3 Global existence and uniqueness in the α > 1/2 case

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Dissipative quasi-geostrophic equations with L ^p data ^∗

Jiahong Wu

Abstract

We seek solutions of the initial value problem for the 2D dissipative quasi-geostrophic (QG) equation withL^pinitial data. The 2D dissipative QG equation is a two dimensional model of the 3D incompressible Navier- Stokes equations. We prove global existence and uniqueness of regular solutions for the dissipative QG equation with sub-critical powers. For the QG equation with critical or super-critical powers, we establish explicit globalL^p bounds for its solutions and conclude that any possible finite time singularity must occur in the first order derivative.

1 Introduction

We study in this paper the 2D dissipative quasi-geostrophic (QG) equation

∂tθ+u· ∇θ+κ(−∆)^αθ=f, x∈R², t >0, (1.1) where κ >0 is the diffusivity coefficient, α ∈ [0,1] is a fractional power, and u= (u1, u2) is the velocity field determined from θby a stream functionψvia the auxiliary relations

u= (u1, u2) =

−∂ψ

∂x2

, ∂ψ

∂x1

and (−∆)^1/2ψ=−θ. (1.2) A fractional power of the Laplacian (−∆)^β is defined by

(−\∆)^βf(ξ) = (2π|ξ|)^2βfb(ξ),

wherefbdenotes the Fourier transform off. One may consult the book of Stein [6, p.117] for more details. For notational convenience, we will denote (−∆)^1/2 by Λ. The relation in (1.2) can then be identified with

u= ∂x₂Λ⁻¹θ,−∂x₁Λ⁻¹θ

= (−R2θ,R1θ),

∗Mathematics Subject Classifications: 35Q35, 76U05, 86A10.

Key words: 2D quasi-geostrophic equation, initial-value problem, existence, uniqueness.

2001 Southwest Texas State University.c

Submitted June 18, 2001. Published August 3, 2001.

1

whereR1 andR2 are the Riesz transforms [6, p.57].

Equation (1.1) is the dissipative version of the inviscid QG equation derived by reducing the general QG models describing atmospheric and oceanic fluid flow under special circumstances of physical interest ([4],[1]). Physically, the scalar θrepresents the potential temperature,uis the fluid velocity andψcan be identified with the pressure. Mathematically, the 2D QG equation serves as a lower dimensional model of the 3D Navier-Stokes equations because of the striking similarity between the behavior of its solution and that of the potentially singular solutions of the 3D hydrodynamic equations.

Our aim of this paper is to establish global existence and uniqueness results for the initial-value problem (IVP) for the QG equation (1.1) with the initial condition

θ(x,0) =θ0(x), x∈R². (1.3) We seek solutions of the IVP (1.1) and (1.3) inL^q([0, T];L^p) for initial data θ₀ ∈ L^r(R²). The notation L^r is standard while L^q([0, T];L^p) stands for the space of functionsf ofxandtsatisfying

kfkL^q([0,T];L^p)=Z T 0

Z

R²

|f(x, t)|^pdxq/p

dt^1/q

<∞.

We distinguish between two cases: α >1/2 (the “sub-critical” case) andα≤1/2 (the “critical” or “ super-critical” case). In theα >1/2 case, we establish that the IVP (1.1) and (1.3) has a unique global (in time) and regular solution in L^q([0, T];L^p). Precise statements are presented in Section 3. It is not clear in theα≤ ¹₂ case whether regular solutions develop finite time singularities. But We show in Section 4 that any singularity must occur in the first derivative if there is a singularity. This is achieved by obtaining explicit L^p bounds for all high order derivatives of any function solving the IVP (1.1) and (1.3).

In preparation, we provide in Section 2 properties of the solution operator for the linear QG equation and show its boundedness when acting onL^p spaces.

2 The solution operator for the linear equation

Consider the solution operator for the linear QG equation

∂tθ+κΛ^2αθ= 0, x∈R², t >0,

whereκ >0, Λ denotes (−∆)^1/2 andα∈[0,1]. For a given initial data θ₀, the solution of this equation is given by

θ=G_α(t)θ₀=e^−κΛ2αtθ₀,

whereGα(t)≡e^−κΛ2αtis a convolution operator with its kernelgαbeing defined through the Fourier transform

gcα(ξ, t) =e^{−κ|ξ|2αt}.

