small enough. Here, α is defined by (4.10). Proof: Multiply (3.13) with (w + 2wt) and integrate it by part over R:
d dt kwtk2+ 1 2kwk 2+ε2 4kwxxk 2+ Z ∞ −∞ p0(W )wx2dx + Z ∞ −∞ wtw dx +ε 2 4 kwxxk 2+ kw tk2+ Z ∞ −∞ p0(W )wx2dx = Z ∞ −∞ ∂t(p0(W ))w2xdx + Z ∞ −∞ (w + 2wt)(g1+ g2+ g3)dx = I0+ I1+ I2+ I3.
From Lemma 2.1, the term I0 has the following estimate:
I0= Z ∞
−∞
p0(W )tw2xdx = O(δ)kwxk2. (4.12)
From Lemma 4.6, integration-by-part and Cauchy’s inequality, we get Z ∞ −∞ wg1dx ≤ − Z ∞ −∞ a1w2xdx − Z ∞ −∞ (∂xa1)wxw dx
+ C(δ + δT)kwtk2+ αkwk2+ Cδ2(1 + t)−7/2 ≤ C(δ + δT)(kwtk2+ kwxk2+ kwk2) + αkwk2+ Cδ2(1 + t)−7/2, Z ∞ −∞ 2wtg1dx ≤ − Z ∞ −∞ a12wxwxtdx + C(δ + δT)kwtk2+ αkwtk2+ Cδ2(1 + t)−7/2 ≤ −d dt Z ∞ −∞ a1w2xdx + O(δ + δT)(kwtk2+ kwxk2) + αkwtk2+ Cδ2(1 + t)−7/2.
Here, we have used
k∂xa1kL∞, k∂ta1kL∞ = O(δ + δT), Z ∞ −∞ δr4· (w + wt) dx ≤ α(kwk2+ kw tk2) + O(δ2)(1 + t)−7/2. For g2, we get Z ∞ −∞ wg2dx ≤ Z ∞ −∞ w · [(a2wx)x+ (b2wt)x] dx + O(δT + δ)(kwk2+ kwxk2+ kwtk2) + αkwk2+ Cδ2(1 + t)−7/2 ≤ O(δT+ δ)(kwxk2+ kwk2+ kwtk2) + αkwk2+ Cδ2(1 + t)−7/2, 2 Z ∞ −∞ wtg2dx ≤ Z ∞ −∞ 2wt· [(a2wx)x+ (b2wt)x] dx + O(δT + δ)(kwk2+ kwxk2+ kwtk2) + αkwtk2+ Cδ2(1 + t)−7/2 ≤ −d dt Z ∞ −∞ a2wx2dx + O(δT + δ)(kwk2+ kwxk2+ kwtk2) + αkwtk2+ Cδ2(1 + t)−7/2.
Here, we have used Lemma 4.3 and the estimates
k∂xa2kL∞, k∂ta2kL∞, k∂xb2kL∞, = O(δ + δT), which also follow from Lemma 4.3. For g3, we get
Z ∞ −∞ (w + 2wt)g3dx ≤ − d dt Z ∞ −∞ a3w2xdx + O(δT + δ)(kwxk2+ kwk2+ kwtk2).
Here, we have used
k∂xa3kL∞, k∂ta3kL∞ = O(δ + δT). We combine the above estimates to get
d dt kwtk2+ 1 2kwk 2+ε 2 4 kwxxk 2+ Z ∞ −∞ (p0(W ) + a1+ a2+ a3) wx2dx + Z ∞ −∞ wtw dx +ε 2 4 kwxxk 2+ (1 − 2α − O(δ + δ T))kwtk2+ Z ∞ −∞ (p0(W ) − O(δ + δT)) w2xdx ≤ (α + O(δ + δT))kwk2
Integrating it in time from 0 to t, then applying Cauchy’s inequality forR wwt, using p0(W ) ≥ ν > 0, and choosing α and δ + δT to satisfy (4.10), we can obtain (4.11).
Proposition 4.10 For local in time classical solution, it holds that kz(t)k2 H2+ kzt(t)k2+ kw(t)k2H2+ kwt(t)k2 + Z t 0 kzt(s)k2+ kw(s)k2H2+ kwt(s)k2 ds ≤ C(δ0+ δ)2. (4.13)
provided δT+ δ is small enough.
Proof: We add (4.9) and (4.11) together. The terms on its right-hand side areRt
0(kzxx(s)k 2+ kzxt(s)k2) ds and R
t 0kw(s)k
2ds, which can be estimated through the relations in Lemma 4.2 as the follows: Z t 0 kw(s)k2≤ Z t 0 O(1)kzx(s)k2, and Z t 0 kzxtk2≤ Z t 0 O(δ + δT)2(kwk2+ kwtk2) ≤ Z t 0 O(δ + δT)2(kzxk2+ kwtk2) Z t 0 kzxxk2≤ Z t 0 O(δ + δT)2(kwk2+ kwxk2) ≤ Z t 0 O(δ + δT)2(kzxk2+ kwxk2)
These terms can be absorbed into the damping and diffusion terms of z and w on the left-hand side, provided δ + δT is sufficiently small.
4.3.2 Higher order estimates
Applying the similar procedure in proving Lemma 4.9, we further estimate higher order derivatives of w as the follows. We performR∞
−∞∂ j
x(3.13) · ∂xj(w + 2wt) dx. After integrating by part, we get d dt 1 2k∂ j xwk 2+ k∂j xwtk2+ ε2 4k∂ j+2 x wk 2+Z ∞ −∞ ∂xjwt· ∂xjw + p0(W )|∂ j+1 x w| 2 dx + Z ∞ −∞ p0(W )|∂j+1x w|2+ k∂xjwtk2+ ε2 4 k∂ j+2 x wk 2 = I0+ Z ∞ −∞ (∂xjw + 2∂jxwt)∂xj(g1+ g2+ g3)dx := I0+ I1+ I2+ I3, (4.14) where I0:= Z ∞ −∞ −∂j x(p0(W )wx)∂xj+1(w + 2wt) + (1 + ∂t) p0(W )(∂xj+1w)2 dx ≤O(δ + δT) kwk2Hj+1+ kwtk2Hj .
