This paper is concerned with optimal lower bounds of decay rates for solutions to the Navier-Stokes equations inÊn

126 (2001) MATHEMATICA BOHEMICA No. 2, 443–455

ON OPTIMAL DECAY RATES FOR WEAK SOLUTIONS TO THE NAVIER-STOKES EQUATIONS IN ^Ên

Tetsuro Miyakawa, Rokko,Maria Elena Schonbek, Santa Cruz

Dedicated to Professor Jindřich Nečas on his 70th birthday

Abstract. This paper is concerned with optimal lower bounds of decay rates for solutions to the Navier-Stokes equations in^Ên. Necessary and sufficient conditions are given such that the corresponding Navier-Stokes solutions are shown to satisfy the algebraic bound

u(t)(t+ 1)⁻ⁿ⁺⁴² .

Keywords: decay rates, Navier-Stokes equations MSC 2000: 35Q10

1. Introduction and the results

Consider the Navier-Stokes equations in ^Ên,n2, which will be treated in this paper in the form of the integral equation

(NS) u(t) = e^−tAa−

_t

0 ∇ ·e^−(t−s)AP(u⊗u)(s) ds,

for prescribed initial velocitya(x) = (a₁(x), . . . , a_n(x)),x= (x₁, . . . , x_n)∈^Ên, and unknown velocity u(x, t) = (u₁(x, t), . . . , u_n(x, t)). Here, A =−∆ is the Laplacian on^Ên; {e^−tA}_t0 is the heat semigroup;P = (P_jk) is the bounded projection onto divergence-free vector fields; u⊗v is the matrix with entries (u⊗v)_jk = u_jv_k;

∇= (∂₁, . . . , ∂_n) with∂_j=∂/∂x_j; and

(∇ ·e^−tAP(u⊗u))_j= n k,=1

∂e^−tAP_jk(uu_k), j = 1, . . . , n.

It is well known that for each a ∈ L² with∇ ·a = 0, (NS) has a weak solutionu defined for allt0, satisfying the energy inequality

u(t)²₂+ 2 _t

0 ∇u²₂dsa²₂ for allt0.

Hereafter · r denotes theL^r-norm.

As shown in [10], there exists a weak solutionusuch that (1.1) u(t)2C(1 +t)⁻ⁿ⁺²⁴ , whenever

(1.2) a∈L², ∇ ·a= 0 and

(1 +|y|)|a(y)|dy <∞.

Assumption (1.2) impliesa∈L¹; so the divergence-free condition gives (see [4]) (1.3)

a(y) dy= 0.

Furthermore, it is shown in [2] that in this case the solutionusatisfies

(1.4)

t→∞lim tⁿ⁺²⁴ u_j(t) + (∂_kE_t)(·)

y_ka_j(y) dy +F_,jk(·, t)

_∞

0

(uu_k)(y, s) dyds

2= 0 forj= 1, . . . , n, where

E_t(x) = (4t)^−n/2e^−|x|2/4t, F_,jk(x, t) =∂E_t(x)δ_jk+ _∞

t

∂∂_j∂_kE_s(x) ds.

(Hereafter, we use the summation convention). Equation (NS) is then written in the form

u_j(x, t) =

E_t(x−y)a_j(y) dy− _t

0

F_,jk(x−y, t−s)(uu_k)(y, s) dyds, j= 1, . . . , n,

as proved in [2]; and the integrals in (1.4) are finite, due to (1.1) and (1.2). Assertion (1.4) was first proved in [1] for smooth solutions when n = 3, and then extended in [2] to the case of weak solutions in all space dimensions by applying the spectral method as given in [3, 5].

The argument of [10] suggests that the decay property (1.1) will be optimal in general. So we are interested in finding a class of weak solutions u satisfying the reverse estimate

u(t)₂Ct⁻ⁿ⁺²⁴ at least for larget.

In this paper we discuss this kind oflower bound problem. Theorem A. Under the assumption(1.2), let

b_k =

ya_k(y) dy, c_k = _∞

0

(uu_k)(y, s) dyds.

(i)We have

(1.5) lim

t→∞tⁿ⁺²⁴ u(t)2= 0

if and only if(b_k) = 0and(c_k) = (cδ_k)for some constantc0.