The kernel gα possesses similar properties as the heat kernel does. For example, for α∈[0,1] and t >0, gα(x, t) is a nonnegative and non-increasing radial function, and satisfies the dilation relation

g_α(x, t) =t^−1/αg_α(xt^−1/(2α),1). (2.1) Furthermore, the operatorsGαand∇Gαare bounded onL^p. To prove this fact, we need the following lemma.

Lemma 2.1 Fort >0,kg_α(·, t)kL¹ = 1and for1≤p <∞

|gα(·, t)∗f|^p≤gα(·, t)∗ |f|^p.

Proof. For anyt >0,kg_α(·, t)kL¹ =cg_α(0, t) = 1. By H¨older’s inequality,

|gα(·, t)∗f|^p = Z

R²

g^1/q_α (x−y, t)·g_α^1/p(x−y, t)f(y)dy

p

≤ kg_α(·, t)k

p q

L¹

Z

R²

g_α(x−y, t)|f(y)|^pdy=g_α(·, t)∗ |f|^p, where (1/q) + (1/p) = 1.

Proposition 2.2 Let 1≤p≤q≤ ∞. For any t >0, the operatorsGα(t)and

∇Gα(t) are bounded operators from L^p to L^q. Furthermore, we have for any f ∈L^p,

kGα(t)fkL^q≤Ct^−1α(¹_p−¹q)kfkL^p (2.2) and

k∇Gα(t)fkL^q ≤Ct⁻(_2α¹⁺_α¹(_p¹−¹q))kfkL^p (2.3) where C is a constant depending onα,pandqonly.

Proof. We first prove (2.2). Forp=q=∞, we have kGα(t)fkL^∞≤ kgα(·, t)kL¹kfkL^∞ =kfkL^∞.

Forp=q <∞, we combine Lemma 2.1 and Young’s inequality to obtain kGα(t)fk^p_Lp = kgα(·, t)∗fk^p_Lp≤

Z

R²

gα(·, t)∗ |f|^pdx

≤ kg_α(·, t)kL¹kfk^pL^p=kfk^pL^p

To prove the general case, we first estimate kGα(t)fkL^∞. Without loss of gen- erality, we considerGα(t)f atx= 0.

|(Gα(t)f)(0)|^p ≤ Z

R²

gα(|x|, t)|f(x)|^pdx= Z _∞

0

gα(ρ, t)dr(ρ)

≤ Z ∞

0

|g⁰_α(ρ, t)|r(ρ)dρ≤ kfk^pL^p· Z ∞

0

|g⁰_κ(ρ, t)|dρ (2.4)

where r(ρ) =R

|y|≤ρ|f(y)|^pdy and g⁰_α = ^∂g_∂ρ^α. Using (2.1), one easily sees that for some constantC

Z ∞ 0

|g_κ⁰(ρ, t)|dρ=C t^−1/α and therefore (2.4) becomes (sincex= 0 is not special !)

kG_α(t)fkL^∞ ≤C t^−pα1 kfkL^p.

We now estimatekG_α(t)fkL^q in terms ofkfkL^p for 1≤p≤q <∞. kG_α(t)fk^q_Lq≤CkG_α(t)fk^q_L⁻∞^pkG_α(t)fk^p_Lp≤Ct^{−pα1 (q−p)}kfk^q_L⁻p^p· kfk^p_Lp. That is,kGα(t)fkL^q ≤Ct^−α1(¹p−q¹)kfkL^p.

Estimate (2.3) can be proved similarly by using the identity

∂_xg_α(x, t) =t^−1/(2α)˜g_α(x, t)

where ˜g_α is another radial function enjoying the same properties asg_αdoes.

The following lemma provides point-wise bounds for ∇gα.

Lemma 2.3 Let α∈(0,1]. Then for anyx∈R²\ {0},t >0,j= 1 or2,

|∂_xjg_α(x, t)| ≤











C

|x|tα¹

,

C

|x|²t2α¹

,

C

|x|³t,

(2.5)

whereC is an explicit constant depending on αonly.