Here, we have used Lemmas 2.1, 2.2, 4.3. The rest terms on the right-hand side are estimated as follows. I1≤ − d dt Z ∞ −∞ a1(∂xj+1w) 2dx
+ C(δT + δ)(kwtk2Hj + kwxk2Hj) + α(k∂xj+1wk2+ k∂xjwtk2) + Cδ2(1 + t)−j−7/2. I2≤ − d dt Z ∞ −∞ a2(∂xj+1w) 2dx + O(δT+ δ)(kwtk2Hj+ kwk 2 Hj+1) + α(k∂xjwk2+ k∂xjwtk2) + O(δ2)(1 + t)−j−7/2. I3≤O(δT + δ)(kwtk2Hj + kwk 2 Hj+1).
The substitution of these estimates into (4.14) leads to d dt 1 2k∂ j xwk 2+ k∂j xwtk2+ ε2 4 k∂ j+2 x wk 2 + d dt Z ∞ −∞ ∂xjwt· ∂xjw + (p0(W ) + a1+ a2+ a3)|∂xj+1w| 2 dx +ε 2 4 k∂ j+2 x wk 2+ (1 − α − O(δ + δ T))k∂xjwtk2 + Z ∞ −∞ (p0(W ) − α − O(δ + δT))|∂xj+1w| 2dx ≤O(δT+ δ)(kwtk2Hj+ kwk 2 Hj+1) + O(δ 2)(1 + t)−j−7/2, j = 1, 2, 3. (4.15) Integrating this inequality from 0 to t and taking summation of it (4.15) with respect to j = 0, 1, 2, 3, we get kw(t)k2 H5+ kwt(t)k2H3+ Z t 0 (kwx(s)k2H4+ kwt(s)k2H3)ds ≤O(δT + δ) Z t 0 kwt(s)k2H3+ kwx(s)k2H3 ds + O(δ + δT) Z t 0 kw(s)k2ds + O(δ0+ δ)2+ O(δ0+ δ)2. (4.16)
The first term on the right-hand side can be absorbed into the the damping energy and dissipation energy on the left-hand side. The termRt
0kw(s)k 2ds = O(δ + δ 0)2by (4.13). Thus, we obtain kw(t)k2 H5+ kwt(t)k2H3+ Z t 0 (kwx(s)k2H4+ kwt(s)k2H3)ds ≤ C(δ0+ δ)2, (4.17)
provided that δT + δ is small enough. Similarly, by performming Z 3 X j=0 ∂t∂jx(3.13) · ∂t∂xj(w + 2wt) dx, we can get kwt(t)k2H3+ kwtt(t)k2H1+ Z t 0 kwtx(s)k2H2+ kwtt(s)k2H1 ds ≤ C(δ0+ δ)2. (4.18)
Theorem 4.11 For local in-time solutions (z, w), it holds for 0 ≤ t ≤ T that kz(t)k2+ kz t(t)k2+ kw(t)k2H5+ kwt(t)k2H3+ kwttk2H1 + Z t 0 (kzt(s)k2+ kwx(s)k2H4+ kwt(s)kH23+ kwtt(s)k2H1)ds ≤ C(δ0+ δ)2, (4.19) for 0 ≤ t ≤ T , provided δT + δ 1. Where
δ0=: kz(0)k + kzt(0)k + kw(0)kH5+ kwt(0)kH3.
4.4
Proof of global existence
Proof of Theorem 1.1: global existence. The local existence of (classical) solutions can be done by using the same argument as in [11]. The Theorem 4.11 shows that the local solutions satisfy the uniform bounds for (any) short time (and therefore satisfies (4.1) too) when initial perturbations are small enough. By using the continuous argument, we extend the local solution globally in time, which also satisfies Theorem 4.11 for any time. The proof is completed.
Proof of Theorem 1.2: global existence. From (1.24) and (1.25), we find (S + τ h(W )) satisfies
(S + τ h(W ))t+ 1 2u 2+ h(ρ) − h(W ) = 1 2ε 2( √ ρ )xx √ ρ − 1 τ(S + τ h(W )) + τ h(W )t, where W = W (· + x0, t). We express this equation in (w, η):
(S + τ h(W ))t− 1 τ(S + τ h(W )) = − (P (˜n)x− η)2 (˜n + w)4 − (h((˜n + w) 2 ) − h(˜n2)) +1 2ε 2(˜n + w)xx ˜ n + w + τ h(W )t. Multiply above equation with (S + τ h(W )) and integrate over R. Using Theorem 1.1, Lemma 2.1 and Cauchy’s inequality, we have
((S + τ h(W ))2)t− 1 2τ(S + τ h(W )) 2 ≤ O(1)(kηk2+ kwk2H2) + O(δ 2)(1 + t)−3/2 This leads to kS(·, t) + τ h(W (· + x0, t))k2≤kS0+ τ h (W (· + x0, 0)) k2e−t/2τ+ Cδ(1 + t)−3/2 + C(kηk2+ kwk2H2) (4.20) and kS(·, t) + τ h(W (· + x0, t))k2H3≤kS0+ τ h (W (· + x0, 0)) k2H3e−t/2τ+ Cδ2(1 + t)−3/2 + C(kηk2H3+ kwk2H5). (4.21) Thus, the proof is completed.
