Regularity of Hölder continuous solutions of the supercritical quasi-geostrophic equation

Regularity of Hölder continuous solutions of the supercritical quasi-geostrophic equation

Peter Constantin

, Jiahong Wu

aDepartment of Mathematics, University of Chicago, 5734 S. University Avenue Chicago, IL 60637, USA bDepartment of Mathematics, Oklahoma State University, Stillwater, OK 74078, USA Received 8 February 2007; received in revised form 22 October 2007; accepted 22 October 2007

Available online 1 November 2007

Abstract

We present a regularity result for weak solutions of the 2D quasi-geostrophic equation with supercritical (α <1/2) dissipa- tion(−)^α: If a Leray–Hopf weak solution is Hölder continuousθ∈C^δ(R²)withδ >1−2αon the time interval[t0, t], then it is actually a classical solution on(t₀, t].

©2007 Elsevier Masson SAS. All rights reserved.

MSC:76D03; 35Q35

Keywords:2D quasi-geostrophic equation; Supercritical dissipation; Regularity; Weak solutions

1. Introduction

We discuss the surface 2D quasi-geostrophic (QG) equation

∂tθ+u· ∇θ+κ(−)^αθ=0, x∈R², t >0, (1.1) whereα >0 andκ0 are parameters, and the 2D velocity fieldu=(u₁, u₂)is determined fromθ by the stream functionψvia the auxiliary relations

(u1, u2)=(−∂x₂ψ, ∂x₁ψ ), (−)^1/2ψ= −θ. (1.2)

Using the notationΛ≡(−)^1/2and∇^⊥≡(∂_x2,−∂_x1), the relations in (1.2) can be combined into

u= ∇^⊥Λ⁻¹θ=(−R2θ,R1θ ), (1.3)

whereR1andR2are the usual Riesz transforms inR². The 2D QG equation withκ >0 andα=¹₂ arises in geo- physical studies of strongly rotating fluids (see [5,16] and references therein) while the inviscid QG equation ((1.1) withκ=0) was derived to model frontogenesis in meteorology, a formation of sharp fronts between masses of hot and cold air (see [7,10,16]).

* Corresponding author.

E-mail addresses:const@cs.uchicago.edu (P. Constantin), jiahong@math.okstate.edu (J. Wu).

0294-1449/$ – see front matter ©2007 Elsevier Masson SAS. All rights reserved.

doi:10.1016/j.anihpc.2007.10.001

The problem at the center of the mathematical theory concerning the 2D QG equation is whether or not it has a global in time smooth solution for any prescribed smooth initial data. In the subcritical caseα > ¹₂, the dissipative QG equation has been shown to possess a unique global smooth solution for every sufficiently smooth initial data (see [8,17]). In contrast, whenα¹₂, the issue of global existence and uniqueness is more difficult and has still unanswered aspects. Recently this problem has attracted a significant amount of research ([1–6,9,11–15,18–24]). In Constantin, Córdoba and Wu [6], we proved in the critical case (α=¹₂) the global existence and uniqueness of classical solutions corresponding to any initial data withL^∞-norm comparable to or less than the diffusion coefficientκ. In a recently posted preprint in arXiv [14], Kiselev, Nazarov and Volberg proved that smooth global solutions exist for anyC^∞ periodic initial data, by removing the L^∞-smallness assumption on the initial data of [6]. Caffarelli and Vasseur (arXiv reference [1]) establish the global regularity of the Leray–Hopf type weak solutions (in L^∞((0,∞);L²)∩ L²((0,∞);H˚^1/2)) of the critical 2D QG equation withα=¹₂in generalRⁿ.

In this paper we present a regularity result of weak solutions of the dissipative QG equation with α < ¹₂ (the supercritical case). The result asserts that if a Leray–Hopf weak solution θ of (1.1) is in the Hölder class C^δ with δ >1−2αon the time interval[t0, t], then it is actually a classical solution on(t0, t]. The proof involves representing the functions in Hölder space in terms of the Littlewood–Paley decomposition and using Besov space techniques.