Proof. Consider the Fourier transform ofF(x, t) =x_i∂_xjg_α(x, t):

F(ξ, t)b = i ∂

∂ξi

(iξjcgα(ξ, t)) = (−1) ∂

∂ξi

ξje^{−κ|ξ|2αt}

= −δ_ij+ 2κα t ξ_iξ_j|ξ|^2α−2

e^{−κ|ξ|2αt}, whereδ_ij is the Kronecker delta. Therefore, forx∈R²andt >0,

|xi∂x_jgα(x, t)| = |F(x, t)| ≤ kFb(·, t)kL¹ ≤ Z

R²

(1 + 2κα|ξ|^2αt)e^{−κ|ξ|2αt}dξ

= 2π Z ∞

0

(1 + 2καρ^2αt)e^−κρ2αtρ dρ=C t^−1/α. where C = ^π_αR∞

0 (1 + 2κα r)r^α1−1e^−κrdr. This proves the first inequality in (2.5). The next two inequalities can be established in a similar fashion by consid- eringF(x, t) =xixk∂x_jgα(x, t) andF(x, t) =xlxixk∂x_jgα(x, t), respectively, where the indicesi, j, k, l= 1 or 2.

We will need the Hardy-Littlewood-Sobolev inequality, which we now recall.

It states that the fractional integral T f(x) =

Z

R²

f(y)

|x−y|^n−γdy, 0< γ < n is a bounded operator fromL^pto L^q ifpandqsatisfies

1≤p < q <∞, 1 q+γ

n =1 p.

One can find the Hardy-Littlewood-Sobolev inequality in [6, p.119].

3 Global existence and uniqueness in the α > 1/2 case

In this section we consider the IVP for the dissipative QG equation θt+u· ∇θ+κΛ^2αθ=f, (x, t)∈R²×[0,∞),

u= (u1, u2) = (−R2θ,R1θ), (x, t)∈R²×[0,∞), (3.1) θ(x,0) =θ0(x), x∈R²,

where κ > 0 and α ∈ [0,1]. Our major result is that the IVP (3.1) with α >1/2,θ0∈L^r andf ∈L^q0([0, T];L^r1) has a unique global (in time) solution inL^q([0, T];L^p) for properp, q, q⁰, r andr1. Furthermore, the solution is shown to be smooth if θ0 and f are sufficiently smooth. Precise statements will be presented in Theorem 3.4 and Theorem 3.5.

The theorems of this section are proved by the method of integral equations and the contraction mapping argument. To proceed, we write the QG equation into the integral form

θ(t) =Gα(t)θ0+ Z t

0

Gα(t−τ) (f−u· ∇θ)(τ)dτ, (3.2) We observe thatu· ∇θ =∇ ·(uθ) because∇ ·u= 0. The nonlinear term can then be alternatively written as

B(u, θ)(t)≡ Z t

0

∇G_α(t−τ)(uθ)(τ)dτ.

We will solve (3.2) inL^p([0, T];L^q) and the following estimates for the operator B acting on this type of spaces will be used.

Proposition 3.1 Let α > 1/2 and T > 0. Assume that u and θ are in L^q([0, T];L^p)withpandq satisfying

p > 2

2α−1, 1 p+α

q =α−1 2.

Then the operator B is bounded inL^q([0, T];L^p)with

kB(u, θ)kL^q([0,T];L^p)≤CkukL^q([0,T];L^p(R²))· kθkL^q([0,T];L^p(R²)). whereC is a constant depending on α,pandq only.

Proof. Forp > _2α²₋₁≥2, we obtain after applying (2.3) of Proposition 2.2

kB(u, θ)kL^p ≤ Z t

0

k∇G_α(t−τ)(u θ)(τ)kL^pdτ

≤ C Z t

0

1

|t−τ|^2α1+α1(p²−¹p) ku θ(·, τ)kL^p/2dτ (3.3)

≤ C Z t

0

1

|t−τ|^{2α1+p α1} ku(·, τ)kL^pkθ(·, τ)kL^pdτ

for some constant C depending on αand ponly. For α >1/2 and p > _2α²₋₁, we have

0< 1 2α+ 1

p α <1.