5
Time decay rate
5.1
A priori decay assumption and the main result
We shall use the idea in [22, 21] to obtain the explicit time decay rate for the global classical solutions and we need more estimates on higher order (both space and time) derivatives. It is not difficult to
verify that Theorem 1.1–1.2 are also valid for solution with arbitrarily higher Sobolev regularity. In this section, we consider that the solutions satisfy
z ∈ Ck(0, ∞; H7−2k), w ∈ Ck(0, ∞; H6−2k), k = 0, 1, 2.
To perform a priori decay estimate, let us assume that for the global classical solution it holds a-priori that δT := max 0≤t≤T kz(t)k + (1 + t)kzt(t)k + 1 X k=0 5−2k X j=0 (1 + t)(j+2k+1)/2k∂j x∂ k tw(t)k + (1 + t) 3k∂6 xw(t)k 1. (5.1) Notice that when T = 0,
δ0= kz(0)k + kzt(0)k + 1 X k=0 6−2k X j=0 k∂k t∂ j xw(0)k 1. (5.2)
Under the assumption δ0 1, we can repeat the same argument in the previous section to get the existence of global classical solution with the following energy estimate
kz(t)k + kzt(t)k + kw(t)kH6+ kwt(t)kH4+ kwttkH2 +
Z T 0
(kzt(s)k + kw(s)kH6+ kwt(s)kH4+ kwtt(s)kH2) ds ≤ C(δ0+ δ). (5.3) The main result in this section is
Theorem 5.1 Under the assumption (5.1), it holds for the global solutions (z, w) that 2 X k=0 5−2k X i=0 (1 + t)i+2k+1k∂k t∂ i xw(t)k 2+ Z t 0 (1 + s)i+2kk∂k t∂ i xw(s)k 2ds + 3 X k=0 6−2k X i=1 (1 + t)i+2kk∂k t∂ i xz(t)k 2+ Z t 0 (1 + s)i+2k−1k∂k t∂ i xz(s)k 2ds ≤ O(δ0+ δ)2, (5.4)
for 0 ≤ t ≤ T , provided δ + ˆδT is small enough. Here δ0 denotes the initial perturbation (5.2).
Proof of Theorems 1.1–1.2: decay rate. In terms of the Sobolev Embedding theorem
kf kL∞≤ kf k1/2· kfxk1/2, (5.5) and (4.20)–(4.21), we can infer from Theorem 5.1 that
kn(., t) −pW (. + x0, t) kL∞ ≤ C(δ0+ δ)(1 + t)−3/4, (5.6) kJ (., t) + τ p (W (. + x0, t))xkL∞ ≤ C(δ0+ δ)(1 + t)−5/4. (5.7) k (S(., t) + τ h (W (. + x0, t)))xkL∞ ≤ C(δ0+ δ)(1 + t)−3/4, (5.8) by which we complete the proofs of Theorems 1.1–1.2.
Strategy to prove Theorem 5.1: We shall obtain decay estimates through the following procedures:
Pz(k, j; i) =: Z t 0 (1 + s)i Z ∞ −∞ (∂k t∂ j x(z-equation)) · (∂ k t∂ j xz) dx ds (5.9)
Pw(k, j; i) =: Z t 0 (1 + s)i Z ∞ −∞ (∂k t∂ j x(w-equation)) · (∂ k t∂ j xw) dx ds. (5.10) Let us define N2= 2 X k=0 5−2k X i=0 (1 + t)i+2k+1k∂tk∂ i xw(t)k 2+Z t 0 (1 + s)i+2kk∂tk∂ i xw(s)k 2ds + kz(t)k2+ 3 X k=1 (1 + t)2kk∂k tz(t)k 2+ Z t 0 (1 + s)2k−1k∂k tz(s)k 2ds (5.11)
From Lemma 5.3 below, N is equivalent to the right-hand side of (5.4). So, our goal is to show N ≤ O(δ + δ0). (5.12)
5.2
Basic estimates
Lemma 5.2 Under the assumption (5.1), it holds that for 0 ≤ j + 2k ≤ 5, 2 ≤ p ≤ ∞ k∂k
t∂xjw(t)kLp= O(δT)(1 + t)−3/4+1/2p−j/2−k for 0 ≤ t ≤ T. (5.13) Proof: For k = 0, 1, this basically follows from assumption 5.1. The estimate for wttfollows from w-equation (3.13).
From the relation zx= (2˜n + w)w, and using (2.3) for ˜n, assumption (5.1) and Lemma 5.2 for w, we can obtain the following relations between zx and w.
Lemma 5.3 Under the assumption (5.1), it holds that
k∂k t∂ j xzxk = k X l=0 j X i=0 O(δT+ δ)(1 + t)(l−k)+(i−j)/2k∂tl∂ i xwk, 0 ≤ 2k + j ≤ 5. (5.14) The assumption 5.1 also implies the following estimates for z, η and J .
Lemma 5.4 Under the assumption (5.1), we have for 0 ≤ t ≤ T , 2 ≤ p ≤ ∞, k∂k t∂ j xz(t)kLp ≤ O(δT+ δ)(1 + t)−1/4+1/2p−j/2−k, 0 ≤ j + 2k ≤ 6, k∂k t∂xjη(t)kLp ≤ O(δT+ δ)(1 + t)−5/4+1/2p−j/2−k, 0 ≤ j + 2k ≤ 4, k∂k t∂ j xJ (t)kLp ≤ O(δT+ δ)(1 + t)−1/2+1/2p−j/2−k, 0 ≤ j + 2k ≤ 4. (5.15) Proof: The first estimate follows from Lemma 5.2 and assumption (5.1). The second estimate comes from η = −zt. From J = η − P (˜n)x and (2.3), we obtain the last estimate.