(ii)There existsc>0 such that

(1.6) u(t)2ct⁻ⁿ⁺²⁴ for larget >0,

if and only if (b_k) = 0 or (c_k)= (cδ_k). In particular, u satisfies (1.6) whenever (b_k)= 0.

. Theorem A (i) implies only that

(1.5) lim sup

t→∞ tⁿ⁺²⁴ u(t)₂>0

if and only if (b_k)= 0 or (c_k)= (cδ_k). Note, however, that our second assertion (1.6) is more stringent than (1.5). Moreover, (1.6) holds for all large t > 0 and forall space dimensions, althoughu(t)2is only known to be lower semicontinuous when n 3. We know nothing about the characterization of solutions satisfying (c_k) = (cδ_k).

We next consider weak solutionsusatisfying

(1.7) u(t)₂C(1 +t)⁻ⁿ⁴.

As shown in [3, 6, 10], such solutions exist for alla∈L² satisfying (1.8) ∇ ·a= 0, e^−tAa2C(1 +t)⁻ⁿ⁴.

Theorem B. Suppose a satisfies (1.8) and let u be a weak solution satisfying (1.7). Then

(1.9) u(t)₂ct⁻ⁿ⁴ for larget >0, if and only if

(1.10) e^−tAa₂ct⁻ⁿ⁴ for larget >0.

The lemma below gives simple examples ofasatisfying (1.10).

Lemma. Leta∈L²,∇ ·a= 0, and suppose that (1.11)

Sⁿ⁻¹|a(r, ω)ˆ |²dω∈L^∞(^Ê₊), lim inf

r→0

Sⁿ⁻¹|ˆa(r, ω)|²dω >0, where the Fourier transformaˆis defined by

ˆa(ξ) =

e^−ix·ξa(x) dx, i =√

−1,

Sⁿ⁻¹ is the unit sphere of^Ên, and ξ= (r, ω)in polar coordinates. Then,

(1.12) e^−tAa₂C(1 +t)⁻ⁿ⁴ for allt >0; e^−tAa₂ct⁻ⁿ⁴ for larget >0, with constantsC >0andc>0independent oft.

. Parseval’s relation gives e^−tAa²₂= (2)⁻ⁿ

e^−2t|ξ|2|ˆa(ξ)|²dξ= (8²t)⁻ⁿ²

e^−|η|2|ˆa(η(2t)⁻¹²)|²dη so that

(8²t)ⁿ²e^−tAa²₂=

e^−|η|2|ˆa(η(2t)⁻¹²)|²dη.

The assumption and Fatou’s lemma together imply lim inf

t→∞ (8²t)ⁿ²e^−tAa²₂= lim inf

t→∞

e^−|η|2|a(η(2t)ˆ ⁻¹²)|²dη

_∞

0

e^−r2

lim inf

t→∞

Sⁿ⁻¹|ˆa(r(2t)⁻¹², ω)|²dω

rⁿ⁻¹dr >0.

This proves the second estimate of (1.12). The first estimate follows frome^−tAa2 a2 and

e^−tAa²₂= (8²t)⁻ⁿ²

e^−|η|2|ˆa(η(2t)⁻¹²)|²dη Ct⁻ⁿ²

Sⁿ⁻¹|ˆa(·, ω)|²dω

∞

_∞

0

e^−r2rⁿ⁻¹dr.

The proof is complete.

. (i) Condition (1.11) implies that ˆais discontinuous atξ= 0. Indeed, since∇ ·a= 0, we haveξ·a(ξ) = 0; so if ˆˆ ais continuous atξ= 0, we getω·ˆa(0) = 0 for all unit vectorsω, and ˆa(0) = 0. (For this reason,a∈L¹ implies (1.3)).

(ii) The assumption of Lemma is not vacuous. Indeed, suppose ˆais written in the form

ˆa(ξ) =f(|ξ|)g(ξ/|ξ|), in terms of functionsf(r) andg(ω) such that

g∈L²(Sⁿ⁻¹), g≡0, ω·g(ω)≡0 (ω∈Sⁿ⁻¹) and

f ∈BC([0,∞)), _∞

0 |f(r)|²rⁿ⁻¹dr <∞, f(0)= 0.

Then, ˆasatisfies condition (1.11).