Applying the Hardy-Littlewood-Sobolev inequality to (3.3) with 1

q + 1−_2α¹ −_{p α}¹

1 = 2

q, i.e., 1 p+α

q =α−1 2, we obtain

kB(u, θ)kL^q([0,T];L^p) ≤ Ck(ku(·, t)kL^pkθ(·, t)kL^p)kL^q/2([0,T])

≤ CkukL^q([0,T];L^p(R²))· kθkL^q([0,T];L^p(R²)).

The next two lemmas detail how Gα behaves when acting onθ0 andf. Lemma 3.2 Let 1/2< α≤1, T >0, and pandqsatisfy

p > 2

2α−1, 1 p+α

q =α−1 2. Assume thatθ0∈L^r(R²) with _2α²₋₁ < r≤p. Then we have

kGα(t)θ0kL^q([0,T];L^p)≤C T^1−α1(¹2+¹_r)kθ0kL^r, whereC is a constant depending on α,p,qandr only.

Proof. By (2.2),

kGα(t)θ0kL^q([0,T];L^p) = hZ T 0

kGα(t)θ0k^qL^pdti^1/q

≤ hZ T 0

t^−α1(¹r−¹p)·q

kθ₀k^q_Lrdti1/q

= CT^1−α1(¹₂⁺¹_r)kθ₀kL^r. Lemma 3.3 Let1/2< α≤1, T >0, andpandqsatisfy

p > 2

2α−1, 1 p+α

q =α−1 2.

Assume f ∈L^q0([0, T];L^r1)withq⁰ being the conjugate of q (i.e., 1/q’+1/q=1) andr₁ satisfying _2α²

−1 < r₁≤p. Then

Z t 0

G_α(t−τ)f(τ)dτ

_Lq([0,T];Lp)≤CT^1+1q−α1

1 2+_r¹

1

kfk_Lq0([0,T];L^r1),

where C is a constant depending onα,p,q andr1 only.

Proof. The result is a consequence of direct computation. By (2.2) and then H¨older’s inequality,

Z t 0

G_α(t−τ)f(τ)dτ

L^q([0,T];L^p)

≤ hZ T 0

Z t 0

(t−τ)^−α1

1 r1−¹p

kf(·, τ)kL^r1dτ^q dti^1/q

≤ hZ T 0

Z t 0

(t−τ)^−α1

1 r1−¹_p

·q

dτ ·Z t 0

kf(·, τ)k^q_L^0r1dτq/q⁰

dti1/q

≤ CT^1+1q−α1

1 2+_r¹

1

kfk_Lq0([0,T];L^r1). Now we state and prove the main theorem.

Theorem 3.4 Let 1/2< α≤1,T >0, andpandqsatisfy p > 2

2α−1, 1 p+α

q =α−1 2. Assume that θ0 ∈ L^r(R²) with _2α²

−1 < r ≤ p and f ∈ L^q0([0, T];L^r1) with

2

2α−1 < r1≤p, where q⁰ denotes the conjugate of q (i.e., 1/q’+1/q=1). Then there exists a constant C such that for anyθ0 andf satisfying

T^1−α1(¹₂⁺¹_r)kθ₀kL^r+T^1+1q−α1

1 2+_r¹

1

kfkL^q0([0,T];L^r1)≤C,

there exists a unique strong solution θ∈L^q([0, T];L^p)for the IVP (3.1) in the sense of (3.2).

Proof. We write the integral equation (3.2) symbolically as θ = Aθ. The operatorAis seen as a mapping of the space E≡L^q([0, T];L^p) into itself. Let

b=T^1−α1(¹2+¹_r)kθ0kL^r+ T^1+1q−α1

1 2+_r¹

1

kfkL^q0([0,T];L^r1)

and setR= 2b. DefineB_R to be the closed ball with radiusR centered at the origin inE. We now show that ifbis bounded by an appropriate constant, then Ais a contraction map on B_R. Let θand ¯θbe any two elements of B_R. Then we have