Next, we use Lemmas 5.2 and 5.4 to give a priori estimates for the nonlinear terms fi and gi as the follows. |f1| = 1 4ε 2(Wx+ zxx)2 W + zx − p(W )t− 1 4ε 2W xx ≤ O(δT + δ)(1 + t)−1/2|zxx| + O(δ)r2. |∂xf1| ≤ O(δT + δ) h (1 + t)−1/2|zxxx| + (1 + t)−1|zxx| i + O(δ)r3 |∂tf1| ≤ O(δT + δ) h (1 + t)−1/2|zxxt| + (1 + t)−1|zxt| + (1 + t)−3/2|zxx| i + O(δ)r4
|f2| = (p(W )x+ zt)2 W + zx ≤ O(δ + δT)(1 + t)−1/2|zt| + O(δ)r2 |∂xf2| ≤ O(δ + δT) h (1 + t)−1/2|ztx| + (1 + t)−1|zxx| + (1 + t)−1|zt| i + O(δ)r3 |∂tf2| ≤ O(δ + δT) h (1 + t)−1/2|ztt| + (1 + t)−1|zxt| + (1 + t)−3/2|zt| i + O(δ)r4 |f3| = |p(W + zx) − p(W ) − p0(W )zx| = O(|zx|2) ≤ O(δT)(1 + t)−1/2|zx| |∂xf3| ≤ O(δT)(1 + t)−1/2|zxx| |∂tf3| ≤ O(δT)(1 + t)−1/2|zxt|. g1= (˜nt+ wt)2 ˜ n + w + ε2 4 (˜nxx+ wxx)2 ˜ n + w − ε2 4n˜xxxx− ˜ntt = a1wxx+ O(1)(δ + ˆδT)(1 + t)−1wt+ O(1)δr4 (5.16) ∂xg1= a1wxxx+ O(δ)r5 + O(δ + δT) h (1 + t)−3/2wxx+ (1 + t)−2wx+ (1 + t)−3/2wt+ (1 + t)−1wtx i ∂tg1= a1wxxt+ O(δ)r6 + O(δ + δT) h (1 + t)−2wxx+ (1 + t)−5/2wx+ (1 + t)−2wt+ (1 + t)−3/2wtx+ (1 + t)−1wtt i g2= 1 2(˜n + w) (P (˜n)x− η)2 (˜n + w)2 xx = a2wxx+ b2wxt+ O(δ)r4+ O(δ + ˆδT) h (1 + t)−3/2wx+ (1 + t)−1wt+ (1 + t)−2w i ∂xg2= a2wxxx+ b2wxxt+ O(δ)r5 + O(δ + ˆδT) h (1 + t)−1wxt+ (1 + t)−3/2wxx+ (1 + t)−2wx+ (1 + t)−3/2wt+ (1 + t)−5/2w i ∂tg2= a2wxxt+ b2wxtt+ O(δ)r6 + O(δ + ˆδT) h (1 + t)−2wxx+ (1 + t)−3/2wxt+ (1 + t)−1wtt+ (1 + t)−5/2wx+ (1 + t)−2wt+ (1 + t)−3w i g3(x, t) =(p0((˜n + w)2) − p0(˜n2))(˜nx+ wx)x= a3wxx+ O(δ + δT) h (1 + t)−1/2wx+ (1 + t)−1w i ∂xg3= a3wxxx+ O(δ + δT) h (1 + t)−1/2wxx+ (1 + t)−1wx+ (1 + t)−3/2w i ∂tg3= a3wxxt+ O(δ + δT) h (1 + t)−1/2wxt+ (1 + t)−3/2wx+ (1 + t)−1wt+ (1 + t)−2w i ∂x[p0(W )wx]x= [p0(W )wxx]x+ O(δ) h (1 + t)−1wx+ (1 + t)−1/2wxx i ∂t[p0(W )wx]x= [p0(W )wtx]x+ O(δ) h (1 + t)−3/2wx+ (1 + t)−1/2wxt i
Here, we recall that
a1= ε2 4 2˜nxx+ wxx ˜ n + w , a2= − J2 ρ2, b2= 2J ρ , a3= (p 0((˜n + w)2) − p0(˜n2).