(iii) In this connection, we note that under condition (1.2) we have (1.10) e^−tAa2ct⁻ⁿ⁺²⁴ for larget >0

if and only if (b_k)= 0. Indeed, using (1.2) and (1.3), we have (see Section 4)

(1.4) lim

t→∞tⁿ⁺²⁴ e^−tAa_k+∂E_tb_k2= 0, k= 1, . . . , n.

Suppose (b_k)= 0. Then (

k ∂E_tb_k²₂)^1/2=Ct⁻ⁿ⁺²⁴ withC >0; so we get e^−tAa2

k ∂E_tb_k²_21/2

− k e^−tAa_k+∂E_tb_k²_21/2

ct⁻ⁿ⁺²⁴

for larget >0. Conversely, if we assume (1.10), then (1.4) implies

k ∂E_tb_k²_21/2

e^−tAa₂−

k e^−tAa_k+∂E_tb_k²_21/2

ct⁻ⁿ⁺²4

for larget >0. Hence

k ∂E_tb_k²₂>0 for larget >0, which implies (b_k)= 0.

TheL²decay problem for weak solutions of the Navier-Stokes equations was suc- cessfully studied for the first time by [5] and the result was then systematically devel- oped by [3, 6, 10]. Estimates (1.6) and (1.9) are studied in [6]–[9] in casen= 2,3, and some sufficient conditions are obtained. Our Theorems A and B providenecessary and sufficient conditions for those estimates to hold. We further note that our lower bound estimates (1.6) and (1.9) hold in all space dimensions n 2, although the

functionu(t)2 is known only to be lower semicontinuous whenn3. As will be seen in the proof below, this is due to (1.4) and the fact that the functions∂E_t(x) and F_,jk(x, t) are written in the form t⁻ⁿ⁺¹2 K(xt⁻¹2) in terms of some bounded, integrable and uniformly continuous functionsK.

We finally consider an example of two-dimensional flowsuwith (b_k) = 0, (c_k) = (cδ_k), which was first treated by [7].

Theorem C. Whenn= 2, there is a smooth weak solutionusuch that(b_k) = 0, (c_k) = (cδ_k), and, with some constantγ >0,

(1.13) u(t)qC_qe^−γt and |u(x, t)|C_me^−γt(1 +|x|)^−m for all1q∞and all integersm0.

The above example was studied by [7, 8, 9], in which is given the exponential decay ofu(t)q for 2q∞. Our estimates (1.13) include the case 1q <2 as well as the decay estimates in the spatial direction. Theorem C is proved in [2].

In what follows we prove Theorems A and B, and conclude the paper with the proof of (1.4) which was given also in [2].

2. Proof of Theorem A

We begin with the following

Proposition 2.1. Let (b_k) and (c_k) be real n×n matrices and let (c_k) be symmetric. Then

(2.1) b_k∂E_t(x)δ_jk+c_kF_,jk(x, t) = 0, j= 1, . . . , n, for allx∈^Ên and for somet >0, if and only if

(2.2) (b_k) = 0 and (c_k) = (cδ_k) for somec∈^Ê.

Furthermore,(2.2) implies that(2.1)holds for allxand for allt >0.

. Assumption (2.1) implies, via the Fourier transformation, b_kξe^−t|ξ|2δ_jk=−c_kξ

e^−t|ξ|2δ_jk−ξ_jξ_{k ∞}

t

e^−s|ξ|2ds

=−(c_j− |ξ|⁻²c_kξ_jξ_k)ξe^−t|ξ|2

for somet >0, and we get|ξ|²(b_j+c_j)ξ=ξ_jc_kξ_kξ. Takingξ_j = 0 for any fixed j, ξ = 1 for any fixed =j, and ξ_k = 0 for all k such that k =j andk =, we easily obtainb_j+c_j = 0 wheneverj =, and so

|ξ|²(b_jj+c_jj)ξ_j=ξ_jc_kξ_kξ, j= 1, . . . , n.

We letξ_j= 1 and ξ_k = 0 fork=j, to getb_jj+c_jj =c_jj; sob_jj = 0. This implies (2.3) |ξ|²c_jjξ_j=ξ_jc_kξ_kξ, j = 1, . . . , n.