kA θ−Aθ¯kE =

Z t 0

G_α(t−τ)(u· ∇θ)dτ− Z t

0

G_α(t−τ)(¯u· ∇θ)dτ¯ _E, where u and ¯u are determined by θ and ¯θ, respectively, through the second relation in (3.1). Recalling the notationB, we have

kA θ−Aθ¯kE =

B(u−u, θ) +¯ B(¯u, θ−θ)¯ _E

≤ kB(u−u, θ)¯ kE+kB(¯u, θ−θ)¯kE. It then follows from applying Proposition 3.1 that

kA θ−Aθ¯kE≤Cku−u¯kEkθkE+Cku¯kEkθ−θ¯kE,

where C is a constant depending onα, pandq only. Since uand ¯uare Riesz transforms of θ and ¯θ, respectively, the classical Calderon-Zygmund singular integral estimates imply that

kukE≤CkθkE, ku¯kE≤Ckθ¯kE.

One can consult the book of Stein [6] for more details on Riesz transforms.

Therefore,

kA θ−Aθ¯kE≤C(kθkE+kθ¯kE)kθ−θ¯kE≤C Rkθ−θ¯kE. We now estimatekAθkE. By Lemma 3.2 and Lemma 3.3, the norm of

A0 =G_α(t)θ₀+ Z t

0

G_α(t−τ)f(τ)dτ inE can be bounded by

kA0kE≤C T^1−α1(¹₂⁺¹_r)kθ0kL^r+ CT¹⁺

1 q−α¹

₁

2+_r¹

1

kfkL^q0([0,T];L^r1)=b Therefore,

kA θkE=kA θ−A0 +A0kE≤ kA θ−A0kE+kA0kE≤C RkθkE+b.

If 2C b≤ ¹₂, thenC R= 2C b≤¹₂ and we have kA θ−Aθ¯kE≤ 1

2kθ−θ¯kE, and kA θkE≤R.

It follows from the contraction mapping principle that there exists a unique θ∈E=L^q([0, T];L^p) solving (3.2). This finishes the proof of Theorem 3.4.

We now show that the solution obtained in the previous theorem is actually smooth. We introduce a notation. For a non-negative multi-indexk= (k1, k2), we define

D^k = ∂

∂_x1

^k1 ∂

∂_x2 ^k2

. and|k|=k₁+k₂.

Theorem 3.5 Let 1/2< α≤1,T >0, andpandqsatisfy p > 2

2α−1, 1 p+α

q =α−1 2. Assume that for a non-negative multi-index k

D^kθ₀∈L^r(R²) and D^kf ∈L^q0([0, T];L^r1(R²)), (3.4) where _2α²₋₁ < r ≤p, _2α²₋₁ < r1 ≤pand q⁰ denotes the conjugate of q. Then for any non-negative multi-index j with |j| ≤ |k|

D^jθ∈L^q([0, T];L^p). (3.5) Furthermore, for each j with 0≤ |j| ≤ |k| −2 and almost everyt∈[0, T]

∂tD^jθ∈L^p(R²). (3.6)

Proof. The basic tool of establishing (3.5) is still the contraction mapping argument. We first consider the case|j|= 1. TakingD of (3.2), we obtain

D θ(t) = Gα(t)(Dθ0) + Z t

0

Gα(t−τ)(Df(τ))dτ+B(D u, θ) +B(u, D θ). (3.7) This integral equation can then be viewed as (Dθ) = ˜A(Dθ) and ˜Ais seen as a mapping of the spaceE consisting of functionsθsuch that

θ∈L^q([0, T];L^p) and Dθ∈L^q([0, T];L^p).

The norm in Eis given by

kθkE =kθkL^q([0,T];L^p)+kDθkL^q([0,T];L^p).

For θ₀ and f satisfying (3.4), the first two terms are bounded in E. The two nonlinear terms acting onEhave similar bounds as stated in Proposition 3.1. As in the proof of Theorem 3.4, we can then show that ˜Ais a contraction mapping of E into itself. Therefore ˜A has a fixed point in E. The uniqueness result of Theorem 3.4 indicates that this θ is just the original θ. Thus we have shown that Dθ∈L^q([0, T];L^p). The proof of (3.5) for |j|= 2,3,· · ·,|k|is similar and we thus omit details.

