JAM Journal of Applied Mathematics 1687-0042 1110-757X Hindawi Publishing Corporation 348513 10.1155/2013/348513 348513 Research Article Well-Posedness of the Two-Dimensional Fractional Quasigeostrophic Equation http://orcid.org/0000-0001-7561-3790 Xu Yongqiang Li Wan-Tong School of Mathematics and Statistics Minnan Normal University Zhangzhou 363000 China mnnu.net 2013 11 11 2013 2013 18 07 2013 20 10 2013 2013 Copyright © 2013 Yongqiang Xu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This paper is concerned with the fractional quasigeostrophic equation with modified dissipativity. We prove the local existence of solutions in Sobolev spaces for the general initial data and the global existence for the small initial data when 1/2α<1.

1. Introduction

This paper is concerned with the nonlocal quasigeostrophic β-plane model with modified dissipativity [1, 2] (1)(t+ψxy-ψyx)q=1Re(-Δ)1+αψ, where (x,y)  Ω can be either the 2D torus 𝕋2 or the whole space 2, t0,q=Δψ-Fψ+βy, and (1/Re)(-Δ)1+αψ with α(0,1) being the modified dissipative term. Let J(f,g)=fxgy-fygx denote the Jacobian operator; (1) can be notationally simplified as (2)t[Δψ-Fψ]+J(ψ,Δψ)+βψx=1Re(-Δ)1+αψ. In this model, ψ is the geostrophic pressure, also called the geostrophic stream function, ξ=Δψ is the vertical component of the relative vorticity, ψ=(-ψ/y,ψ/y) is a zeroth-order balance in the momentum equation, and F,β, and Re are the rotational Froude number, the Coriolis parameter, and the Reynolds number, respectively. Usually, ν=1/Re is also called viscosity parameter. It has some features in common with the much studied two-dimensional surface quasigeostrophic equation (SQGE) (see  and references therein). However the quasi-geostrophic β-plane model has a number of novel and distinctive features.

Recently, this equation has been intensively investigated because of both its mathematical importance and its potential applications in meteorology and oceanography. The quasi-geostrophic β-plane model is a simplified model for the shallow water β-plane model [2, 10, 11] when the Rossby number is small under several assumptions on the magnitude of the bottom topography variations, which is used to understand the atmospheric and oceanic circulation, the gulf stream, and the variability of this circulation on time scales from several months to several years. In this regime, quasi-geostrophic theory is an adequate approximation to describe the flow and is developed for the simulation of large-scale geophysical currents in the middle latitudes.

When α=1, this is the standard quasi-geostrophic model studied in , which was put forward as a simplified model of the shallow water model (see also  for a review). In , the author studied a multilayer quasi-geostrophic model, which is a generalization of the single layer model in the case α=1. The general fractional power α was considered by Pu and Guo . The equation is (3)t[Δψ-Fψ]+J(ψ,Δψ)+βψx=1Re(-Δ)1+αψ+f,(4)ψ(x,y,0)=ψ0(x,y). In , they proved the global existence of weak solutions by employing the Galerkin approximation method for initial data belonging to the (inhomogeneous) Sobolev space H2(Ω). If the initial data is in the (homogeneous) Sobolev space H˙s(Ω)(s>2), it is natural for us to ask whether (3) has regular solutions.

In this paper, we only consider the 2D torus 𝕋2 with periodic boundary conditions. And we will prove the well-posedness results of (3) under certain condition on initial data which belong to the (homogeneous) Sobolev space H˙s(𝕋2)(s>3-2α). In Section 3, the local existence and uniqueness of the solutions of the problem are proved in H˙s(𝕋2) when s>3-2α for 1/2<α<1. That is, for any initial data ψ0H˙s(𝕋2) and fL2(0,T;H˙s-α-2(𝕋2)), there exists (5)T=(fL2(0,T;H˙s-α-2),ψ0H˙s,Re), such that (3) has a uniqueness solution on [0,T], satisfying (6)ψL(0,T;H˙s(𝕋2))L2(0,T;H˙s+α(𝕋2)). However, we may not obtain the global existence of solutions from energy (34), if the initial data has large H˙s norm. The main reason is that in the H˙s energy estimate for (3), the integral (Λ2(s-1),J(ψ,Δψ))0 for s>2, where (u,v) denotes the integral 𝕋2u(x,y)v(x,y)dxdy as usual. Thus, it is necessary to control it. To overcome this essential difficulty, we will make use of the properties of the product estimates (Proposition 2) as well as those of the Sobolev embedding inequality.