and we have used ka1k∞= O(δ + δT)(1 + t)−1, ka2k∞= O(δT)(1 + t)−1 kb2k∞= O(δT)(1 + t)−1/2, ka3k∞= O(δT)(1 + t)−3/4 ka1,xk∞= O(δ + δT)(1 + t)−3/2, ka2,xk∞= O(δ + δT)(1 + t)−3/2 kb2,xk∞= O(δT)(1 + t)−1, ka3,xk∞= O(δ + δT)(1 + t)−5/4 ka1,tk∞= O(δ + δT)(1 + t)−2, ka2,tk∞= O(δ + δT)(1 + t)−2 kb2,tk∞= O(δT)(1 + t)−3/2, ka3,tk∞= O(δ + δT)(1 + t)−7/4. which follow from assumption (5.1). We summarize the above estimates as the following lemma. Lemma 5.5 Under the assumption (5.1), we have for 0 ≤ t ≤ T , 2k + j ≤ 4,
|∂k t∂xjf1| ≤ O(δ + δT) k X l=0 j+2k−2l X i=0 (1 + t)−1/2−k+l−(j−i)/2|∂l t∂xizxx| + O(δ)r2+2k+j |∂k t∂ j xf2| ≤ O(δ + δT) k X l=0 j+2k−2l X i=0 (1 + t)−1/2−k+l−(j−i)/2|∂l t∂ i xzt| + O(δ)r2+2k+j |∂k t∂ j xf3| ≤ O(δ + δT) k X l=0 j+2k−2l X i=0 (1 + t)−1/2−k+l−(j−i)/2|∂l t∂ i xzx| ∂tk∂xjg1= [a1∂tk∂ j xwx]x+ O(δ)r4+2k+j+ O(δ + δT) k X l=0 j+2k−2l X i=0 (1 + t)−1−k+l−(j−i)/2∂tl∂ixwt ∂tk∂ j xg2= [a2∂tk∂ j xwx]x+ [b2∂tk∂ j xwt]x+ O(δ)r4+2k+j+ O(δ + δT) k+1 X l=0 j+2k−2l+1 X i=0 (1 + t)−2−k+l−(j−i)/2∂tl∂ i xw ∂tk∂xjg3= [a3∂tk∂ j xwx]x+ O(δ + δT) k X l=0 j+2k−2l+1 X i=0 (1 + t)−1−k+l−(j−i)/2∂tl∂xiw ∂tk∂xj[p0(W )wx]x= [p0(W )∂tk∂ j xwx]x+ O(δ + δT) k X l=0 j+2k−2l X i=0 (1 + t)−1/2−k+l−(j−i)/2∂tl∂xiwx
5.3
Decay estimates for w
We have seen that by performmingP
0≤j≤4Pw(0, j; 0) + Pz(0, 0; 0), we have got the following energy estimate for (z, w):
Proposition 5.6 Under the assumption (5.1), it holds for the global solution (z, w) that kz(t)k2 H2+ kzt(t)k2+ kw(t)k2H6+ kwt(t)k2H4 Z t 0 (kzx(s)k2H1+ kzt(s)k2+ kw(s)k2H6+ kwt(s)k2H4) ds ≤ C(δ0+ δ)2, (5.17) provided δ0+ δ 1.
Proposition 5.7 Under the assumption (5.1), it holds for the global solutions w that 4 X j=0 (1 + t)j+1 k∂j xw(t)k 2 H2+ k∂xjwt(t)k2 + 4 X j=0 Z t 0 (1 + s)j+1 k∂j+1 x w(s)k 2 H1+ k∂xjwt(s)k2 ds ≤ O(N1), (5.18)
provided that δ + ˆδT is small enough. Where
N1:= (δ + δ0)2+ (δ + δT)δ2T. (5.19) Proof: We perform R∞ −∞∂ j x(3.13) · ∂ j
x(w + 2wt) dx for j = 0, ..., 4. Using integration by part, we obtain d dt k∂j xwtk2+ 1 2k∂ j xwk 2+ Z ∞ −∞ p0(W )(∂xj+1w)2dx +ε 2 4 k∂ j+2 x wk 2+ Z ∞ −∞ ∂xjwt· ∂xj+1w dx + ε 2 4 k∂ j+2 x wk 2+ k∂j xwtk2+ Z ∞ −∞ p0(W )(∂j+1x w)2dx ≤ Z ∞ −∞ ∂xj(g1+ g2+ g3) · ∂jx(w + 2wt) dx + I0+ I1, (5.20) where I0:= Z ∞ −∞ p0(W )∂xj+1w − ∂xj(p0(W )wx) · (∂xj+1w) dx, I1:= Z ∞ −∞ ∂t p0(W )(∂j+1x w) 2 − 2∂j x(p0(W )wx) · ∂xj+1wt dx. (5.21) By using Lemma 5.5, the terms on the right-hand side of (5.20) are estimated as the follows.
|I0| ≤ O(δ)k∂j+1x wk 2+ O(δ) j X i=1 (1 + t)−1−j+ik∂ixwk 2 |I1| = Z ∞ −∞ p0(W )t(∂xj+1w)2− 2p0(W )∂jxwx− ∂jx(p0(W )wx) x· ∂ j xwt dx ≤ O(δ)(1 + t)−1k∂j+1 x wk2+ αk∂jxwtk2+ [p0(W )∂xjwx− ∂xj(p0(W )wx)]x 2 ≤ O(δ)(1 + t)−1k∂j+1 x wk 2+ αk∂j xwtk2+ O(δ) j+1 X i=1 (1 + t)−2−j+ik∂i xwk 2 ≤ αk∂j xwtk2+ O(δ) j+1 X i=1 (1 + t)−2−j+ik∂i xwk2 (5.22) Z ∞ −∞ ∂xj(g1+ g2) · ∂xjw dx ≤ − Z ∞ −∞ (a1+ a2)(∂j+1x w)2dx + α(1 + t)−1k∂xjwk2 + (1 + t) " O(δ + δT)2 1 X l=0 j+1−2l X i=0 (1 + t)−4+2l−j+ik∂l t∂xiwk2+ O(δ2)kr4+jk2 # ≤ − Z ∞ −∞ (a1+ a2)(∂xj+1w) 2dx + α(1 + t)−1k∂j xwk 2