We now prove (3.6) and start with the case |j|= 0. Becauseθ satisfies

∂_tθ= f −u· ∇θ−κΛ^2αθ,

it suffices to show that the terms on the right are inL^pfor almost everyt. Since for almost everyt

f ∈L^r1, Df ∈L^r1, u∈L^p, Du∈L^p, Dθ∈L^p, D²θ∈L^p, we obtain by applying the Gagliardo-Nirenberg inequality

kf(·, t)kL^p≤Ckf(·, t)k¹L^−r1^σ kDf(·, t)k^σL^r1, σ= 2 r1 −2

p.

k(u· ∇θ)(·, t)kL^p ≤ Cku(·, t)kL^2p kDθ(·, t)kL^2p

≤ Cku(·, t)k¹_L⁻pkDu(·, t)kL^p kDθ(·, t)k¹_L⁻pkD²θ(·, t)kL^p, where= 1/p. Therefore, for almost everyt

f(·, t)∈L^p, u· ∇ ∈L^p, and Λ^2αθ∈L^p

and this in turn implies ∂_tθ ∈ L^p. The proof for ∂_tD^jθ ∈ L^p with |j| >0 is similar. This completes the proof of (3.6).

4 L

bounds in the α ≤ 1/2 case

Forα >1/2, the issue of global existence, uniqueness and regularity concerning the IVP (3.1) with L^r initial data is resolved in Section 3. Our major interest of this section is in theα≤ ¹₂ case although all theorems to be presented hold for any α∈[0,1]. We conclude that any possible finite time singularity must occur in the first derivative. This is achieved by bounding the L^p norms of all high order derivatives ofθby the initialL^p norms and a magic quantity.

Lemma 4.1 Let α ∈ [0,1], p ∈ (1,∞) and k be a nonnegative multi-index.

Then for any sufficiently smooth θ, we have for anyt≥0 Z

R²

|D^kθ|^p−2(x, t) (D^kθ(x, t)) Λ^2αD^kθ(x, t)dx≥0.

Proof. Let gα(x, s) be the kernel of the solution operator for the linear QG equation, as defined in the previous section. Then Θ(x, s) ≡gα(·, s)∗(D^kθ) satisfies the equation

∂_sΘ +κΛ^2αΘ = 0 (4.1)

and Θ(x, s)→D^kθ ass→0. Multiplying both sides of (4.1) byp|Θ|^p−2Θ and integrate overR², we obtain

d ds

Z

R²

|Θ|^pdx+ p κ Z

R²

|Θ|^p−2ΘΛ^2αΘdx= 0.

Integrating the above over [s1, s2] with respect tos, we have Z

R²

|Θ|^p(x, s2)dx− Z

R²

|Θ|^p(x, s1)dx=−p κ Z s₂

s₁

Z

R²

|Θ|^p−2ΘΛ^2αΘdxds, (4.2) wheres1ands2are arbitrarily fixed. Applying (2.2) of Proposition 2.2, we have

Z

R²

|Θ|^p(x, s₂)dx = kg_α(·, s₂)∗(D^kθ)k^p_Lp

= kg_α(·, s₂−s₁)∗[g_α(·, s₁)∗(D^kθ)]k^p_Lp

≤ kgα(·, s1)∗(D^kθ)k^p_Lp = Z

R²

|Θ|^p(x, s1)dx.

That is, the left hand side of (4.2) is not positive. Therefore Z s₂

s1

Z

R²

|Θ|^p−2ΘΛ^2αΘdxds≥0.

The arbitrariness of s₁ands₂ then implies that for anys >0, Z

R²

|Θ|^p−2(x, s) Θ(x, s) Λ^2αΘ(x, s)dxds≥0. (4.3) Lettings→0 and recalling the definition of Θ, we obtain for anyt≥0

Z

R²

|D^kθ|^p−2(x, t) (D^kθ(x, t)) Λ^2αD^kθ(x, t)dx≥0.

One consequence of the previous lemma is that the L^p-norm (p ∈ (1,∞]) of any solutionθ of the IVP (3.1) is uniformly bounded by theL^p norm of the initial data. Thus finite-time singularity is only possible in the derivatives ofθ.