In Section 4, global existence and uniqueness for small initial data in H˙s(𝕋2) are also proved when s>3-2α. More precisely, we just need the following condition: (7)ψ0H˙xs+fLt2H˙xs-α-2+(ψ0Hx1γ+fLt2Lx2γ)(ψ0H˙xs1-γ+fLt2H˙xs-α-21-γ)<ε, where γ=1-((3-2α)/s).

For the cases, α=1/2 and s>3, we also obtain the unique global solution in Hs proved by(8)(ψ0Hx1γ1+fLt2H˙xs-5/2γ1)(ψ0H˙xs1-γ1+fLt2H˙xs-5/21-γ1)+(ψ0Hx1γ2+fLt2Lx2γ2)(ψ0H˙xs1-γ2+fLt2H˙xs-5/21-γ2)<ε, where γ1=1-2/s and γ2=1-3/s.

We conclude this introduction by mentioning the global existence result of weak solutions obtained .

Proposition 1.

Let α(0,1),T>0,ψ0H2(Ω), and fL2(0,T;L2(Ω)). There exists a weak solution of (3)-(4) which satisfies (9)ψL(0,T;H2(Ω))L2(0,T;H2+α(Ω)).

2. Notations and Preliminaries

We now review the notations used throughout the paper. Let us denote Λ=(-Δ)1/2. The Fourier transform f^ of a tempered distribution f(x) on 𝕋2 is defined as (10)f^(k)=1(2π)2𝕋2f(x)e-ik·xdx. Generally, Λβf for β can be identified with the Fourier series (11)k2|k|βf^(k)eik·x.Lp(𝕋2) denotes the space of the pth-power integrable functions normed by (12)fLp=(𝕋2|f|pdx)1/p,fL=esssupx𝕋2|f(x)|. For any tempered distribution f on Ω and s, we define (13)fH˙s=ΛsfL2=(k2|k|2s|f^(k)|2)1/2.H˙s denotes the homogeneous Sobolev space of all f for which fH˙s is finite. The homogeneous counterparts of H˙s are denoted by Hs.

Next, this section contains a few auxiliary results used in the paper. In particular, we recall, by now, the classical, product, and commutator estimates, as well as the Sobolev embedding inequalities. Proofs of these results can be found for instance, in .

Proposition 2 (product estimate).

If s>0, then, for all f,gHsL, one has the estimates (14)Λs(fg)LpC(fLp1ΛsgLp2+ΛsfLp3gLp4), where 1/p=1/p1  +1/p2=1/p3+1/p4 and p1,p2, and p3(1,). In particular (15)Λs(fg)L2C(fLΛsgL2+ΛsfL2gL).

In the case of a commutator we have the following estimate.

Proposition 3 (commutator estimate).

Suppose that s>0 and p(1,). If f,gS, then (16)Λs(fg)-fΛs(g)LpC(fLp1fΛs-1gLp2+Λs(f)Lp3gLp2), where s>0, 1/p=1/p1+1/p2=1/p3+1/p4, and p1,p2, and p3(1,).

We will use as well the following Sobolev inequality.

Proposition 4 (Sobolev inequality).

Suppose that q>1, p[q,), and (17)1p=1q-sd. Suppose that ΛsfLq; then fLp and there is a constant C>0 such that (18)fLpCΛsfLq.

The following result is from Henry  with extensions for nonintegral order derivatives like in, for example, Triebel [18, 19].

