uniqueness of the solution

Quantitative estimates of the stationary Navier-Stokes equations at infinity and

uniqueness of the solution

Ching-Lung Lin

Jenn-Nan Wang

Abstract

In this paper we are interested in the asymptotic behavior of in- compressible fluid around a bounded obstacle. Under certain a priori decaying assumptions, we derive a quantitative estimate of the de- caying rate of the difference of any two velocity functions at infinity.

This quantitative estimate gives us a sufficient condition, expressed in terms of integrability, to guarantee that the solution of the Navier- Stokes equations is unique.

1 Introduction

Let B be a bounded domain in Rⁿ and Ω = Rⁿ\ ¯B with n ≥ 2. Without loss of generality, we let 0 belong to interior of B and B ⊂ B₁(0) = {x :

|x| < 1}. Assume that Ω is filled with an incompressible fluid described by the stationary Navier-Stokes equations

( −∆u + u · ∇u + ∇p = f in Ω,

∇ · u = 0 in Ω. (1.1)

We are interested in the following question: let u₁ and u₂ be two solutions of (1.1) satisfying some pre-described assumptions such as boundedness or

∗Department of Mathematics, NCTS, National Cheng Kung University, Tainan 701, Taiwan. Email:cllin2@mail.ncku.edu.tw

†Department of Mathematics, NCTS (Taipei), National Taiwan University, Taipei 106, Taiwan. Email: jnwang@math.ntu.edu.tw

decaying conditions, then find a sufficient condition which guarantees that u₁ ≡ u₂ in Ω. In this paper, we answer this question by deriving a minimal decay rate of u1− u2 at infinity if u1 6= u2.

This question is motivated by the following problem. It was shown by Finn [2] that when n = 3 and f = 0, if u|_∂B = 0 and u = o(|x|⁻¹), then u is trivial. Inspired by Finn’s result, we would like to ask the following question:

when n = 3, if we know a priori that u = O(|x|⁻¹), what is the minimal decaying rate of any nontrivial u satisfying (1.1)? It should be remarked that the boundary value of u on ∂B is irrelevant in this problem. Moreover, the asymptotic behavior u = O(|x|⁻¹) characterizes the so-called physically reasonable solutions introduced by Finn [3].

To answer the main question of the paper, we simply subtract two equa- tions for u₁ and u₂ and obtain

(−∆v + v · ∇v + v · ∇u₂+ u₂· ∇v + ∇p_v = 0 in Ω,

∇ · v = 0 in Ω,

where v = u₁ − u₂ and p_v = p₁ − p₂. Therefore, to solve the problem, it suffices to consider the generalized Navier-Stokes equations

(−∆v + v · ∇v + v · ∇α + α · ∇v + ∇p = 0 in Ω,

∇ · v = 0 in Ω (1.2)

with ∇ · α = 0. To describe the main theorem, we denote I(x) =

Z

|y−x|<1

|v(y)|²dy and

M (t) = inf

|x|=tI(x).

Then we prove that

Theorem 1.1 Let v ∈ (H_loc¹ (Ω))ⁿ be a nontrivial solution of (1.2) with an appropriate p ∈ H_loc¹ (Ω). Assume that for 0 ≤ κ₁ ≤ ¹₄, 0 ≤ κ₂ ≤ ¹₂, 0 < δ ≤ ¹₈ and λ ≥ 1

(|v(x)| + |α(x)| + |∇v(x)| ≤ λ(1 + |x|²)^−κ1−δ

|∇α(x)| ≤ λ(1 + |x|²)^−κ2−δ. (1.3)

Then there exist ˜t depending on λ, n, κ₁, κ₂, δ and positive constants C₁, C₂ such that

M (t) ≥ exp −(C1/δ²)t^κ(log t)^C2/δ

for t ≥ ˜t, (1.4) where κ = max{2 − 4κ₁, 2 − 2κ₂} and the constant C₂ depends on λ and n, while C₁ is explicitly given by

C₁ = k     

log min{ inf

˜t<|x|<˜t^(1−δ)−1

Z

|y−x|<1

|v(y)|²dy, 1}

!    

+ γ, where k is an absolute constant and γ depends on λ and n.