Hence,c₁₁=. . .=c_nn =c_kξ_kξ|ξ|⁻². We then set j = 1, ξ₁ =ξ₂ = 1 andξ_k = 0 fork3 in (2.3), to get 2c₁₁=c₁₁+c₂₂+c₁₂+c₂₁= 2(c₁₁+c₁₂) sincec_k =c_k by assumption. Therefore,c₁₂ = 0. We thus obtainc_j = 0 =−b_j wheneverj =; so (b_k) = 0 and (c_k) = (cδ_k). That (2.2) implies (2.1) forall t >0 is easily seen from

F_k,jk=∂_jE_t+ _∞

t

∂_j∆E_sds=∂_jE_t+ _∞

t

∂_j∂_sE_sds=∂_jE_t−∂_jE_t= 0, where∂_s=∂/∂s. The proof of Proposition 2.1 is complete.

To establish Theorem A, it suffices in view of (1.4) to prove the following Proposition 2.2. Letasatisfy(1.2)and define

b_k=

ya_k(y) dy, c_k= _∞

0

(uu_k)(y, s) dyds.

Then we have

(2.4) either (b_k)= 0 or (c_k)= (cδ_k), if and only if a corresponding weak solutionusatisfies (2.5) u(t)2ct⁻ⁿ⁺²⁴ for larget >0 with a constantc >0 indenpendent oft.

. In what follows we write

b= (b₁, . . . , b_n), F,k= (F_,1k, . . . , F_,nk).

Assume first (2.4). By Proposition 2.1, we have ∂E_tb+F,kc_k2 =Ct⁻ⁿ⁺²⁴ for allt >0 with someC >0, and so (1.4) implies

u(t)2∂E_tb+F,kc_k2− u(t) +∂E_tb+F,kc_k2

=Ct⁻ⁿ⁺²⁴ −o(t⁻ⁿ⁺²⁴ )ct⁻ⁿ⁺²⁴

for larget >0. Assume next (2.5). By (1.4) we have

∂E_tb+F,kc_k2u(t)2− u(t) +∂E_tb+F,kc_k2ct⁻ⁿ⁺²⁴ −o(t⁻ⁿ⁺²⁴ ), and so

∂E_tb+F,kc_k2>0 for larget >0.

We thus obtain (2.4) by Proposition 2.1. This proves Proposition 2.2.

3. Proof of Theorem B

Suppose thatn3. We have c_k =

_∞

0

(uu_k)(y, s) dyds <∞;

so the argument given in [2, Sect. 5] applies to our present situation, implying

(3.1) lim

t→∞tⁿ⁺²⁴ u(t)−e^−tAa+F_,kc_k2= 0.

Suppose (1.9) holds. SinceF,kc_k2=Ct⁻ⁿ⁺²⁴ , it follows from (3.1) that e^−tAa₂u(t)₂− −u(t) + e^−tAa−F_,kc_k+F_,kc_k2

u(t)2− u(t)−e^−tAa+F,kc_k2− F,kc_k2

ct⁻ⁿ⁴ −Ct⁻ⁿ⁺²⁴ ct⁻ⁿ⁴

for larget >0. This proves (1.10). Conversely, if (1.10) holds, then (3.1) implies u(t)2e^−tAa2− F,kc_k2− u(t)−e^−tAa+F,kc_k2

ct⁻ⁿ⁴ −Ct⁻ⁿ⁺²⁴ ct⁻ⁿ⁴ for larget >0. This proves (1.9) in casen3.

Whenn= 2, we introduce c_k(t) =

_t/2

0

(uu_k)(y, s) dyds

instead ofc_k. The argument of [2, Sect. 5] is then modified to yield (3.1) u(t)−e^−tAa+F,kc_k(t)2Ct⁻¹log(1 +t).

See also Section 4 below. Since F_,kc_k(t)₂Ct⁻¹

_t/2

0 u(s)²₂dsCt⁻¹log(1 +t),

this implies u(t)−e^−tAa2Ct⁻¹log(1 +t). Now we can prove the result in the same way as in the casen3. Indeed, (1.10) implies

u(t)2e^−tAa2− u(t)−e^−tAa2ct⁻¹² −Ct⁻¹log(1 +t)ct⁻¹² for larget >0, while (1.9) yields

e^−tAa₂u(t)₂− u(t)e^−tA₂ct⁻¹² −Ct⁻¹log(1 +t)ct⁻¹² for larget >0. The proof of Theorem B is complete.