Whenθis inC^δ, it also belongs to the Besov space ˚B_p,^δ(1_∞^−2/p)for anyp2. By takingpsufficiently large, we have θ∈C^δ1∩B˚_p,^δ1_∞forδ₁>1−2α. The idea is to show thatθ∈C^δ2∩B˚_p,^δ2_∞withδ₂> δ₁. Through iteration, we establish thatθ∈C^γ withγ >1. Thenθbecomes a classical solution.

The results of this paper can be easily extended to a more general form of the quasi-geostrophic equation in which x∈Rⁿanduis a divergence-free vector field determined byθthrough a singular integral operator.

The rest of this paper is divided into two sections. Section 2 provides the definition of Besov spaces and necessary tools. Section 3 states and proves the main result.

2. Besov spaces and related tools

This section provides the definition of Besov spaces and several related tools. We start with a some notation. Denote byS(Rⁿ)the usual Schwarz class andS(Rⁿ)the space of tempered distributions.fˆdenotes the Fourier transform off, namely

f (ξ )ˆ =

Rⁿ

e^−ix·ξf (x) dx.

The fractional Laplacian(−)^α can be defined through the Fourier transform (−)^αf = |ξ|^2αf (ξ ).ˆ

Let S0=

φ∈S,

Rⁿ

φ(x)x^γdx=0, |γ| =0,1,2, . . .

.

Its dualS₀is given by S0=S/S0^⊥=S/P,

whereP is the space of polynomials. In other words, two distributions inS are identified as the same inS₀if their difference is a polynomial.

It is a classical result that there exists a dyadic decomposition ofRⁿ, namely a sequence{Φ_j} ∈S(Rⁿ)such that suppΦˆ_j⊂A_j, Φˆ_j(ξ )= ˆΦ₀(2^−jξ ) or Φ_j(x)=2^{j n}Φ₀(2^jx)

and ∞ k=−∞

Φˆk(ξ )=

1 ifξ∈Rⁿ\ {0}, 0 ifξ=0,

where Aj=

ξ∈Rⁿ: 2^j−1<|ξ|<2^j+1 . As a consequence, for anyf ∈S₀,

∞ k=−∞

Φ_k∗f=f. (2.1)

For notational convenience, set

_jf=Φ_j∗f, j =0,±1,±2, . . . . (2.2) Definition 2.1.Fors∈Rand 1p, q∞, the homogeneous Besov space ˚B_p,q^s is defined by

B˚_p,q^s =

f ∈S₀: fB˚_p,q^s <∞ ,

where

fB˚_p,q^s =

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

j

2^{j s}jfL^p

q1/q

forq <∞, sup

j

2^{j s}_jfL^p forq= ∞. For_j defined in (2.2) andS_j≡

k<j_k,

jk=0 if|j−k|2 and j(Sk−1f kf )=0 if|j−k|3.

The following proposition lists a few simple facts that we will use in the subsequent section.

Proposition 2.2.Assume thats∈Randp, q∈ [1,∞].

(1) If1q₁q₂∞, thenB˚_p,q^s

1⊂B˚_p,q^s

2.

(2) (Besov embedding)If1p₁p₂∞ands₁=s₂+n(_p¹

1 −_p¹₂), thenB˚_p^s1_1,q(Rⁿ)⊂B˚_p^s2_2,q(Rⁿ).

(3) If1< p <∞, then

B˚_p,min(p,2)^s ⊂W˚^s,p⊂B˚_p,max(p,2)^s ,

whereW˚^s,pdenotes a standard homogeneous Sobolev space.

We will need a Bernstein type inequality for fractional derivatives.