Proposition 5.

If 0a1,1p,q,r, and m,k are nonnegative with (19)k-nq=a(m-np)+(1-a)(-nr),1qap+1-ar except that one requires a1 when m-(n/p)=k,1<p<, then there is a constant C such that (20)DkuLqCDmuLpauLr1-a, for all uCc.

3. Local Existence and Large Data

In , the authors studied and established the existence and uniqueness of local and global solutions to the two-dimensional SQGE. It is natural that (3) is more complex than SQGE. However, we also establish an analogue. In this section we will prove that (3) is locally well-posed in H˙s(𝕋2) when s>3-2α for 1/2<α<1. Regarding arbitrarily large initial data, we obtain the following result.

Theorem 6 (local existence).

Let α(1/2,1) and fix s>3-2α. Assume that ψ0H˙s(𝕋2) and fL2(0,T;H˙s-α-2(𝕋2)) have zero mean on 𝕋2. Then there exist a time T>0 and a unique smooth solution (21)ψL(0,T;H˙s(𝕋2))L2(0,T;H˙s+α(𝕋2)) of the Cauchy problem (3)-(4).

Proof.

First of all, multiplying (3) by Λ2(s-1)ψ, we get the following energy inequality: (22)12ddt[ΛsψL22+FΛs-1ψL22]+1ReΛs+αψL22|(f,Λ2(s-1)ψ)|+|(J(ψ,Δψ),Λ2(s-1)ψ)|+|(βψx,Λ2(s-1)ψ)|. Integration by parts gives us the following estimate: (23)(βψx,Λ2(s-1)ψ)=0. Then we get the inequality (24)12ddt[ΛsψL22+FΛs-1ψL22]+1ReΛs+αψL22|(f,Λ2(s-1)ψ)|+|(J(ψ,Δψ),Λ2(s-1)ψ)|. We estimate the first term on the right side by (25)|(f,Λ2(s-1)ψ)|Re2Λs-α-2fL22+12ReΛs+αψL22. To handle the second term, we proceed as follows.

First note that (26)|(J(ψ,Δψ),Λ2(s-1)ψ)|=|(ψ·Δψ,Λ2(s-1)ψ)|=|Λs-α-2(ψ·Δψ),Λs+αψ|Λs-α-2(ψ·Δψ)L2Λs+αψL2Λs-α-1(ψ·Δψ)L2Λs+αψL2. The estimate of the product term follows from Proposition 2. Hence, we have (27)Λs-α-1(ψ·Δψ)L2C(Λs-αψL2ΔψL2+Λs-α+1ψLpψL2p/(p-2)). We now fix an arbitrary p such that (28)2s-1<p<21+(1-2α)=11-α.

Note that p>2 since s>2 and the range for p is nonempty since s>3-2α. For α(1/2,1), our choice of p and Proposition 5 give (29)Λs-α+1ψLpΛsψL21-ξΛs+αψL2ξ, where ξ(0,1) may be computed explicitly from ξα=2-α-(2/p).

In order to estimate Λs-αψL2ΔψL2 in (27), we split it into two cases.