4. Proof of (1.4)

Here we present the proof of (1.4) given in [2]. The same method can be applied to the proof of (3.1) and (3.1) with no essential change. Leta satisfy (1.2) and so (1.3). We first prove

(4.1) lim

t→∞tⁿ⁺²⁴ e^−tAa+ (∂_kE_t)(·)

y_ka(y) dy

2= 0.

Direct calculation gives e^−tAa=

[E_t(x−y)−E_t(x)]a(y) dy=− ¹

0

(∂_kE_t)(x−yθ)y_ka(y) dθdy

=−(∂_kE_t)(x)

y_ka(y) dy− ¹

0

[(∂_kE_t)(x−yθ)−(∂_kE_t)(x)]y_ka(y) dθdy, so

e^−tAa+ (∂_kE_t)(x)

y_ka(y) dy=− ¹

0

[(∂_kE_t)(x−yθ)−(∂_kE_t)(x)]y_ka(y) dθdy.

We can write (∂_kE_t)(x) =t⁻ⁿ⁺¹² (∂_kE₁)(xt⁻¹²), to obtain e^−tAa+ (∂_kE_t)(·)

y_ka(y) dy

2Ct^{−n+24 1}

0

ϕ_t(y, θ)|y||a(y)|dθdy.

Here ϕ_t(y, θ) = (∇E₁)(· −yθt⁻¹²)−(∇E₁)(·)2 is bounded and lim

t→∞ϕ_t(y, θ) = 0 for any fixed (y, θ). Since|y||a(y)|is integrable by (1.2), the dominated convergence theorem yields

t→∞lim

1

0 ϕ_t(y, θ)|y||a(y)|dθdy= 0.

This proves (4.1). Now let usatisfy (1.1). We next show that the function w(t) =u(t)−e^−tAa=−

_t

0

F,k(x−y, t−s)(uu_k)(y, s) dyds satisfies

(4.2) lim

t→∞tⁿ⁺²⁴ w(t) +F,k(·, t) _∞

0

(uu_k)(y, s) dyds

2= 0.

Indeed, we have w(t) +F,k(x, t)

_∞

0

(uu_k)(y, s) dyds

=F,k(x, t) _∞

t/2

(uu_k)(y, s) dyds

− _t/2

0

[F_,k(x−y, t−s)−F_,k(x, t−s)](uu_k)(y, s) dyds

− _t/2

0

[F_,k(x, t−s)−F_,k(x, t)](uu_k)(y, s) dyds

− _t

t/2

F,k(x−y, t−s)(uu_k)(y, s) dyds

≡I₁+I₂+I₃+I₄. It is easy to see that

(4.3) tⁿ⁺²⁴ I₁₂C _∞

t/2

(1 +s)⁻¹⁻ⁿ² ds→0 ast→ ∞. We writeI₃ in the form

I₃= _t/2

0 1 0

s(∂_tF,k)(x, t−sθ)(uu_k)(y, s) dθdyds to get

I₃₂C _t/2

0 1 0

s(t−sθ)^−1−n+24 |u(y, s)|²dθdyds Ct^−1−n+24

_t/2

0

su(s)²₂ds

and so

(4.4) tⁿ⁺²⁴ I₃2Ct⁻¹ _t

0

(1 +s)⁻ⁿ² ds→0 as t→ ∞.

To estimateI₂, note that we can writeF,k(x, t) =t⁻ⁿ⁺¹² K(xt⁻¹²), to get I₂2Ct⁻ⁿ⁺²⁴

_t/2

0

K(· −y(t−s)⁻¹²)−K(·)2|u(y, s)|²dyds

≡Ct⁻ⁿ⁺²⁴ _t/2

0

ϕ_t(y, s)|u(y, s)|²dyds≡Ct⁻ⁿ⁺²⁴ _t/2

0 ψ_t(s) ds.

Sinceψ_t(s)Cu(s)²₂, the dominated convergence theorem implies

t→∞lim _M

0

ψ_t(s) ds= 0 for any fixedM >0.

Givenε >0, chooseM >0 so that _M^∞u(s)²₂ds < ε. Then fort >2M, _t/2

0

ψ_t(s) ds _M

0

ψ_t(s) ds+C _∞

M u(s)²₂ds _M

0

ψ_t(s) ds+Cε.

This implies that

(4.5) lim

t→∞tⁿ⁺²⁴ I₂2= 0.