It is interesting to compare Theorem 1.1 with the result obtained in [6]

where we showed that for the standard stationary Navier-Stokes equations (i.e., α = 0 in (1.2)) if v is bounded (for n = 2) or C¹ bounded (for n ≥ 3) in Ω, then

M (t) ≥ exp(−Ct²⁺).

We would also like to mention some related results for the Schr¨odinger opera- tor with potential in [1] and [7] where the unique continuation property were proved under some integrability conditions similar to those in Corollary 1.3 and 1.4.

We can immediately deduce several consequences from Theorem1.1. As- sume that n = 3 and f = O(|x|⁻³) at infinity. Let u₁, u₂ be two solutions of (1.1) satisfying u1 = O(|x|⁻¹) and u2 = O(|x|⁻¹). It was proved by Sverak and Tsai [9] that both ∇u₁and ∇u₂ are O(|x|⁻²). So we can choose κ₁ = 1/4, κ₂ = 1/2 (then κ = 1), and fix δ = 1/8 in Theorem 1.1. Due to Sverak and Tsai’s result, we can also relax condition (1.3). Setting v = u1 − u2 and α = v₁, we obtain from Theorem 1.1 that

Corollary 1.2 Let u₁, u₂ ∈ (H_loc¹ (Ω))³ be solutions of (1.1) with appropriate pressures p₁, p₂ ∈ H_loc¹ (Ω). Assume that f (x) = O(|x|⁻³), u₁(x) = O(|x|⁻¹), and u₂ = O(|x|⁻¹), at infinity. Then there exist ˜t and positive constants s₁, s₂ such that

inf

|x|=t

Z

|y−x|<1

|(u₁− u₂)(y)|²dy ≥ exp (−s₁t(log t)^s2) for t ≥ ˜t, where s₁ depends linearly on

log



min{ inf

˜t<|x|<˜t^8/7

Z

|y−x|<1

|(u₁− u₂)(y)|²dy, 1}

    .

Corollary 1.2 immediately implies the following qualitative uniqueness results.

Corollary 1.3 Let u₁, u₂ ∈ (H_loc¹ (Ω))³ be solutions of (1.1) with appropriate pressures p₁, p₂ ∈ H_loc¹ (Ω). Assume that f (x) = O(|x|⁻³), u₁(x) = O(|x|⁻¹), and u₂ = O(|x|⁻¹), at infinity. Then there exist R and positive constants s₁, s₂ such that if

Z

Ω∩{|x|≥R}

exp(s|x|(log |x|)^s2)|(u₁− u₂)(x)|²dx < ∞

for all s > s₁, then u₁ ≡ u₂ in Ω, where s₁’s dependence is described in Corollary 1.2.

In particular, let u₂ = 0 and f = 0, we have that

Corollary 1.4 Let n = 3, f = 0, and u ∈ (H_loc¹ (Ω))³ be a solution of (1.1) with an appropriate p ∈ H_loc¹ (Ω). Assume that u(x) = O(|x|⁻¹). Then there exist R and positive constants s₁, s₂ such that if

Z

Ω∩{|x|≥R}

exp(s|x|(log |x|)^s2)|u(x)|²dx < ∞

for all s > s₁, then u ≡ 0 in Ω, where s₁ depends linearly on the quantity 

log min{ inf

R<|x|<R⁸⁷

Z

|y−x|<1

|u(y)|²dy, 1}

!     .

As in [6], we prove our result along the line of Carleman’s method. Some useful techniques used in [6] are collected in the next Section. The proof of the main theorem is given in Section 3.

2 Reduced system and Carleman estimates

Fixing x₀ with |x₀| = t >> 1, we define

w(x) = (at)v(atx + x₀), ˜α(x) = (at)α(at + x₀), and ˜p(x) = (at)²p(atx + x₀), where r₁ is the constant given in Lemma 2.1 and a ≥ 8/r₁ which will be determined in the proof of Theorem 1.1. Likewise, we denote

Ωt:= B¹

a− ¹

20atδ

(0) = {x : |x| < 1 a − 1

20at^δ}.