Case 1 (3-2α<2<s). From Proposition 5 and Sobolev inequality, we have (30)Λs-αψL2Λs+αψL2θΛs-1ψL21-θΛs+αψL2θΛsψL21-θ, where θ=(1-α)/(1+α). In addition, since ψ has zero mean and p>2/(s-1), from the Sobolev embedding we obtain (31)ΛψL2p/(p-2)CΛsψL2,ΔψL2CΛsψL2. Combining estimates (27)–(31) gives (32)12ddt[ΛsψL22+FΛs-1ψL22]+1ReΛs+αψL22ReΛs-α-2fL22+C(Λs+αψL21+θΛsψL22-θ+Λs+αψL21+ξΛsψL22-ξ), where 0<ξ<1 and θ=(1-α)/(1+α) is as defined earlier. The second term on the right side of (32) is bounded using the ε-Young inequality as (33)12ReΛs+αψL22+CRe-(1+θ)/(1-θ)ΛsψL2(2-θ)/(1-θ)+CRe-(1+ξ)/(1-ξ)ΛsψL2(2-ξ)/(1-ξ) and we finally obtain the following estimate: (34)ddt[ΛsψL22+FΛs-1ψL22]+12ReΛs+αψL22ReΛs-α-2fL22+C(ΛsψL22(2-θ)/(1-θ)+ΛsψL22(2-ξ)/(1-ξ)). Using Gronwall’s inequality, from estimate (34) we may deduce the existence of a positive time (35)T=(fL2(0,T;H˙s-α-2),ψ0H˙s,Re) such that (36)ψL(0,T;H˙s(𝕋2))L2(0,T;H˙s+α(𝕋2)). Note that we have 2(2-θ)/(1-θ)>2,2(2-ξ)/(1-ξ)>2, and hence we may not obtain the global existence of solutions from the energy (34), if the initial data has large Hs norm. These a priori estimates can be made formal using a standard approximation procedure. We omit further details

Case 2 (3-2α<s<2). Using Proposition 5, we obtain (37)ΔψL2CΛsψL21-θ1Λs+αψL2θ1, where θ1=(2-s)/α. From Sobolev embedding, we have (38)Λs-αψL2CΛsψL2. Then, using the same method as in Case 1, we can complete Theorem 6.

4. Global Existence and Small Data

The main result of this section concerns global well-posedness in case of small initial data.

Theorem 7 (global existence).

Let α(1/2,1), fL2(0,;H˙s-α-2(𝕋2))L2(0,;L2(𝕋2)) and let ψ0H˙s(𝕋2) have zero mean on 𝕋2, where s>3-2α. There exists a small enough constant ε>0 depending on Re, such that if (39)ψ0H˙xs+fLt2H˙xs-α-2+(ψ0Hx1γ+fLt2Lx2γ)(ψ0H˙xs1-γ+fLt2H˙xs-α-21-γ)<ε, where γ=1-(3-2α)/s, then the unique smooth solution ψ of the Cauchy problem (3)-(4) is global in time; that is, ψL(0,;H˙s(𝕋2)).

Proof.

We proceed as in the proof of Theorem 6. The product term in (27) is now estimated by (40)Λs-α-1(ψ·Δψ)L2C(Λs-αψL2ΔψL2+Λs-α+1ψLpψL2p/(p-2)), where p=1/(1-α), so that (41)Λs-α+1ψLpΛs+αψL2.

Similarly, in order to estimate Λs-αψL2ΔψL2 in (40), we split it into two cases.

Case 3 (3-2α<2<s). From Sobolev imbedding, we have (42)Λs-αψL2CΛs+αψL2,ΔψL2CΛsψL2.

Case 4 (3-2α<s<2). Using Sobolev imbedding, we have (43)Λs-αψL2CΛsψL2,ΔψL2CΛs+αψL2.

So, we can always obtain the following estimate (44)Λs-αψL2ΔψL2CΛsψL2Λs+αψL2.

With this choice of p and the above embedding, the product estimate gives us (45)Λs-α-1(ψ·Δu)L2C(Λs+αψL2ΛsψL2+Λs+αψL2ψL2p/(p-2))CΛs+αψL2(ΛsψL2+ψL2p/(p-2)). Combining (24) with (45) and proceeding as in (34) we obtain (46)12ddt[ΛsψL22+FΛs-1ψL22]+1ReΛs+αψL22Λs-α-2fL2Λs+αψL2+CΛs+αψL22(ΛsψL2+ψL2p/(p-2)) which in turn implies (47)12ddt[ΛsψL22+FΛs-1ψL22]+12ReΛs+αψL22Re2Λs-α-2fL22+CΛs+αψL22(ΛsψL2+ψL2p/(p-2)).

