On the Serrin’s Regularity Criterion for the β-Generalized Dissipative Surface Quasi-geostrophic Equation?

Jihong ZHAO Qiao LIU

1 Introduction

In this paper,we study the two dimensionalβ-generalized surface quasi-geostrophic equation as follows:

Hereα∈(0,1],β∈[1,2),κ>0 is the dissipative coefficient,andθ=θ(t,x,y):is a real-valued function of a time variabletand two space variables(x,y),and represents the potential temperature of the fluid,whileu=(u1,u2):is the velocity field of the fluid which is de fined by

where the fractional power of the Laplacian Λα=is de fined by the Fourier transformandR1,R2are Riesz transforms de fined byforj=1,2.

Theβ-generalized surface quasi-geostrophic equation(1.1)was introduced by Kiselev in[21].Forβ=1,(1.1)reduces to the following dissipative surface quasi-geostrophic equation:

(1.3)is an important model in geophysical fluid dynamics used in meteorology and oceanography,and they are special cases of the general quasi-geostrophic approximations for atmosphere and oceanic fluid flow with small Rossy and Ekman numbers(see[12,27]for more details about its physical background).Due to its analogy with 3D incompressible Navier-Stokes/Euler equations,in the last two decades,(1.3)attracted enormous attention and many important results were obtained.For the global well-posedness of(1.3)in the subcritical caseα>1,we refer the readers to[2,13,28].For the global well-posedness with small initial data in various functional spaces(e.g.,Sobolev spaces,Besov spaces,H?lder spaces,etc.)of(1.3)in the critical caseα=1,we refer the readers to[1,7,9–10,14,24].Recently,the global regularity of weak solutions in the critical caseα=1 was addressed by the following two mathematical groups:Kiselev,Nazarov and Volberg[22]proved global well-posedness of(1.3)with periodicC∞data by using a certain non-local maximum principle for a suitable chosen modulus of continuity;Caf f arelli and Vasseur[4]obtained a global regular weak solution to(1.3)with merelyL2initial data by using the modified De Georgi interation.For the global regularity of the supercritical caseα<1,we refer the readers to[3,8,29,35].Parts of the above global well-posedness results were subsequently extended to(1.1)withβ∈[1,2)by[11,26,31–32].

Although the global existence of smooth solutions to(1.1)with suitable choices ofαandβwas established(see[32]),the regularity issue of weak solutions in the supercritical case is still an open problem,so the development of the regularity criterion of weak solutions is of major importance for both theoretical and practical purposes.Forβ=1,Constantin,Majda and Tabak[12]proved that the maximum norm ofcontrols the breakdown of the smooth solution to(1.3)in both viscous and invisid cases,i.e.,they proved that if

then the solutionθcan be extended beyond timeT.Chae[6]established that if

then there is no singularity up to timeT.For some improvements of(1.5),we refer the readers to[15–17,19,30,34].For theβ-generalized surface quasi-geostrophic equation(1.1),under the hypothesis thatα+β=2,Yamazaki[33]established that if

then there is no singularity up to timeT.

Motivated by the above cited results,the first purpose of this paper is to establish a similar Serrin’s regularity criterion(1.5)for theβ-generalized surface dissipative quasi-geostrophic equation(1.1).In the sequel,ifβ=1,we let

Theorem 1.1Let α∈(0,1]and β∈[1,2),such that α+2β<4.Assume that θ is a smooth solution to(1.1)with initial dataAssume further that for some T>0,

Then the solution θ can be smoothly extended after time T.

Remark 1.1(i)Theorem 1.1 is clearly a generalization of(1.5).

(ii)The conditionsα+2β<4 andappear due to the Gagliardo-Nirenberg inequalities and the Hardy-Littlewood-Sobolev inequalities which we will use in the proof of Theorem 1.1.

The second purpose of this paper is based on the observation that the velocity fielduis divergence free,i.e.,+=0,so we can establish the following regularity criterion in terms of partial derivatives of the solutionθ.

Theorem 1.2Let α∈(0,1]and β∈[1,2),such that α+β≥2and α+2β<4.Assume that θis a smooth solution to(1.1)with initial data θ0Assume further that for some T>0,

Then the solution θ can be smoothly extended after time T.

Remark 1.2(i)The role ofcan be replaced byin Theorem 1.2.This implies that one direction of the derivative of the solutionθcontrols the regularity of the solutionθ.

(ii)Theorem 1.2 covers the supercritical case,and the distinction between Theorem 1.2 and the regularity result of Yamazaki[33]is that we improve the conditionα+β=2 toα+β≥2.

(iii)Using a single partial derivative of the solution to control the regularity of weak solutions was observed in many equations in fluid dynamics,e.g.,for the Navier-Stokes equations(see[18,23,36]),for the MHD equations(see[5]),and for the nematic liquid crystal flows(see[25]).

The remaining part of this paper is organized as follows.In Section 2,we give the proof of Theorem 1.1.Section 3 is devoted to the proof of Theorem 1.2.Throughout this paper,Cstands for a generic positive constant which may vary from line to line,anddenotes the norm of the Banach spaceX.

2 The Proof of Theorem 1.1

In this section,we present the proof of Theorem 1.1.Multiplying(1.1)byθ,integrating overand using the fact=0,one obtains

and it follows that

Applying Λ3θto(1.1),multiplying the resulting identity by Λ3θ,and integrating overwe have

Thanks to the fact that?·u=0,we have

Thus we get

To estimate the right-hand side of(2.4),we need to use the following well-known commutator estimate(see[20]):Fors>1,we have

with 1

For the case ofby using(2.5),we see that

where we used the Hardy-Littlewood-Sobolev inequalities(α+2β<4)

the boundedness of Riesz operators inwith 1

For the case ofby using(2.5)again,we obtain

where we used the Hardy-Littlewood-Sobolev’s inequalities

and the Gagliardo-Nirenberg’s inequality

Letq=It is easy to verify thatThen,by(2.6)–(2.7),one obtains that for

where we used the following Sobolev interpolation inequality:

Hence,we obtain from(2.8)that

Applying Gronwall’s inequality to(2.9)on the time interval[0,T]and using the condition(1.7),we can easily see that

Combining(2.10)with the energy inequality(2.1),we get the boundednesson the time interval[0,T].The proof of Theorem 1.1 is complete.

3 The Proof of Theorem 1.2

In this section,we present the proof of Theorem 1.2.Applyingto(1.1),multiplying the resultant byand integrating overwe see that

Since?·u=0,it follows that

Hence,

Similarly,

Hence,by(3.3)–(3.4),we obtain

For the case ofwe proceed in the same way as the proof of(2.6)to estimate the termsas follows:

For the case ofin a way similar to the proof of(2.7),we estimate the terms Ii(i=1,2,3,4)as follows:

Note that if we setwhich satis fiesthen by putting the above estimates(3.6)–(3.13)together,we get for all

Dividing both sides of(3.14)bywe get

Applying Gronwall’s inequality to(3.15),it follows from the condition(1.8)that

Going back to(3.14),and integrating on the time interval[0,T],we obtain

In particular,we notice that

Now we are in a position to derive the desired estimate of Λ3θ.In a way similar to the proof of Theorem 1.1,by using(2.5),we have

where we used,under the assumptionsα+β≥2 andα+2β<4,the following Gagliardo-Nirenberg inequalities:

Sinceand≤2,it follows from Gronwall’s inequality that

Combining this with(2.1)yields the boundedness ofon the time interval[0,T].We complete the proof of Theorem 1.2.

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