From (1.2), it is easy to get that

( −∆w + w · ∇w + w · ∇ ˜α + ˜α · ∇w + ∇˜p = 0 in Ω_t,

∇ · w = 0 in Ω_t. (2.1)

In view of (1.3), we have that







k ˜αk_L^∞_(Ωt)+ kwk_L^∞_(Ωt) ≤ C₀aλt^{1−2κ1−δ}, k∇wk_L^∞_(Ωt) ≤ C₀a²λt^{2−2κ1−δ},

k∇ ˜αk_L^∞_(Ωt)≤ C₀a²λt^{2−2κ2−34δ},

(2.2)

where we can choose C₀ = (20)^5/4.

To prove Theorem1.1, we use the reduced system containing the vorticity equation derived in [6]. Let us define the vorticity q of the velocity w by

q = curl w := 1

√2(∂_iw_j − ∂_jw_i)_1≤i,j≤n.

The formal transpose of curl is given by (curl^>v)_1≤i≤n := 1

√2 X

1≤j≤n

∂_j(v_ij − v_ji),

where v = (v_ij)_1≤i,j≤n. It is easy to see that

∆w = ∇(∇ · w) − curl^>curlw (see, for example, [8] for a proof), which implies

∆w + curl^>q = 0 in Ω_t. (2.3) Next we observe that

w · ∇ ˜α + ˜α · ∇w = ∇(w · ˜α) −√

2(curl w) ˜α −√

2(curl ˜α)w

= ∇(w · ˜α) −√

2q ˜α −√

2(curl ˜α)w and in particular

w · ∇w = ∇(1

2|w|²) −√

2(curl w)w = ∇(1

2|w|²) −√ 2qw.

Thus, applying curl on the first equation of (2.1), we have that

−∆q+Q(q)(w+ ˜α)+q(∇w+∇ ˜α)^>−(∇w+∇ ˜α)q^>−divF = 0 in Ω_t, (2.4) where

(Q(q)w)_ij = X

1≤k≤n

(∂_jq_ik− ∂_iq_jk)w_k and

(divF )_ij =

n

X

k=1

∂_kF_ijk with F_ijk = X

1≤m≤n

(curl ˜α)_jmw_mδ_kⁱ − (curl ˜α)_imw_mδ_k^j
.

Putting together (2.3), (2.4), and using (1.3), to prove the main theorem, it suffices to consider

( ∆q + A(x) · ∇q + B(x)q + divF = 0 in Ω_t,

∆w + curl^>q = 0 in Ω_t (2.5)

with

kAk_L^∞_(Ωt)≤ C₀λat^{1−2κ1−δ}, kBk_L^∞_(Ωt)≤ C₀λa²t^{2−2κ1−δ}, and

|F (x)| ≤ C₀λa²t^{2−2κ2−δ}|w(x)|, ∀ x ∈ Ω_t.

Our proof relies on appropriate Carleman estimates. Here we need two Carleman estimates with weights ϕβ = ϕβ(x) = exp(−β ˜ψ(x)), where β > 0 and ˜ψ(x) = log |x| + log((log |x|)²).

Lemma 2.1 There exist a sufficiently small number r₁ > 0 depending on n and a sufficiently large number β₁ > 3, a positive constant C, depending on n such that for all v ∈ U_r1 and f = (f₁, · · · , f_n) ∈ (U_r1)ⁿ, β ≥ β₁, we have that

Z

ϕ²_β(log |x|)²(β|x|⁴⁻ⁿ|∇v|²+ β³|x|²⁻ⁿ|v|²)dx

≤ C Z

ϕ²_β(log |x|)⁴|x|²⁻ⁿ[(|x|²∆v + |x|divf )²+ β²kf k²]dx, (2.6) where U_r1 = {v ∈ C₀^∞(Rⁿ\ {0}) : supp(v) ⊂ B_r1}.

Lemma2.1 is a modified form of [5, Lemma 2.4]. For the sake of brevity, we omit the proof here. Replacing β of Lemma 2.1 with β + 1 and choosing f = 0 implies

Lemma 2.2 There exist a sufficiently small number r1 > 0, a sufficiently large number β₁ > 1, a positive constant C, such that for all v ∈ U_r1 and β ≥ β₁, we have

Z

ϕ²_β(log |x|)⁻²|x|⁻ⁿ(β|x|²|∇v|²+ β³|v|²)dx ≤ C Z

ϕ²_β|x|⁻ⁿ(|x|⁴|∆v|²)dx.