Observe that (48)ψL2p/(p-2)CψL2γΛsψL21-γ, where γ=1-(3-2α)/s. Therefore, if (49)C(ΛsψL2+ψL2γΛsψL21-γ)14CRe estimate (47) combined with Sobolev imbedding inequality Λs+αψL2ΛsψL2 shows that (50)ddt[ΛsψL22+FΛs-1ψL22]+12ReΛsψL22RefH˙xs-α-22 and hence (51)sup0t<{ΛsψL22+FΛs-1ψL22}Λsψ0L22+FΛs-1ψ0L22+RefLt2H˙xs-α-22. By Sobolev imbedding, we have (52)Λs-1ψ0L2CΛsψ0L2. Combining (51) and (52), we get (53)ΛsψL22CΛsψ0L22+RefLt2H˙xs-α-22. Note that taking the L2-product of (3) with ψ gives for any t>0(54)ddt[ψL22+FΛs-1ψL22]+1Re0tΛ1+αψL22RefLx22. Thus, there exists some constant K (dependent on F) such that (55)sup0t<{ψL22+FψL22}Kψ0H12+KfLt2Lx2 which gives us a basic uniform estimate of ψ in LtHx1.

Hence, from (53) and (55) we obtain that condition (49) is satisfied for all t>0 as long as we have (56)ψ0H˙xs+fLt2H˙xs-α-2+(ψ0Hx1γ+fLt2Lx2γ)×(ψ0H˙xs1-γ+fLt2H˙xs-α-21-γ)<ε, where ε is sufficiently small, thereby concluding the proof of the theorem.

Note also that the proof of Theorem 7 fails for the value α=1/2. Thus, α=1/2 indeed is the limit of the local well-posedness theory. Nonetheless, we still can prove that the considered system is globally well-posed for small data.

Theorem 8 (global existence for small data).

Let s>3 and assume that the initial data ψ0H˙s(𝕋2) and fL2(0,;H˙s-(5/2)(𝕋2))L2(0,;L2(𝕋2)) have zero mean on 𝕋2. There exists a sufficiently small constant ε>0 depending on Re, such that if (57)(ψ0Hx1γ1+fLt2H˙xs-(5/2)γ1)(ψ0H˙xs1-γ1+fLt2H˙xs-(5/2)1-γ1)+(ψ0Hx1γ2+fLt2Lx2γ2)(ψ0H˙xs1-γ2+fLt2H˙xs-(5/2)1-γ2)<ε, where γ1=1-(2/s) and γ2=1-(3/s), then the unique smooth solution (58)ψL(0,;H˙s(𝕋2)) of the Cauchy problem (3)-(4) is global in time.

Proof.

We proceed as in the proof of Theorem 7 and obtain the energy estimate (59)12ddt[ΛsψL22+Λs-1ψL22]+1ReΛs+(1/2)ψL22Λs-(5/2)fL2Λs+(1/2)ψL2+Λs-(3/2)(ψ·Δψ)L2Λs+(1/2)ψL2.

The second term on the right side is estimated using the product estimate in Proposition 2. Thus we obtain, similar to (45), (60)Λs-(3/2)(ψ·Δψ)L2C(ψLΛs+(1/2)ψL2+Λs-(1/2)ψL2ΔψL)C(ψLΛs+(1/2)ψL2+Λs+(1/2)ψL2ΔψL). By interpolation inequality, we have (61)ψLΛsψL21-γ1ψL2γ1,ΔψLΛsψL21-γ2ψL2γ2, where γ1=1-(2/s) and γ2=1-(3/s). Combining estimates (61) gives (62)12ddt[ΛsψL22+Λs-1ψL22]+1ReΛs+(1/2)ψL2212ReΛs-(5/2)fL22+CRe2Λs+(1/2)ψL22×(ΛsψL21-γ1ψL2γ1+ΛsψL21-γ2ψL2γ2).

We obtain the desired result as in the proof of Theorem 7.

Remark 9.

When 2<s3, the result of Theorem 8 is still open.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is supported by the Doctoral Starting-up Foundation of Minnan Normal University, China-NSAF (Grant no. L21228).

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