1. Introduction
This paper is concerned with the nonlocal quasigeostrophic β-plane model with modified dissipativity [1, 2]
(1)(∂∂t+∂ψ∂x∂∂y-∂ψ∂y∂∂x)q=1Re(-Δ)1+αψ,
where (x,y)∈ Ω can be either the 2D torus 𝕋2 or the whole space ℝ2, t≥0, 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)+β∂ψ∂x=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 [3–9] 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 [1], which was put forward as a simplified model of the shallow water model (see also [2] for a review). In [12], 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 [13]. The equation is
(3)∂∂t[Δψ-Fψ]+J(ψ,Δψ)+β∂ψ∂x=1Re(-Δ)1+αψ+f,(4)ψ(x,y,0)=ψ0(x,y).
In [13], 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 ψ0∈H˙s(𝕋2) and f∈L2 (0,T;H˙s-α-2(𝕋2)), there exists
(5)T=(∥f∥L2(0,T;H˙s-α-2),∥ψ0∥H˙s,Re),
such that (3) has a uniqueness solution on [0,T], satisfying
(6)ψ∈L∞(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)dx dy 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)∥ψ0∥H˙xs+∥f∥Lt2H˙xs-α-2 +(∥ψ0∥Hx1γ+∥f∥Lt2Lx2γ)(∥ψ0∥H˙xs1-γ+∥f∥Lt2H˙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)(∥ψ0∥Hx1γ1+∥f∥Lt2H˙xs-5/2γ1)(∥ψ0∥H˙xs1-γ1+∥f∥Lt2H˙xs-5/21-γ1) +(∥ψ0∥Hx1γ2+∥f∥Lt2Lx2γ2)(∥ψ0∥H˙xs1-γ2+∥f∥Lt2H˙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 [13].

Proposition 1.
Let α∈(0,1),T>0, ψ0∈H2(Ω), and f∈L2(0,T;L2(Ω)). There exists a weak solution of (3)-(4) which satisfies
(9)ψ∈L∞(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)2∫𝕋2f(x)e-ik·xdx.
Generally, Λβf for β∈ℝ can be identified with the Fourier series
(11)∑k∈ℤ2|k|βf^(k)eik·x.Lp(𝕋2) denotes the space of the pth-power integrable functions normed by
(12)∥f∥Lp=(∫𝕋2|f|pdx)1/p, ∥f∥L∞=ess supx∈𝕋2|f(x)|.
For any tempered distribution f on Ω and s∈ℝ, we define
(13)∥f∥H˙s=∥Λsf∥L2=(∑k∈ℤ2|k|2s|f^(k)|2)1/2.H˙s denotes the homogeneous Sobolev space of all f for which ∥f∥H˙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 [14–16].

Proposition 2 (product estimate).
If s>0, then, for all f,g∈Hs∩L∞, one has the estimates
(14)∥Λs(fg)∥Lp≤C(∥f∥Lp1∥Λsg∥Lp2+∥Λsf∥Lp3∥g∥Lp4),
where 1/p=1/p1 +1/p2=1/p3+1/p4 and p1,p2, and p3∈(1,∞). In particular
(15)∥Λs(fg)∥L2≤C(∥f∥L∞∥Λsg∥L2+∥Λsf∥L2∥g∥L∞).

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

Proposition 3 (commutator estimate).
Suppose that s>0 and p∈(1,∞). If f,g∈S, then
(16)∥Λs(fg)-fΛs(g)∥Lp ≤C(∥∇f∥Lp1∥fΛs-1g∥Lp2+∥Λs(f)∥Lp3∥g∥Lp2),
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 Λsf∈Lq; then f∈Lp and there is a constant C>0 such that
(18)∥f∥Lp≤C∥Λsf∥Lq.

The following result is from Henry [17] with extensions for nonintegral order derivatives like in, for example, Triebel [18, 19].

Proposition 5.
If 0≤a≤1,1≤p, q, r≤∞, and m,k are nonnegative with
(19)k-nq=a(m-np)+(1-a)(-nr),1q≤ap+1-ar
except that one requires a≠1 when m-(n/p)=k,1<p<∞, then there is a constant C such that
(20)∥Dku∥Lq≤C∥Dmu∥Lpa∥u∥Lr1-a,
for all u∈Cc∞.