(2.7) In addition to Carleman estimates, we also need the following interior estimate.

Lemma 2.3 For any 0 < a₁ < a₂ such that B_a2 ⊂ Ω_t for t > 1, let X = B_a2\ ¯B_a1 and d(x) be the distant from x ∈ X to Rⁿ\X. Then we have

Z

X

d(x)²|∇w|²dx + Z

X

d(x)⁴|∇q|²dx + Z

X

d(x)²|q|²dx

≤ C

1 + a²t^−3δ2 2Z

X

|w|²dx. (2.8)

where the constant C depends on n, λ.

The proof of this lemma is given in [6].

3 Proof of Theorem 1.1

This section is devoted to the proof of the main theorem, Theorem 1.1.

Since (w, p) ∈ (H¹(Ω_t))ⁿ⁺¹, the regularity theorem implies w ∈ H_loc² (Ω_t).

Therefore, to use estimate (2.7), we simply cut-off w. So let χ(x) ∈ C₀^∞(Rⁿ) satisfy 0 ≤ χ(x) ≤ 1 and

χ(x) =







0, |x| ≤ _8at¹ ,

1, _4at¹ < |x| < ¹_a− _20at³ δ, 0, |x| ≥ ¹_a −_20at² δ. It is easy to see that for any multiindex α

(|D^αχ| = O((at)^|α|) if _8at¹ ≤ |x| ≤ _4at¹ ,

|D^αχ| = O((at^δ)^|α|) if ¹_a− _20at³ δ ≤ |x| ≤ ¹_a− _20at² δ. (3.1)

To apply Carleman estimates above, it suffices to take 1/a ≤ r₁. Now apply- ing (2.7) to χw gives

Z

(log |x|)⁻²ϕ²_β|x|⁻ⁿ(β|x|²|∇(χw)|²+ β³|χw|²)dx

≤ C Z

ϕ²_β|x|⁻ⁿ|x|⁴|∆(χw)|²dx. (3.2) Here and after, C and ˜C denote general constants whose value may vary from line to line. The dependence of C and ˜C will be specified whenever necessary. Next applying (2.6) to v = χq and f = |x|χF yields that

Z

ϕ²_β(log |x|)²(|x|⁴⁻ⁿβ|∇(χq)|² + |x|²⁻ⁿβ³|χq|²)dx

≤ C Z

ϕ²_β(log |x|)⁴|x|²⁻ⁿ[(|x|²∆(χq) + |x|div(|x|χF ))²+ β²k|x|χF k²]dx.

(3.3) Combining β×(3.2) and (3.3), we obtain that

Z

W

(log |x|)⁻²ϕ²_β|x|⁻ⁿ(β²|x|²|∇w|²+ β⁴|w|²)dx +

Z

W

(log |x|)²ϕ²_β|x|⁻ⁿ(β|x|⁴|∇q|²+ |x|²β³|q|²)dx

≤ Z

ϕ²_β(log |x|)⁻²|x|⁻ⁿ(β²|x|²∇(χw)|²+ β⁴|χw|²)dx +

Z

(log |x|)²ϕ²_β|x|⁻ⁿ(β|x|⁴|∇(χq)|²+ β³|x|²|χq|²)dx

≤ Cβ Z

ϕ²_β|x|⁻ⁿ|x|⁴|∆(χw)|²dx +C

Z

ϕ²_β(log |x|)⁴|x|²⁻ⁿ[ |x|²∆(χq) + |x|div(|x|χF )
2

+ β²k|x|χF k²]dx, (3.4) where W denotes the domain {x : _4at¹ < |x| < _a¹ − _20at³ δ}. To simplify the notations, we denote Y = {x : _8at¹ ≤ |x| ≤ _4at¹ } and Z = {x : ¹_a − _20at³ δ ≤