Proposition 2.3.Letα0. Let1pq∞. (1) Iff satisfies

suppfˆ⊂

ξ∈Rⁿ: |ξ|K2^j , for some integerjand a constantK >0, then

(−)^αf

L^q(Rⁿ)C12^{2αj+j n(p1−1q)}fL^p(Rⁿ). (2) Iff satisfies

suppfˆ⊂

ξ∈Rⁿ: K₁2^j|ξ|K₂2^j

(2.3) for some integerjand constants0< K₁K₂, then

C₁2^2αjfL^q(Rⁿ)(−)^αf

L^q(Rⁿ)C₂2^{2αj+j n(1p−1q)}fL^p(Rⁿ), whereC₁andC₂are constants depending onα, pandq only.

The following proposition provides a lower bound for an integral that originates from the dissipative term in the process ofL^pestimates (see [21,4]).

Proposition 2.4.Assume eitherα0andp=2or0α1and2< p <∞. Letj be an integer andf∈S. Then

Rⁿ

|_jf|^p−2_jf Λ^2α_jf dxC2^2αj_jf^p_Lp

for some constantCdepending onn,αandp.

3. The main theorem and its proof

Theorem 3.1.Letθbe a Leray–Hopf weak solution of (1.1), namely θ∈L^∞

[0,∞);L²(R²)

∩L²

[0,∞);H˚^α(R²)

. (3.1)

Letδ >1−2αand let0< t₀< t <∞. If θ∈L^∞

[t₀, t];C^δ(R²)

, (3.2)

then

θ∈C^∞

(t₀, t] ×R² .

Proof. First, we notice that (3.1) and (3.2) imply that θ∈L^∞

[t0, t];B˚_p,^δ1_∞(R²) ,

for anyp2 andδ₁=δ(1−_p²). In fact, for anyτ∈ [t₀, t], θ (·, τ )

B˚^δ_p,¹_∞=sup

j

2^δ1j_jθL^p

sup

j

2^δ1j_jθ¹_L⁻∞^p2_jθ_L^p22

θ (·, τ )^1−p2

C^δ θ (·, τ )^2p

L². Sinceδ >1−2α, we haveδ₁>1−2αwhen

p > p₀≡ 2δ δ−(1−2α). Next, we show that

θ∈L^∞

[t₀, t];B˚_p,^δ1_∞∩C^δ1 implies

θ (·, t )∈B˚_p,^δ2_∞∩C^δ2

for someδ₂> δ₁to be specified. Letj be an integer. Applying_j to (1.1), we get

∂tjθ+κΛ^2αjθ= −j(u· ∇θ ). (3.3)

By Bony’s notion of paraproduct, j(u· ∇θ )=

|j−k|2

j(Sk−1u· ∇kθ )+

|j−k|2

j(ku· ∇Sk−1θ )

+

kj−1

|k−l|1

_j(_ku· ∇_lθ ). (3.4)

Multiplying (3.3) byp|_jθ|^p−2_jθ, integrating with respect tox, and applying the lower bound

R^d

|jf|^p−2jf Λ^2αjf dxC2^2αjjf^p_L^p

of Proposition 2.4, we obtain d

dt_jθ^p_Lp+Cκ2^2αj_jθ^p_LpI₁+I₂+I₃, (3.5)

whereI₁,I₂andI₃are given by I1= −p

|j−k|2

|jθ|^p−2jθ·j(Sk−1u· ∇kθ ) dx,

I2= −p

|j−k|2

|jθ|^p−2jθ·j(ku· ∇Sk−1θ ) dx,

I3= −p

kj−1

|_jθ|^p−2_jθ·

|k−l|1

_j(_ku· ∇_lθ ) dx.

We first boundI₂. By Hölder’s inequality I2Cjθ^p_L^−p1

|j−k|2

kuL^p∇Sk−1θL∞.