3. Local Existence and Large Data
In [7], 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 ψ0∈H˙s(𝕋2) and f∈L2(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)ψ∈L∞(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[∥Λsψ∥L22+F∥Λs-1ψ∥L22]+1Re∥Λs+αψ∥L22 ≤|(f,Λ2(s-1)ψ)|+|(J(ψ,Δψ),Λ2(s-1)ψ)| +|(β∂ψ∂x,Λ2(s-1)ψ)|.
Integration by parts gives us the following estimate:
(23)(β∂ψ∂x,Λ2(s-1)ψ)=0.
Then we get the inequality
(24)12ddt[∥Λsψ∥L22+F∥Λs-1ψ∥L22]+1Re∥Λs+αψ∥L22 ≤|(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)|≤Re2∥Λs-α-2f∥L22+12Re∥Λs+αψ∥L22.
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)∥L2∥Λs+αψ∥L2 ≤∥Λs-α-1(∇⊥ψ·Δψ)∥L2∥Λs+αψ∥L2.
The estimate of the product term follows from Proposition 2. Hence, we have
(27)∥Λs-α-1(∇⊥ψ·Δψ)∥L2 ≤C(∥Λs-αψ∥L2∥Δψ∥L2+∥Λs-α+1ψ∥Lp∥∇ψ∥L2p/(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+1ψ∥Lp≤∥Λsψ∥L21-ξ∥Λs+αψ∥L2ξ,
where ξ∈(0,1) may be computed explicitly from ξα=2-α-(2/p).

In order to estimate ∥Λs-αψ∥L2∥Δψ∥L2 in (27), we split it into two cases.

Case
1 (3-2α<2<s). From Proposition 5 and Sobolev inequality, we have
(30)∥Λs-αψ∥L2≤∥Λs+αψ∥L2θ∥Λs-1ψ∥L21-θ≤∥Λs+αψ∥L2θ∥Λsψ∥L21-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)∥Λψ∥L2p/(p-2)≤C∥Λsψ∥L2,∥Δψ∥L2≤C∥Λsψ∥L2.
Combining estimates (27)–(31) gives
(32)12ddt[∥Λsψ∥L22+F∥Λs-1ψ∥L22]+1Re∥Λs+αψ∥L22 ≤Re∥Λs-α-2f∥L22+C(∥Λs+αψ∥L21+θ∥Λsψ∥L22-θ +∥Λs+αψ∥L21+ξ∥Λsψ∥L22-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)12Re∥Λs+αψ∥L22+CRe-(1+θ)/(1-θ)∥Λsψ∥L2(2-θ)/(1-θ) +CRe-(1+ξ)/(1-ξ)∥Λsψ∥L2(2-ξ)/(1-ξ)
and we finally obtain the following estimate:
(34)ddt[∥Λsψ∥L22+F∥Λs-1ψ∥L22]+12Re∥Λs+αψ∥L22 ≤Re∥Λs-α-2f∥L22 +C(∥Λsψ∥L22(2-θ)/(1-θ)+∥Λsψ∥L22(2-ξ)/(1-ξ)).
Using Gronwall’s inequality, from estimate (34) we may deduce the existence of a positive time
(35)T=(∥f∥L2(0,T;H˙s-α-2),∥ψ0∥H˙s,Re)
such that
(36)ψ∈L∞(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)∥Δψ∥L2≤C∥Λsψ∥L21-θ1∥Λs+αψ∥L2θ1,
where θ1=(2-s)/α. From Sobolev embedding, we have
(38)∥Λs-αψ∥L2≤C∥Λsψ∥L2.
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), f∈L2(0,∞;H˙s-α-2(𝕋2))∩L2(0,∞;L2(𝕋2)) and let ψ0∈H˙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)∥ψ0∥H˙xs+∥f∥Lt2H˙xs-α-2 +(∥ψ0∥Hx1γ+∥f∥Lt2Lx2γ)(∥ψ0∥H˙xs1-γ+∥f∥Lt2H˙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, ψ∈L∞(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)∥L2 ≤C(∥Λs-αψ∥L2∥Δψ∥L2+∥Λs-α+1ψ∥Lp∥∇ψ∥L2p/(p-2)),
where p=1/(1-α), so that
(41)∥Λs-α+1ψ∥Lp≤∥Λs+αψ∥L2.

Similarly, in order to estimate ∥Λs-αψ∥L2∥Δψ∥L2 in (40), we split it into two cases.

Case
3 (3-2α<2<s). From Sobolev imbedding, we have
(42)∥Λs-αψ∥L2≤C∥Λs+αψ∥L2,∥Δψ∥L2≤C∥Λsψ∥L2.

Case
4 (3-2α<s<2). Using Sobolev imbedding, we have
(43)∥Λs-αψ∥L2≤C∥Λsψ∥L2,∥Δψ∥L2≤C∥Λs+αψ∥L2.

So, we can always obtain the following estimate
(44)∥Λs-αψ∥L2∥Δψ∥L2≤C∥Λsψ∥L2∥Λs+αψ∥L2.