|x| ≤ ¹_a− _20at² δ}. By (2.4) and estimates (3.1), we deduce from (3.4) that Z

W

(log |x|)⁻²ϕ²_β|x|⁻ⁿ(β²|x|²|∇w|²+ β⁴|w|²)dx +

Z

W

(log |x|)²ϕ²_β|x|⁻ⁿ(β|x|⁴|∇q|²+ |x|²β³|q|²)dx

≤ Cβ Z

W

ϕ²_β|x|⁻ⁿ|x|⁴|∇q|²dx +Ca²t^{2−4κ1−2δ}

Z

W

(log |x|)⁴ϕ²_β|x|⁻ⁿ|x|⁶|∇q|²dx +Ca⁴t^{4−4κ1−2δ}

Z

W

(log |x|)⁴ϕ²_β|x|⁻ⁿ|x|⁶|q|²dx +Cβ²a⁴t^{4−4κ2−34δ}

Z

W

(log |x|)⁴ϕ²_β|x|⁻ⁿ|x|⁴|w|²dx +C(at)⁴β

Z

Y ∪Z

ϕ²_β|x|⁻ⁿ| ˜U |²dx +C(at)⁴β²

Z

Y ∪Z

(log |x|)⁴ϕ²_β|x|²⁻ⁿ| ˜U |²dx, (3.5) where | ˜U (x)|² = |x|⁴|∇q|² + |x|²|q|²+ |x|²|∇w|²+ |w|² and C depends on n, λ.

Now we can choose a > a₀ ≥ 8/r₁ such that (log |x|)² ≥ 2C for all x ∈ W . Then the first term on the right hand side of (3.5) can be absorbed by the left hand side of (3.5). Now, let β ≥ β2 = t^κ and choose t ≥ t0 with t0

depending on a, λ, δ such that the second term to the fourth term on the right hand side of (3.5) can be removed. With the choices described above, we obtain from (3.5) that

β⁴(b₁)⁻ⁿ(log b₁)⁻²ϕ²_β(b₁) Z

1 at<|x|<b1

|w|²dx

≤ β⁴ Z

W

(log |x|)⁻²ϕ²_β|x|⁻ⁿ|w|²dx

≤ Cβ(at)⁴ Z

Y ∪Z

(log |x|)⁴ϕ²_β|x|⁻ⁿ| ˜U |²dx

≤ Cβ²(at)⁴(log b₂)⁴b⁻ⁿ₂ ϕ²_β(b₂) Z

Y

| ˜U |²dx +Cβ²(at)⁴(log b₃)⁴b⁻ⁿ₃ ϕ²_β(b₃)

Z

Z

| ˜U |²dx, (3.6)

where b₁ = ¹_a −_20at⁸ δ, b₂ = _8at¹ and b₃ = _a¹ − _20at³ δ.

Using (2.8), we can control | ˜U |² terms on the right hand side of (3.6).

Indeed, let X = Y₁ := {x : _16at¹ ≤ |x| ≤ _2at¹ }, then we can see that d(x) ≥ C|x| for all x ∈ Y,

where C an absolute constant. Therefore, (2.8) implies Z

Y

|x|²|∇w|² + |x|⁴|∇q|²+ |x|²|q|²
dx

≤ C Z

Y1

d(x)²|∇w|²+ d(x)⁴|∇q|² + d(x)²|q|²
dx

≤ C

1 + a²t^−3δ2

2Z

Y1

|w|²dx

≤ Ca⁴ Z

Y1

|w|²dx. (3.7)

Here C depends on n, λ. On the other hand, let X = Z₁ := {x : _2a¹ ≤ |x| ≤

1

a −_20at¹ δ}, then

d(x) ≥ Ct^−δ|x| for all x ∈ Z,

where C another absolute constant. Thus, it follows from (2.8) that Z

Z

|x|²|∇w|²+ |x|⁴|∇q|²+ |x|²|q|²
dx

≤ Ct^4δ Z

Z1

d(x)²|∇w|²+ d(x)⁴|∇q|²dx + d(x)²|q|²
dx

≤ Ct^4δ

1 + a²t^−3δ2

2Z

Z1

|w|²dx

≤ C(at)⁴ Z

Z1

|w|²dx. (3.8)

Combining (3.6), (3.7), and (3.8) leads to b^−2β−n₁ (log b1)^−4β−2

Z

1

2at<|x|<b1

|w|²dx

≤ Ca⁸t⁴(log b₂)⁴b⁻ⁿ₂ ϕ²_β(b₂) Z

Y1

|w|²dx +C(at)⁸(log b₃)⁴b⁻ⁿ₃ ϕ²_β(b₃)

Z

Z1

|w|²dx.

