Well-Posedness of the Two-Dimensional Fractional Quasigeostrophic Equation

Abstract

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 1/2≤𝛼<1.

1. Introduction

This paper is concerned with the nonlocal quasigeostrophic 𝛽-plane model with modified dissipativity [1, 2] 𝜕+𝜕𝑡𝜕𝜓𝜕𝜕𝑥−𝜕𝑦𝜕𝜓𝜕𝜕𝑦1𝜕𝑥𝑞=𝑅_𝑒(−Δ)^1+𝛼𝜓, (1) where (𝑥,𝑦)∈Ω can be either the 2D torus 𝕋² or the whole space ℝ², 𝑡≥0,𝑞=Δ𝜓−𝐹𝜓+𝛽𝑦, and (1/𝑅_𝑒)(−Δ)^1+𝛼𝜓 with 𝛼∈(0,1) being the modified dissipative term. Let 𝐽(𝑓,𝑔)=𝑓_𝑥𝑔_𝑦−𝑓_𝑦𝑔_𝑥 denote the Jacobian operator; (1) can be notationally simplified as 𝜕𝜕𝑡Δ𝜓−𝐹𝜓+𝐽𝜓,Δ𝜓+𝛽𝜕𝜓=1𝜕𝑥𝑅_𝑒(−Δ)^1+𝛼𝜓. (2) 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 𝑅_𝑒 are the rotational Froude number, the Coriolis parameter, and the Reynolds number, respectively. Usually, 𝜈=1/𝑅_𝑒 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 𝜕𝜕𝑡Δ𝜓−𝐹𝜓+𝐽𝜓,Δ𝜓+𝛽𝜕𝜓=1𝜕𝑥𝑅_𝑒(−Δ)^1+𝛼𝜓𝜓+𝑓,𝑥,𝑦,0=𝜓₀𝑥,𝑦. (3)(4) 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 𝐻̇^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 𝐻̇^𝑠(𝕋²)(𝑠>3−2𝛼). In Section 3, the local existence and uniqueness of the solutions of the problem are proved in 𝐻̇^𝑠(𝕋²) when 𝑠>3−2𝛼 for 1/2<𝛼<1. That is, for any initial data 𝜓₀∈𝐻̇^𝑠(𝕋²) and 𝑓∈𝐿²(0,𝑇;𝐻̇^{𝑠−𝛼−2}(𝕋²)), there exists 𝑇=𝑓_{𝐿2̇𝐻(0,𝑇;𝑠−𝛼−2)},𝜓_0̇𝐻𝑠,𝑅_𝑒, (5) such that (3) has a uniqueness solution on [0,𝑇], satisfying 𝜓∈𝐿^∞0,𝑇;𝐻̇^𝑠𝕋²∩𝐿²0,𝑇;𝐻̇^𝑠+𝛼𝕋². (6) However, we may not obtain the global existence of solutions from energy (34), if the initial data has large 𝐻̇^𝑠 norm. The main reason is that in the 𝐻̇^𝑠 energy estimate for (3), the integral (Λ^2(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 𝐻̇^𝑠(𝕋²) are also proved when 𝑠>3−2𝛼. More precisely, we just need the following condition: 𝜓_{0̇𝐻𝑠𝑥}+𝑓_{𝐿2𝑡̇𝐻𝑥𝑠−𝛼−2}+𝜓₀^𝛾_𝐻1𝑥+𝑓^𝛾_{𝐿2𝑡𝐿2𝑥}𝜓_0̇𝐻^1−𝛾_𝑠𝑥+𝑓_𝐿^1−𝛾_{2𝑡̇𝐻𝑥𝑠−𝛼−2}<𝜀, (7) where 𝛾=1−((3−2𝛼)/𝑠).

For the cases, 𝛼=1/2 and 𝑠>3, we also obtain the unique global solution in 𝐻^𝑠 proved by𝜓₀^𝛾1_𝐻1𝑥+𝑓^𝛾1_{𝐿2𝑡̇𝐻𝑥𝑠−5/2}𝜓₀^1−𝛾1_{̇𝐻𝑠𝑥}+𝑓^1−𝛾1_{𝐿2𝑡̇𝐻𝑥𝑠−5/2}+𝜓₀^𝛾2_𝐻1𝑥+𝑓^𝛾2_{𝐿2𝑡𝐿2𝑥}𝜓₀^1−𝛾2_{̇𝐻𝑠𝑥}+𝑓^1−𝛾2_{𝐿2𝑡̇𝐻𝑥𝑠−5/2}<𝜀, (8) 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 𝜓∈𝐿^∞0,𝑇;𝐻²(Ω)∩𝐿²0,𝑇;𝐻^2+𝛼(Ω). (9)