+ O(δ + δT)2 1 X l=0 j+1−2l X i=0 (1 + t)−3+2l−j+ik∂l t∂ i xwk 2+ O(δ2)(1 + t)−5/2−j Z ∞ −∞ ∂xj(g1+ g2) · 2∂xjwtdx ≤ − d dt Z ∞ −∞ (a1+ a2)(∂xj+1w) 2dx + αk∂xjwtk2+ O(1)kg1+ g2k2 ≤ −d dt Z ∞ −∞ (a1+ a2)(∂xj+1w) 2dx + αk∂xjwtk2 + O(δ + δT)2 1 X l=0 j+1−2l X i=0 (1 + t)−4+2l−j+ik∂l t∂ i xwk 2+ O(δ2)(1 + t)−7/2−j (5.23) Z ∞ −∞ ∂xjg3· ∂xjw dx ≤ − Z ∞ −∞ a3(∂xj+1w) 2dx + O(δ + δT)(1 + t)−1k∂xjwk2+ O(δ + δT)(1 + t) X 1≤i≤j+1 (1 + t)−2−j+ik∂i xwk2 ≤ − Z ∞ −∞ a3(∂xj+1w) 2dx + O(δ + δ T) j+1 X i=0 (1 + t)−1−j+ik∂i xwk 2k2 Z ∞ −∞ ∂xjg3· 2∂xjwtdx ≤ − d dt Z ∞ −∞ a3(∂xj+1w) 2dx + αk∂xjwtk2+ O(δ + δT)2 j+1 X i=0 (1 + t)−2−j+ik∂xiwk 2. (5.24) Hence, we obtain for j = 0, ..., 4
d dt k∂jxwtk2+ 1 2k∂ j xwk 2 + Z ∞ −∞ (p0(W ) − O(δ + δT)) (∂j+1x w) 2 dx +ε 2 4 k∂ j+2 x wk 2 + Z ∞ −∞ ∂jxwt· ∂xj+1w dx + ε 2 4 k∂ j+2 x wk 2+ (1 − α − O(δ + δ T))k∂xjwtk2+ Z ∞ −∞ (p0(W ) − α − O(δT + δ)) (∂xj+1w) 2dx ≤ O(δ + δT+ α) j X i=0 (1 + t)−1−j+ik∂i xwk 2+ O(δ + δ T) j−1 X i=0 (1 + t)−1−j+ik∂i xwtk2+ O(δ2)(1 + t)−5/2−j. (5.25) Now, take j = 0 in the above equation. We performRt
0(1 + s) i(5.25)
j=0ds for i = 1. This yields (1 + t)(kw(t)k2H2+ kwt(t)k2) + Z t 0 (1 + s)(kwx(s)k2H1+ kwt(s)k2) ds ≤ O(δ0+ δ)2+ Z t 0 (kwx(s)k2H1+ kwt(s)k2) ds + O(δ + δT+ α) Z t 0 kw(s)k2ds ≤ O(N 1). (5.26) Here, the last step follows from (5.17).
Next, we perform Rt
0(1 + s)(5.25)j=1ds. When i = 1, using (5.17), we get (1 + t)kwx(t)k2H2+ kwxt(t)k2 + Z t 0 (1 + s)(kwxx(s)k2H1+ kwxt(s)k2) ds ≤ O(δ + δ0)2+ Z t 0 (kwx(s)k2H2+ kwxt(s)k2) ds + Z t 0 O(δ + δT + α) kwx(s)k2+ (1 + s)−1kw(s)k2 ds ≤ O(N1).
Now, we combine this with the result in (5.26) to get Z t 0 (1 + s)(kwx(s)k2H2+ kwxt(s)k2) ds ≤ O(N1). (5.27) Thirdly, we performRt 0(1 + s) 2(5.25) j=1ds. Using (5.27), we obtain (1 + t)2 kwx(t)k2H2+ kwxtk2 + Z t 0 (1 + s)2(kwxx(s)k2H1+ kwxt(s)k2) ds ≤ O(N1). For j = 2, 3, 4, through the same procedures
Z t 0
(1 + s)i(5.25)jds for i = 1, ..., j + 1, we can inductively obtain that for j = 2, 3, 4
(1 + t)j+1(k∂xjw(t)k2H2+ k∂xjwt(t)k2) + Z t
0
(1 + s)j+1(k∂xjwx(s)k2H1+ k∂xjwt(s)k2) ds ≤ O(N1).
Proposition 5.8 Under the assumption (5.1), it holds for the global solutions w that for j = 0, ..., 4, 0 ≤ t ≤ T , (1 + t)j+2(k∂xjwt(t)k2+ k∂xj+1w(t)k 2 H1) + Z t 0 (1 + s)j+2k∂jxwt(s)k2ds ≤ O(N1), (5.28)
provided δ + ˆδT is small enough. Where N1 is defined by (5.19). Proof: For j = 0, ..., 4, we perform
Z ∞ −∞
∂xj(3.13) · 2∂jxwtdx.
By integration by parts, we obtain d dt k∂xjwtk2+ Z ∞ −∞ p0(W )(∂xj+1w)2dx + ε 2 4k∂ j+2 x wk 2 + k∂xjwtk2 ≤ αk∂xjwtk2+ 3 X m=0 k∂xjgmk2+ I1
where I1is defined by (5.21). We use (5.22), (5.23), (5.24) and Z ∞ −∞ δr4+j∂xjwtdx ≤ αk∂xjwtk2+ O(δ2)(1 + t)−7/2−j to get d dt k∂j xwtk2+ Z ∞ −∞ (p0(W ) + a1+ a2+ a3) (∂xj+1w)2+ ε2 4k∂ j+2 x wk2 + (1 − α − O(δ + δT))k∂xjwtk2 ≤O(δ + δT) j+1 X i=0 (1 + t)−2−j+ik∂i xwk 2+ j−1 X i=0 (1 + t)−2−j+ik∂i xwtk2 ! + O(δ2)(1 + t)−7/2−j. (5.29)
Multiplying this equation by (1 + s)j+2and integrating it from 0 to t, using Proposition 5.7, we obtain (5.28).
Similar to the procedures to estimate w and its x-derivatives, we perform the procedures Z ∞ −∞ ∂xj∂tk(w-equation) · ∂jx∂tk(w + 2wt) dx for 0 ≤ j + 2k ≤ 4 and Z ∞ −∞ ∂xj∂kt(w-equation) · ∂xj∂tk2wtdx for 0 ≤ j + 2k ≤ 4. These lead to the following proposition. Its proof is omitted.