(3.9) Notice that (3.9) holds for all β ≥ β₂.

Changing 2β + n to β, (3.9) becomes b^−β₁ (log b₁)^{−2β+2n−2}

Z

1

2at<|x|<b1

|w|²dx

≤ Ca⁸t⁴b^−β₂ (log b₂)^−2β+2n+4 Z

Y1

|w|²dx +C(at)⁸b^−β₃ (log b₃)^−2β+2n+4

Z

Z1

|w|²dx. (3.10)

Dividing b^−β₁ (log b₁)^{−2β+2n−2} on the both sides of (3.10) and noting β ≥ n + 2 > n − 1, i.e., 2β − 2n + 2 > 0, we have for t ≥ t₁ ≥ t₀ that

Z

|x+^b4x0

t |<¹

at

|w(x)|²dx

≤ Z

1

2at<|x|<b1

|w(x)|²dx

≤ Ca⁸t⁴(log(8at))⁶(b₁/b₂)^β Z

Y1

|w|²dx

+C(at)⁸(b₁/b₃)^β(log b₃)⁶[log b₁/ log b₃]^2β−2n+2 Z

Z1

|w|²dx

≤ Ca⁸t⁴(log(8at))⁶(8t)^β Z

|x|<¹

at

|w(x)|²dx

+C(at)⁸(log b₃)⁶(b₁/b₅)^β Z

Z1

|w(x)|²dx, (3.11)

where b₄ = _a¹−_at¹δ and b₅ = ¹_a−_20at⁶ δ. In deriving the third inequality above, we use the fact that

0 ≤ (b₅

b₃)(log b₁

log b₃)² = 1 − 1

2t^δlog a− 3

20t^δ + O(t^−2δ) ≤ 1

for all t ≥ t₂ ≥ t₁ and a > a₁ = max{1, a₀}, where t₂ depends on t₁, δ, and a. From now on we fix a, which depends only on n and r₁. Recall that r₁ is a function of n. Therefore, t₂ depends on n, λ, and δ. Having fixed constant a, | log b₃| can be bounded by a positive constant. Thus, (3.11) is reduced to

Z

|x+^b4x0_t |<_at¹

|w(x)|²dx ≤ Ct⁴(log t)⁶(8t)^β Z

|x|<_at¹

|w(x)|²dx

+Ct⁸(b1/b5)^β Z

Z1

|w(x)|²dx, (3.12) where C depends on n and λ.

From (3.12), (2.2), the definition of w(x), the change of variables y = atx + x₀, and x₀ = ty₀, we have that

I(t^1−δy₀) ≤ Ct⁴(log t)⁶(8t)^β Z

|y−x0|<1

|u(y)|²dy + Ct^8−3δ2

 t^δ t^δ+ ₁₀¹

^β

≤ C(8t)^β+10I(ty₀) + Ct⁸

 t^δ t^δ+₁₀¹

β

≤ C(8t)^2βI(ty₀) + Ct⁸

 t^δ t^δ+₁₀¹

^β

(3.13) provided β ≥ β₂. For simplicity, by denoting

A(t) = 2 log 8t, B(t) = log(t^δ+₁₀¹ t^δ ), (3.13) becomes

I(t^1−δy0) ≤ C n

exp(βA(t))I(ty0) + t⁸exp(−βB(t)) o

. (3.14)

Now, we consider two cases. If

exp(β₂A(t))I(ty₀) ≥ t⁸exp(−β₂B(t)),

then we have

I(x₀) = I(ty₀) ≥ t⁸exp(−β₂(A(t) + B(t))) = t⁸(8t)^−2β2 t^δ+ ₁₀¹ t^δ

^−β2 , that is

I(ty₀) ≥ t^−2β2+8 = t^−2tκ+8 ≥ exp(−2t^κlog t). (3.15) for all t ≥ t₂. Note that we have used the relation β₂ = t^κ in (3.15). On the other hand, if

exp(β₂A(t))I(ty₀) < t⁸exp(−β₂B(t)), then we can pick a ˜β > β₂ such that

exp( ˜βA(t))I(ty₀) = t⁸exp(− ˜βB(t)). (3.16) Solving ˜β from (3.16) and using (3.14), we have that

I(t^1−δy₀) ≤ C exp( ˜βA(t))I(ty₀)

= C (I(ty0))^τ(t⁸)^1−τ

≤ Ct⁸(I(ty₀))^τ, (3.17) where τ = _A(t)+B(t)^B(t) .