Applying Bernstein’s inequality, we obtain I₂C_jθ^p_L⁻p¹

|j−k|2

_kuL^p

mk−1

2^m_mθL^∞

Cjθ^p_L^−p1

|j−k|2

kuL^p2^(1−δ1)k

mk−1

2^{(m−k)(1−δ1)}2^mδ1mθL^∞. Thus, for 1−δ₁>0, we have

I₂C_jθ^p_L⁻p¹θ_C^δ1

|j−k|2

_kuL^p2^(1−δ1)k.

We now estimateI₁. The standard idea is to decompose it into three terms: one with commutator, one that becomes zero due to the divergence-free condition and the rest. That is, we rewriteI₁as

I₁= −p

|j−k|2

|_jθ|^p−2_jθ· [_j, S_k−1u· ∇]_kθ dx−p

|_jθ|^p−2_jθ·(S_ju· ∇_jθ ) dx

−p

|j−k|2

|_jθ|^p−2_jθ·(S_k−1u−S_ju)· ∇_jkθ dx

=I₁₁+I₁₂+I₁₃,

where we have used the simple fact that

|k−j|2_kjθ =_jθ, and the brackets []represent the commutator, namely

[_j, S_k−1u· ∇]_kθ=_j(S_k−1u· ∇_kθ )−S_k−1u· ∇_jkθ.

Sinceuis divergence free,I₁₂becomes zero.I₁₂can also be handled without resort to the divergence-free condition.

In fact, integrating by parts inI₁₂yields I₁₂=

|_jθ|^p∇ ·S_ju dx_jθ^p_Lp∇ ·S_juL^∞. By Bernstein’s inequality,

|I₁₂|_jθ^p_Lp

mj−1

2^m_muL^∞

= _jθ^p_L^p2^(1−δ1)j

mj−1

2^{(1−δ1)(m−j )}2^mδ1_muL^∞. For 1−δ1>0,

|I₁₂|C_jθ^p_Lp2^(1−δ1)ju_C^δ1 C_jθ^p_L⁻p¹2^(1−2δ1)jθ_B˚^δ1

p,∞u_C^δ1. We now boundI₁₁andI₁₃. By Hölder’s inequality,

|I₁₁|p_jθ^p_L^−p1

|j−k|2

[_j, S_k−1u· ∇]_kθ

L^p.

To bound the commutator, we have by the definition of_j [_j, S_k−1u· ∇]_kθ=

Φ_j(x−y)

S_k−1(u)(x)−S_k−1(u)(y)

· ∇_kθ (y) dy.

Using the fact thatθ∈C^δ1 and thus S_k−1(u)(x)−S_k−1(u)(y)

L^∞u_C^δ1|x−y|^δ1, we obtain

[_j, S_k−1u· ∇]_kθ

L^p2^−δ1ju_C^δ12^k_kθL^p. Therefore,

|I₁₁|Cp_jθ^p_L⁻p¹2^−δ1ju_C^δ1

|j−k|2

2^k_kθL^p. The estimate forI13is straightforward. By Hölder’s inequality,

|I₁₃|p_jθ^p_L^−p1

|j−k|2

S_k−1u−S_juL^p∇_jθL^∞

Cpjθ^p_L^−p12^(1−δ1)jθ_C^δ1

|j−k|2

kuL^p.

We now boundI₃. By Hölder’s inequality and Bernstein’s inequality,

|I₃|p_jθ^p_L⁻p¹

_j∇ ·

kj−1

|l−k|1

_lu_kθ L^p

p_jθ^p_L⁻p¹2^ju_C^δ1

kj−1

2^−δ1k_kθL^p. (3.6)

Inserting the estimates forI1,I2andI3in (3.5) and eliminatingp_jθ^p_L⁻p¹from both sides, we get d

dt_jθL^p+Cκ2^2αj_jθL^pC2^(1−2δ1)jθ_B˚^δ1

p,∞u_C^δ1+C2^−δ1ju_C^δ1

|j−k|2

2^k_kθL^p

+Cθ_C^δ1

|j−k|2

_kuL^p2^(1−δ1)k+C2^(1−δ1)jθ_C^δ1

|j−k|2

_kuL^p

+C2^juC^δ1

kj−1

2^−δ1k_kθL^p. (3.7)

The terms on the right can be further bounded as follows.