With this choice of p and the above embedding, the product estimate gives us
(45)∥Λs-α-1(∇⊥ψ·Δu)∥L2 ≤C(∥Λs+αψ∥L2∥Λsψ∥L2+∥Λs+αψ∥L2∥∇ψ∥L2p/(p-2)) ≤C∥Λs+αψ∥L2(∥Λsψ∥L2+∥∇ψ∥L2p/(p-2)).
Combining (24) with (45) and proceeding as in (34) we obtain
(46)12ddt[∥Λsψ∥L22+F∥Λs-1ψ∥L22]+1Re∥Λs+αψ∥L22 ≤∥Λs-α-2f∥L2∥Λs+αψ∥L2 +C∥Λs+αψ∥L22(∥Λsψ∥L2+∥∇ψ∥L2p/(p-2))
which in turn implies
(47)12ddt[∥Λsψ∥L22+F∥Λs-1ψ∥L22]+12Re∥Λs+αψ∥L22 ≤Re2∥Λs-α-2f∥L22 +C∥Λs+αψ∥L22(∥Λsψ∥L2+∥∇ψ∥L2p/(p-2)).

Observe that
(48)∥∇ψ∥L2p/(p-2)≤C∥ψ∥L2γ∥Λsψ∥L21-γ,
where γ=1-(3-2α)/s. Therefore, if
(49)C(∥Λsψ∥L2+∥ψ∥L2γ∥Λsψ∥L21-γ)≤14CRe
estimate (47) combined with Sobolev imbedding inequality ∥Λs+αψ∥L2≥∥Λsψ∥L2 shows that
(50)ddt[∥Λsψ∥L22+F∥Λs-1ψ∥L22] +12Re∥Λsψ∥L22≤Re∥f∥H˙xs-α-22
and hence
(51)sup0≤t<∞{∥Λsψ∥L22+F∥Λs-1ψ∥L22} ≤∥Λsψ0∥L22+F∥Λs-1ψ0∥L22+Re∥f∥Lt2H˙xs-α-22.
By Sobolev imbedding, we have
(52)∥Λs-1ψ0∥L2≤C∥Λsψ0∥L2.
Combining (51) and (52), we get
(53)∥Λsψ∥L22≤C∥Λsψ0∥L22+Re∥f∥Lt2H˙xs-α-22.
Note that taking the L2-product of (3) with ψ gives for any t>0(54)ddt[∥∇ψ∥L22+F∥Λs-1ψ∥L22]+1Re∫0t∥Λ1+αψ∥L22≤Re∥f∥Lx22.
Thus, there exists some constant K (dependent on F) such that
(55)sup0≤t<∞{∥∇ψ∥L22+F∥ψ∥L22}≤K∥ψ0∥H12+K∥f∥Lt2Lx2
which gives us a basic uniform estimate of ψ in Lt∞Hx1.

Hence, from (53) and (55) we obtain that condition (49) is satisfied for all t>0 as long as we have
(56)∥ψ0∥H˙xs+∥f∥Lt2H˙xs-α-2+(∥ψ0∥Hx1γ+∥f∥Lt2Lx2γ) ×(∥ψ0∥H˙xs1-γ+∥f∥Lt2H˙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 ψ0∈H˙s(𝕋2) and f∈L2(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)(∥ψ0∥Hx1γ1+∥f∥Lt2H˙xs-(5/2)γ1)(∥ψ0∥H˙xs1-γ1+∥f∥Lt2H˙xs-(5/2)1-γ1) +(∥ψ0∥Hx1γ2+∥f∥Lt2Lx2γ2)(∥ψ0∥H˙xs1-γ2+∥f∥Lt2H˙xs-(5/2)1-γ2)<ε,
where γ1=1-(2/s) and γ2=1-(3/s), then the unique smooth solution
(58)ψ∈L∞(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[∥Λsψ∥L22+∥Λs-1ψ∥L22]+1Re∥Λs+(1/2)ψ∥L22 ≤∥Λs-(5/2)f∥L2∥Λs+(1/2)ψ∥L2 +∥Λs-(3/2)(∇⊥ψ·Δψ)∥L2∥Λs+(1/2)ψ∥L2.

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)∥L2 ≤C(∥∇ψ∥L∞∥Λs+(1/2)ψ∥L2+∥Λs-(1/2)ψ∥L2∥Δψ∥L∞) ≤C(∥∇ψ∥L∞∥Λs+(1/2)ψ∥L2+∥Λs+(1/2)ψ∥L2∥Δψ∥L∞).
By interpolation inequality, we have
(61)∥∇ψ∥L∞≤∥Λsψ∥L21-γ1∥ψ∥L2γ1,∥Δψ∥L∞≤∥Λsψ∥L21-γ2∥ψ∥L2γ2,
where γ1=1-(2/s) and γ2=1-(3/s). Combining estimates (61) gives
(62)12ddt[∥Λsψ∥L22+∥Λs-1ψ∥L22]+1Re∥Λs+(1/2)ψ∥L22 ≤12Re∥Λs-(5/2)f∥L22+CRe2∥Λs+(1/2)ψ∥L22 ×(∥Λsψ∥L21-γ1∥ψ∥L2γ1+∥Λsψ∥L21-γ2∥ψ∥L2γ2).

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

Remark 9.
When 2<s≤3, the result of Theorem 8 is still open.