It remains to prove

(4.6) lim

t→∞tⁿ⁺²⁴ I₄2= 0.

To do so, we follow the arguments of [3, 5]. The function v(t) =−

_t

τ

F_,k(x−y, t−s)(uu_k)(y, s) dyds=u(t)−e^−(t−τ)Au(τ) defined fortτ >0 satisfies

∂_tv+Av=−P(u· ∇u) (t > τ), v(τ) = 0.

(We may assumev is smooth, replacinguby the approximate solutionsu_N given in [3]). Since (P(u·∇v), v) = (u·∇v, v) = 0, the standard energy integral method gives

∂_tv²₂+ 2A^1/2v²₂=−2(u· ∇u, v) = 2(u· ∇v, u) = 2(u· ∇v, u₀)

and

2|(u· ∇v, u₀)|2u2A^1/2v2u₀∞Cu2A^1/2v2(t−τ)⁻ⁿ⁴τ⁻ⁿ⁺²⁴

CA^1/2v2(t−τ)⁻ⁿ⁺¹² τ⁻ⁿ⁺²⁴ A^1/2v²₂+C(t−τ)⁻ⁿ⁻¹τ⁻¹⁻ⁿ², whereu₀(t) = e^−(t−τ)Au(τ). We thus obtain

∂_tv²₂+A^1/2v²₂C(t−τ)⁻ⁿ⁻¹τ⁻¹⁻ⁿ².

Let{E_λ}λ0 be the spectral measure associated to A. SinceA^1/2v²₂ (v²₂− Ev²₂) for any >0, the above estimate yields

∂_tv²₂+v²₂Ev²₂+C(t−τ)⁻ⁿ⁻¹τ⁻¹⁻ⁿ².

But,Ev²₂C^{n+22 t}

τ u²₂ds₂

as shown in [3, 5]; so

∂_tv²₂+v²₂C^{n+42 t}

τ u²₂ds₂

+C(t−τ)⁻ⁿ⁻¹τ⁻¹⁻ⁿ².

Here we set=m/(t−τ),m >0, and multiply both sides by (t−τ)^m, to obtain

∂_t((t−τ)^mv²₂)C_m(t−τ)^{m−n2−2 t}

τ u²₂ds₂

+C(t−τ)^m−n−1τ⁻¹⁻ⁿ². Now fixm so thatm > n/2 + 2 andm > n+ 1, and integrate the above inequality, to get

v(t)²₂C(t−τ)⁻²⁻ⁿ² _t

τ s

τ u²₂dσ₂

ds+C(t−τ)⁻ⁿτ⁻¹⁻ⁿ².

Insertingτ=t/2 yieldsv(t) =I₄, so tⁿ⁺ⁿ²I₄²₂Ct^{n−1 ∞}

t/2u²₂ds₂

+Ct⁻¹Ct⁻¹→0

ast→ ∞. This proves (4.6).

. We are grateful to Professor Jindřich Nečas for valuable discussions on the problem treated in this paper. We also thank Dr. Kenji Nakanishi for showing us the examples given in Lemma. The first author is partially supported by Grant-in-Aid for Scientific Research, No. 10440045, from the Japan Ministry of Education.

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[7] M. E. Schonbek: Lower bounds of rates of decay for solutions to the Navier-Stokes equa- tions. J. Amer. Math. Soc.4(1991), 423–449.

[8] M. E. Schonbek: Asymptotic behavior of solutions to the three-dimensional Navier-Sto- kes equations. Indiana Univ. Math. J.41(1992), 809–823.

[9] M. E. Schonbek: On decay of solutions to the Navier-Stokes equations. Applied Nonlinear Analysis (A. Sequeira, H. Beirao da Veiga, J. H. Videman, eds.). Kluwer/Plenum, New York, 1999, pp. 505–512.

[10] M. Wiegner: Decay results for weak solutions of the Navier-Stokes equations in^Ên. J.

London Math. Soc.35(1987), 303–313.

Authors’ addresses: Tetsuro Miyakawa, Department of Mathematics, Kobe University, Rokko, Kobe 657-8501, Japan, e-mail:[email protected]; Maria Elena Schon- bek, Department of Mathematics, University of California, Santa Cruz, CA 95064, USA, e-mail:[email protected].