It is time to prove Theorem 1.1. Let |x₀| = t for t ≥ t

1 1−δ

2 and y₀ = ^x_t⁰, then we can write

t = µ(^(1−δ)−s) (3.18)

for some positive integer s and t2 ≤ µ < t

1 1−δ

2 ≤ t²₂. For simplicity, we define d_j = µ(^(1−δ)−j) and τ_j = _A(d^B(dj)

j)+B(dj) for j = 1, 2 · · · s. Define

J = {1 ≤ j ≤ s : exp(d^κ_jA(d_j))I(d^jy₀) ≥ d⁸_jexp(−d^κ_jB(d_j))}.

Now, we divide it into two cases. If J = ∅, we only need to consider (3.17).

Using (3.17) iteratively starting from t = d1, we have that I(µy₀) ≤ C(d⁸₁) (I(d₁y₀))^τ1

≤ C^s(d₁d₂· · · d_s)⁸(I(x₀))^{τ1τ2···τs}. (3.19) By (3.18) and (3.19), we obtain that

I(µy₀) ≤ C(log log t/| log(1−δ)|)t^8/δ(I(x₀))^{τ1τ2···τs}

≤ t^C˜0/δ(I(x₀))^{τ1τ2···τs}, (3.20)

where ˜C₀ depends on λ, n. It is easily to see that 1

τ_j = 2 log(8d_j) + log(1 + 0.1d^−δ_j )

log(1 + 0.1d^−δ_j ) ≤ 4 log(8dj) log(1 + 0.1d^−δ_j ). and thus

1 τ1τ2· · · τs

≤ 4^slog(8^sd₁· · · d_s) log(1 + (0.1)^s(d1· · · ds)^−δ)

≤ 2(40)^st^{1−(1−δ)s}log(8^st^{(1−(1−δ)s)/δ})

≤ 2(40)(log log t/| log(1−δ)|)

t(log t)/δ +2(40)(log log t/| log(1−δ)|)

t(log log t/| log(1 − δ)|) log 8

≤ 2(log t)4/| log(1−δ)|t(log t)/δ

+6(log t)4/| log(1−δ)|t(log log t/| log(1 − δ)|)

≤ ( ˜C₁/δ)t(log t)^C˜1/δ, (3.21) where ˜C1 is an absolute constant. Raising both sides of (3.20) to the power

1

τ1τ2···τs and using (3.21), we obtain that (min{I(µy₀), 1})^{( ˜C1}/δ)t(log t)^C1/δ˜

≤ I(µy₀)^{τ1τ2···τs1}

≤ e^{( ˜C2/δ2)t(log t)C2/δ˜} (I(x₀)) , (3.22) where ˜C₂ depends on n, λ.

Next, if J 6= ∅, let l be the largest integer in J . Then from (3.15) we have I(d_ly₀) ≥ d^−2d_l ^κl+8. (3.23) Iterating (3.17) starting from t = dl+1 yields

I(d_ly₀) ≤ C^s−l(d_l+1· · · d_s)⁸(I(x₀))^{τl+1···τs}

≤ C(log log t/| log(1−δ)|)(t/d_l)^8/δ(I(x₀))^{τl+1···τs}

≤ t^C˜0/δ(I(x₀))^{τl+1···τs}. (3.24) It is enough to assume I(d_ly₀) < 1. Repeating the computations in (3.21), we can see that

1

τ_l+1· · · τ_s ≤ ( ˜C1/δ)(t/dl)(log t)^C˜1/δ ≤ ( ˜C1/δ)(t/dl)^κ(log t)^C˜1/δ. (3.25)

Hence, combining (3.23), (3.24) and using (3.25), we get that

t^{−( ˜C3/δ)tκ(log t)C3/δ˜} ≤ e^{( ˜C2/δ2)t(log t)C2/δ˜} (I(x₀)) , (3.26) where ˜C₃ is an absolute constant. The proof is complete in view of (3.15), (3.22) and (3.26).

2

Acknowledgements

The authors were supported in part by the National Science Council of Tai- wan.

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