Proposition 5.9 Under the assumption (5.1), it holds for the global solutions w, 0 ≤ j + 2k ≤ 4, that (1 + t)2k+j+1 k∂k t∂ j xw(t)k 2 H2+ k∂ k t∂ j xwt(t)k2 + Z t 0 (1 + s)2k+j+1 k∂k t∂ j+1 x w(s)k 2 H1+ k∂kt∂xjwt(s)k2 ds ≤ O(N1), (5.30) (1 + t)2k+j+2 k∂k t∂ j xwt(t)k2+ k∂tk∂ j xwx(t)k2H1 + Z t 0 (1 + s)2k+j+2k∂k t∂ j xwt(s)k2ds ≤ O(N1), (5.31) provided that δ + ˆδT is small enough. Here,
N1:= (δ + δ0)2+ (δ + δT)δ2T.
5.4
Decay estimates for z
t, z
ttand z
tttProposition 5.10 Under the assumption (5.1), it holds for the global solutions z that for k = 0, 1, 2,
(1 + t)2k k∂tkz(t)k 2 H2+ k∂ k tzt(t)k2 + Z t 0 (1 + s)2k k∂tkzx(s)k2H1+ k∂ k tzt(s)k2 ds ≤ O(N1), (5.32) (1 + t)2k+1 k∂k tzx(t)k2H1+ k∂tkzt(t)k2 + Z t 0 (1 + s)2k+1k∂k tzt(s)k2ds ≤ O(N1), (5.33) for 0 ≤ t ≤ T , provided that δ + ˆδT is small enough.
Proof: By performming the procedure Z ∞ −∞ ∂kt(z-equation) · ∂tk(z + 2zt) dx and Z ∞ −∞ ∂tk(z-equation) · ∂tk2ztdx
for k = 0, 1, 2 and integrating by part, we get d dt k∂k tztk2+ 1 2k∂ k tzk 2+Z ∞ −∞ p0(W )(∂ktzx)2dx + ε2 4k∂ k tzxxk2+ Z ∞ −∞ ∂tkzx· ∂tkztdx + k∂tkztk2+ ε2 4k∂ k tzxxk2+ Z ∞ −∞ p0(W )(∂tkzx)2dx = Z ∞ −∞ 3 X m=0 ∂tkfm,x· ∂tk(z + 2zt) ! dx + J0+ J1, (5.34) and d dt k∂k tztk2+ Z ∞ −∞ p0(W )(∂tkzx)2dx + ε2 4k∂ k tzxxk2 + k∂tkztk2
= Z ∞ −∞ 3 X m=0 ∂tkfm,x· ∂tk2zt ! dx + J1, (5.35) where J0:= Z ∞ −∞ p0(W )∂ktzx− ∂tk(p0(W )zx) · (∂tkzx) dx, J1:= Z ∞ −∞ ∂t p0(W )(∂tkzx)2 − 2∂tk(p0(W )zx) · (∂tkzxt) dx. (5.36)
By using Lemma 5.5, we get
|J0| ≤ O(δ)k∂ktzxk2+ O(δ) k−1 X l=0 (1 + t)−2k+2lk∂ltzxk2≤ O(δ) k X l=0 (1 + t)−2k+2lk∂tlwk 2 |J1| = Z ∞ −∞ p0(W )t(∂ktzx)2− 2p0(W )∂tkzx− ∂kt(p0(W )zx) · ∂tkztx dx ≤ O(δ)(1 + t)−1k∂tkzxk2+ O(δ)(1 + t)k∂tk+1zxk2+ δ−1(1 + t)−1 p0(W )∂tkzx− ∂tk(p0(W )zx) 2 ≤ O(δ)(1 + t)−1k∂k tzxk2+ O(δ)(1 + t)k∂tk+1zxk2+ O(δ) k−1 X l=0 (1 + t)−1−2k+2lk∂l tzxk2 ≤ O(δ) k+1 X l=0 (1 + t)−1−2k+2lk∂l twk 2 Z ∞ −∞ ∂tk(f1+ f2+ f3) · ∂tkzxdx ≤ k∂tkzxk2+ O(1) k∂ktf1k2+ k∂ktf2k2+ k∂ktf3k2 Z ∞ −∞ ∂kt(f1+ f2+ f3) · ∂tkztxdx ≤ (1 + t)k∂tk+1zxk2+ O(1)(1 + t)−1 k∂tkf1k2+ k∂tkf2k2+ k∂ktf3k2 ≤ (δ + δT) k+1 X l=0 (1 + t)−1−2k+2lk∂l twk 2+ O(1)(1 + t)−1 k∂k tf1k2+ k∂tkf2k2+ k∂tkf3k2 k∂k tf1k2≤ O(δ + δT) k X l=0 2k−2l X i=0 (1 + t)−1−2k+2l+ik∂l t∂ i xzxxk2+ O(δ2)kr2+2kk2 ≤ O(δ + δT) k X l=0 1+2k−2l X i=1 (1 + t)−2−2k+2l+ik∂l t∂ixwk2+ O(δ2)(1 + t)−3/2−2k k∂tkf2k2≤ O(δ + δT) k X l=0 2k−2l X i=0 (1 + t)−1−2k+2l+ik∂tl∂ i xztk2+ O(δ2)kr2+2kk2 ≤ O(δ + δT) k X l=0 (1 + t)−1−2k+2lk∂tlztk2+ k+1 X l=1 2k−2l−1 X i=0 (1 + t)−2−2k+2l+ik∂tl∂ i xzxk2+ O(δ2)(1 + t)−3/2−2k ≤ O(δ + δT) k+1 X l=1 (1 + t)−3−2k+2lk∂l tzk 2+ k+1 X l=1 2k−2l−1 X i=0 (1 + t)−2−2k+2l+ik∂l t∂ i xwk 2+ O(δ2)(1 + t)−3/2−2k
k∂k tf3k2≤ O(δ + δT) k X l=0 2k−2l X i=0 (1 + t)−1−2k+2l+ik∂l t∂ i xzxk2≤ O(δ + δT) k X l=0 2k−2l X i=0 (1 + t)−1−2k+2l+ik∂l t∂ i xwk 2.