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 𝑓(𝑘)=1(2𝜋)^2𝕋2𝑓(𝑥)𝑒^{−𝑖𝑘⋅𝑥}𝑑𝑥. (10) Generally, Λ^𝛽𝑓 for 𝛽∈ℝ can be identified with the Fourier series _𝑘∈ℤ2|𝑘|^𝛽𝑓(𝑘)𝑒^{𝑖𝑘⋅𝑥}. (11)𝐿^𝑝(𝕋²) denotes the space of the 𝑝th-power integrable functions normed by 𝑓_𝐿𝑝=_𝕋2𝑓^𝑝𝑑𝑥^1/𝑝,𝑓_𝐿∞=esssup_𝑥∈𝕋2𝑓(𝑥). (12) For any tempered distribution 𝑓 on Ω and 𝑠∈ℝ, we define 𝑓_̇𝐻𝑠=Λ^𝑠𝑓_𝐿2=_𝑘∈ℤ2|𝑘|^2𝑠𝑓(𝑘)^21/2. (13)𝐻̇^𝑠 denotes the homogeneous Sobolev space of all 𝑓 for which ‖𝑓‖_̇𝐻𝑠 is finite. The homogeneous counterparts of 𝐻̇^𝑠 are denoted by 𝐻^𝑠.

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∈𝐻^𝑠∩𝐿^∞, one has the estimates Λ^𝑠𝑓𝑔_𝐿𝑝≤𝐶𝑓_𝐿𝑝1Λ^𝑠𝑔_𝐿𝑝2+Λ^𝑠𝑓_𝐿𝑝3𝑔_𝐿𝑝4, (14) where 1/𝑝=1/𝑝₁+1/𝑝₂=1/𝑝₃+1/𝑝₄ and 𝑝₁,𝑝₂, and 𝑝₃∈(1,∞). In particular Λ^𝑠𝑓𝑔_𝐿2≤𝐶𝑓_𝐿∞Λ^𝑠𝑔_𝐿2+Λ^𝑠𝑓_𝐿2𝑔_𝐿∞. (15)

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 Λ^𝑠𝑓𝑔−𝑓Λ^𝑠𝑔_𝐿𝑝≤𝐶∇𝑓_𝐿𝑝1𝑓Λ^𝑠−1𝑔_𝐿𝑝2+Λ^𝑠𝑓_𝐿𝑝3𝑔_𝐿𝑝2, (16) 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 1𝑝=1𝑞−𝑠𝑑. (17) Suppose that Λ^𝑠𝑓∈𝐿^𝑞; then 𝑓∈𝐿^𝑝 and there is a constant 𝐶>0 such that 𝑓_𝐿𝑝≤𝐶Λ^𝑠𝑓_𝐿𝑞. (18)

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 𝑛𝑘−𝑞𝑛=𝑎𝑚−𝑝+()−𝑛1−𝑎𝑟,1𝑞≤𝑎𝑝+1−𝑎𝑟 (19) except that one requires 𝑎≠1 when 𝑚−(𝑛/𝑝)=𝑘,1<𝑝<∞, then there is a constant 𝐶 such that 𝐷^𝑘𝑢_𝐿𝑞≤𝐶𝐷^𝑚𝑢^𝑎_𝐿𝑝‖𝑢‖_𝐿^1−𝑎𝑟, (20) for all 𝑢∈𝐶^∞_𝑐.

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 𝐻̇^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∈𝐻̇^𝑠(𝕋²) and 𝑓∈𝐿²(0,𝑇;𝐻̇^{𝑠−𝛼−2}(𝕋²)) have zero mean on 𝕋². Then there exist a time 𝑇>0 and a unique smooth solution 𝜓∈𝐿^∞0,𝑇;𝐻̇^𝑠𝕋²∩𝐿²0,𝑇;𝐻̇^𝑠+𝛼𝕋² (21) of the Cauchy problem (3)-(4).