The following result was shown in [5] and we now briefly describe it.

Theorem 4.2 Let α∈[0,1]and p∈(1,∞]. Then any solution θ of the IVP (3.1) satisfies for t≥0

kθ(·, t)kL^p(R²)≤ kθkL^p(R²). A sketch of the proof for this theorem is given in [2].

We now state and prove our main theorem, in which we establish estimates to bound theL^p norms of derivatives of any solutionθof the IVP (3.1) in terms of∇u(uis related toθthrough the second relation in (3.1)). Roughly speaking, this means that no finite time singularity in high-order derivatives is possible if

∇udoes not become infinite first. The role of the forcing termf is not crucial, so we set it equal to zero for the sake of clear presentation.

Theorem 4.3 Let α ∈ [0,1]. Assume that θ is a solution of the IVP (3.1).

Then for any p∈(1,∞] and a multi-index kwith |k| ≥1,

kD^kθ(·, t)kL^p≤ kD^kθ0kL^p·e^{R0tk∇u(·,τ)kL∞dτ} (4.4) holds for any t≥0, where u is determined byθ through the second relation in (3.1).

Proof. We start with the case|k|= 1. Forp∈(0,∞), we takeD of the first equation in (3.1), multiply byp|Dθ|^p−2Dθand then integrate overR²to obtain

d dt

Z

R²

|Dθ|^pdx+ p κ Z

R²

|Dθ|^p−2Dθ·Λ^2α(Dθ)dx

= −p Z

R²

|Dθ|^p−2Dθ ·D(u· ∇θ)dx (4.5) The right hand side actually consists of two terms

−p Z

R²

|Dθ|^p−2Dθ ·Du· ∇θdx and −p Z

R²

|Dθ|^p−2Dθ ·u· ∇(Dθ)dx, but one of them is zero

Z

R²

|Dθ|^p−2Dθ ·u· ∇(Dθ)dx= Z

R²

u· ∇(|Dθ|^p)dx= 0 because∇ ·u= 0. Therefore, (4.5) becomes

d dt

Z

R²

|Dθ|^pdx+p κ Z

R²

|Dθ|^p−2Dθ·Λ^2α(Dθ)dx

= −p Z

R²

|Dθ|^p−2Dθ ·Du· ∇θdx,

which in turn implies that d

dt Z

R²

|Dθ|^pdx+p κ Z

R²

|Dθ|^p−2Dθ·Λ^2α(Dθ)dx≤pk∇u(·, t)kL^∞

Z

R²

|Dθ|^pdx.

By Lemma 4.1, the second term on the left hand side is nonnegative. So d

dt Z

R²

|Dθ|^pdx≤pk∇u(·, t)kL^∞

Z

R²

|Dθ|^pdx.

Gronwall’s inequality then implies (4.4). Once we have the bound (4.4) for any p < ∞, we can then take the limit of (4.4) as p → ∞ to establish (4.4) for p=∞.

The inequality (4.4) for general k can be proved by induction. One needs the Calderon-Zygmund inequality for Riesz transforms

kD^ju(·, t)kL^p≤CkD^jθ(·, t)kL^p, p∈(1,∞), |j| ≤ |k|.

Acknowledgments. This research was partially supported by the NSF grant DMS 9971926, by the American Mathematical Society Centennial Fellowship, and by the ORAU Ralph E. Powe Junior Faculty Enhancement Award.

References

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[2] P. Constantin, D. Cordoba and J. Wu, On the critical dissipative quasi- geostrophic equation,Indiana Univ. Math. J., 2001 (in press).

[3] P. Constantin and J. Wu, Behavior of solutions of 2D quasi-geostrophic equations,SIAM J. Math. Anal.30(1999), 937-948.

[4] J. Pedlosky,Geophysical Fluid Dynamics, Springer-Verlag, New York, 1987.

[5] S. Resnick,Dynamical Problems in Non-linear Advective Partial Differen- tial Equations, Ph.D. Thesis, University of Chicago, 1995.

[6] E. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, 1970.

Jiahong Wu

Department of Mathematics Oklahoma State University 401 Mathematical Sciences Stillwater, OK 74078 USA e-mail: [email protected]