Putting all these together, (5.34) becomes d dt k∂tkztk2+ 1 2k∂ k tzk 2+Z ∞ −∞ p0(W )(∂tkzx)2dx + ε2 4 k∂ k tzxxk2+ Z ∞ −∞ ∂tkwx· ∂tkwtdx + (1 − O(δ + δT))k∂tk+1zk2+ ε2 4 k∂ k tzxxk2+ Z ∞ −∞ (p0(W ) − O(δ)) (∂tkzx)2dx ≤ O(δ + δT) " k X l=0 (1 + t)−2k+2lk∂l twk 2+ (1 + t)k∂k+1 t wk 2+ k X l=0 2k−2l X i=1 (1 + t)−1−2k+2l+ik∂l t∂ i xwk 2 + k X l=0 (1 + t)−1k∂l t∂ 2k−2l+1 x wk 2 # + O(δ + δT) k X l=1 (1 + t)−3−2k+2lk∂l tzk 2+ O(δ2)(1 + t)−3/2−2k. (5.37) And (5.35) becomes d dt k∂tkztk2+ Z ∞ −∞ p0(W )(∂tkzx)2dx + ε2 4 k∂ k tzxxk2 + k∂tkztk2 ≤ O(δ + δT) "k+1 X l=0 (1 + t)−2k−1+2lk∂l twk 2+ k X l=0 2k−2l X i=1 (1 + t)−2−2k+2l+ik∂l t∂ i xwk 2 + k X l=0 (1 + t)−2k∂l t∂x2k−2l+1wk2 # + O(δ + δT) k X l=1 (1 + t)−4−2k+2lk∂tlzk 2 + O(δ2)(1 + t)−5/2−2k. (5.38)
We notice that from Theorem Z t 0 (1 + s)2kright-hand-side of (5.37) ds = O(N1) Z t 0 (1 + s)2k+1right-hand-side of (5.38) ds = O(N1) Using these, we proceed the following procedures:
• For k = 0,Rt 0(5.37)k=0ds leads to (5.39) below; • For k = 0,Rt 0(1 + s)(5.38)k=0ds leads to (5.40); • For k = 1,Rt 0 P2k i=0(1 + s) i(5.37) k=1ds leads to (5.41); • For k = 1,Rt 0(1 + s) 2k+1(5.37) k=1ds leads to (5.42); • For k = 2,Rt 0 P2k i=0(1 + s)i(5.37)k=2ds leads to (5.43); • For k = 2,Rt 0(1 + s) 2k+1(5.37) k=2ds leads to (5.44).
kz(t)k2 H2+ kzt(t)k2+ Z t 0 (kzx(s)k2H1+ kzt(s)k2) ds = O(N1) (5.39) (1 + t)(kzx(t)k2H1+ kzt(t)k2) + Z t 0 (1 + s)kzt(s)k2ds = O(N1) (5.40) (1 + t)2 kzt(t)k2H2+ kztt(t)k2 + Z t 0 (1 + s)2(kztx(s)k2H1+ kztt(s)k2) ds = O(N1) (5.41) (1 + t)3(kztx(t)k2H1+ kztt(t)k2) + Z t 0 (1 + s)3kztt(s)k2ds = O(N1) (5.42) (1 + t)4 kztt(t)k2H2+ kzttt(t)k2 + Z t 0 (1 + s)4(kzttx(s)k2H1+ kzttt(s)k2) ds = O(N1) (5.43) (1 + t)5(kzttx(t)k2H1+ kzttt(t)k2) + Z t 0 (1 + s)5kzttt(s)k2ds = O(N1). (5.44)
This completes the proof.
In general, we have the following proposition. Its proof is the same as Proposition 5.10. We shall not repeat it.
Proposition 5.11 Under the assumption (5.1), it holds for the global solutions z that for 0 ≤ j +2k ≤ 4, (1+t)2k+j k∂tk∂ j xz(t)k 2 H2+ k∂ k t∂ j xzt(t)k2+ Z t 0 (1+s)2k+j k∂tk∂ j xzx(s)k2H1+ k∂ k t∂ j xzt(s)k2 ds ≤ O(N1), (1 + t)2k+j+1 k∂k t∂ j xzx(t)k2H1+ k∂tk∂xjzt(t)k2 + Z t 0 (1 + s)2k+j+1k∂k t∂ j xzt(s)k2ds ≤ O(N1), for 0 ≤ t ≤ T , provided that δ + ˆδT is small enough.
Proof of Theorem 5.1: Recall
N2:= 2 X k=0 5−2k X i=0 (1 + t)i+2k+1k∂k t∂ i xw(t)k 2+Z t 0 (1 + s)i+2kk∂k t∂ i xw(s)k 2ds + kz(t)k2+ 2 X k=1 (1 + t)2kk∂tkz(t)k 2+Z t 0 (1 + s)2k−1k∂ktz(s)k 2ds (5.45)
and N1= O(δ + δ0)2+ (δ + δT)δ2T. Combining all estimates in this section, we get N2= O(N1) ≤ O(δ + δ0)2+ (δ + N )N2.
When δ + δ0is small enough, we can get N ≤ O(δ + δ0).
Acknowledgments I.C. is supported by the National Science Council of the Republic of China under the grant NSC94-2115-M002-017. H.L. is supported by the Beijing Nova program and NNSFC No.10431060 respectively.
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