C2^−δ1ju_C^δ1

|j−k|2

2^k_kθL^p=C2^(1−2δ1)ju_C^δ1

|j−k|2

2^δ1k_kθL^p2^{(k−j )(1−δ1)} C2^(1−2δ1)ju_C^δ1θ_B˚^δ1

p,∞,

Cθ_C^δ1

|j−k|2

_kuL^p2^(1−δ1)k=C2^(1−2δ1)jθ_C^δ1

|j−k|2

2^δ1k_kuL^p2^{(k−j )(1−2δ1)} C2^(1−2δ1)jθC^δ1u_B˚^δ1

p,∞,

C2^(1−δ1)jθC^δ1

|j−k|2

_kuL^p=C2^(1−2δ1)jθC^δ1

|j−k|2

2^δ1k_kuL^p2^(j−k)δ1 C2^(1−2δ1)jθ_C^δ1u_B˚^δ1

p,∞

and

C2^ju_C^δ1

kj−1

2^−δ1k_kθL^p=C2^(1−2δ1)ju_C^δ1

kj−1

2^{−2δ1(k−j )}2^δ1k_kθL^p

C2^(1−2δ1)juC^δ1θ_B˚^δ1 p,∞. We can write (3.7) in the following integral form

_jθ (t )

L^pe^{−Cκ22αj(t−t0)}_jθ (t₀)

L^p

+C t t0

e^{−Cκ22αj(t−s)}2^(1−2δ1)j

θ_C^δ1u_B˚^δ1

p,∞+ u_C^δ1θ_B˚^δ1 p,∞

ds.

Multiplying both sides by 2^{(2α+2δ1−1)j} and taking the supremum with respect toj, we get θ (t )

B˚_p,^2δ1_∞^+2α−1sup

j

e^{−Cκ22αj(t−t0)}2^{(δ1+2α−1)j}θ (t₀)_˚

B^δ_p,∞¹

+Cκ⁻¹sup

j

1−e^{−Cκ22αj(t−t0)}

smax∈[t0,t]θ (s)

B˚_p,∞^δ1 θ (s)

C^δ1.

Here we have used the fact that u_C^δ1 θ_C^δ1 and u_B˚^δ1

p,∞θ_B˚^δ1 p,∞. Therefore, we conclude that if

θ∈L^∞

[t0, t];B˚_p,^δ1_∞∩C^δ1 , then

θ (·, t )∈B˚_p,^2δ1_∞^+2α−1. (3.8)

Sinceδ₁>1−2α, we have 2δ1+2α−1> δ₁and thus gain regularity. In addition, according to the Besov embedding of Proposition 2.2,

B˚_p,^2δ_∞^1+2α−1⊂B˚_∞^δ2_,∞, where

δ₂=2δ1+2α−1−2 p =δ₁+

δ₁−

1−2α+2 p

.

We haveδ₂> δ₁when p > p1≡ 2

δ₁−(1−2α).

Noting that

B˚_∞^δ2_,∞∩L^∞=C^δ2,

we conclude that, forp >max{p₀, p₁}, θ (·, t )∈B˚_p,^δ2_∞∩C^δ2

for someδ₂> δ₁. The above process can then be iterated withδ₁replaced byδ₂. A finite number of iterations allow us to obtain that

θ (·, t )∈C^γ

for some γ >1. The regularity in the spatial variable can then be converted into regularity in time. We have thus established thatθ is a classical solution to the supercritical QG equation. Higher regularity can be proved by well- known methods. 2

Acknowledgement

PC was partially supported by NSF-DMS 0504213. JW thanks the Department of Mathematics at the University of Chicago for its support and hospitality.

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