Proof. First of all, multiplying (3) by Λ^2(𝑠−1)𝜓, we get the following energy inequality: 12𝑑Λ𝑑𝑡^𝑠𝜓²_𝐿2+𝐹Λ^𝑠−1𝜓²_𝐿2+1𝑅_𝑒Λ^𝑠+𝛼𝜓²_𝐿2≤𝑓,Λ^2(𝑠−1)𝜓+𝐽𝜓,Δ𝜓,Λ^2(𝑠−1)𝜓+𝛽𝜕𝜓𝜕𝑥,Λ^2(𝑠−1)𝜓. (22) Integration by parts gives us the following estimate: 𝛽𝜕𝜓𝜕𝑥,Λ^2(𝑠−1)𝜓=0. (23) Then we get the inequality 12𝑑Λ𝑑𝑡^𝑠𝜓²_𝐿2+𝐹Λ^𝑠−1𝜓²_𝐿2+1𝑅_𝑒Λ^𝑠+𝛼𝜓²_𝐿2≤𝑓,Λ^2(𝑠−1)𝜓+𝐽𝜓,Δ𝜓,Λ^2(𝑠−1)𝜓. (24) We estimate the first term on the right side by 𝑓,Λ^2(𝑠−1)𝜓≤𝑅_𝑒2Λ^{𝑠−𝛼−2}𝑓²_𝐿2+12𝑅_𝑒Λ^𝑠+𝛼𝜓²_𝐿2. (25) To handle the second term, we proceed as follows.
First note that 𝐽𝜓,Δ𝜓,Λ^2(𝑠−1)𝜓=∇^⟂𝜓⋅∇Δ𝜓,Λ^2(𝑠−1)𝜓=Λ^{𝑠−𝛼−2}∇^⟂𝜓⋅∇Δ𝜓,Λ^𝑠+𝛼𝜓≤Λ^{𝑠−𝛼−2}∇^⟂𝜓⋅∇Δ𝜓_𝐿2Λ^𝑠+𝛼𝜓_𝐿2≤Λ^{𝑠−𝛼−1}∇^⟂𝜓⋅Δ𝜓_𝐿2Λ^𝑠+𝛼𝜓_𝐿2. (26) The estimate of the product term follows from Proposition 2. Hence, we have Λ^{𝑠−𝛼−1}∇^⟂𝜓⋅Δ𝜓_𝐿2≤𝐶Λ^𝑠−𝛼𝜓_𝐿2Δ𝜓_𝐿2+Λ^{𝑠−𝛼+1}𝜓_𝐿𝑝∇𝜓_{𝐿2𝑝/(𝑝−2)}. (27) We now fix an arbitrary 𝑝 such that 22𝑠−1<𝑝<()=11+1−2𝛼.1−𝛼 (28)
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 Λ^{𝑠−𝛼+1}𝜓_𝐿𝑝≤Λ^𝑠𝜓_𝐿^1−𝜉₂Λ^𝑠+𝛼𝜓^𝜉_𝐿2, (29) where 𝜉∈(0,1) may be computed explicitly from 𝜉𝛼=2−𝛼−(2/𝑝).
In order to estimate ‖Λ^𝑠−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 Λ^𝑠−𝛼𝜓_𝐿2≤Λ^𝑠+𝛼𝜓^𝜃_𝐿2Λ^𝑠−1𝜓_𝐿^1−𝜃₂≤Λ^𝑠+𝛼𝜓^𝜃_𝐿2Λ^𝑠𝜓_𝐿^1−𝜃₂, (30) where 𝜃=(1−𝛼)/(1+𝛼). In addition, since 𝜓 has zero mean and 𝑝>2/(𝑠−1), from the Sobolev embedding we obtain Λ𝜓_{𝐿2𝑝/(𝑝−2)}≤𝐶Λ^𝑠𝜓_𝐿2,Δ𝜓_𝐿2≤𝐶Λ^𝑠𝜓_𝐿2. (31) Combining estimates (27)–(31) gives 12𝑑Λ𝑑𝑡^𝑠𝜓²_𝐿2+𝐹Λ^𝑠−1𝜓²_𝐿2+1𝑅_𝑒Λ^𝑠+𝛼𝜓²_𝐿2≤𝑅_𝑒Λ^{𝑠−𝛼−2}𝑓²_𝐿2+𝐶Λ^𝑠+𝛼𝜓_𝐿^1+𝜃₂Λ^𝑠𝜓_𝐿^2−𝜃₂+Λ^𝑠+𝛼𝜓_𝐿^1+𝜉₂Λ^𝑠𝜓_𝐿^2−𝜉₂, (32) 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 12𝑅_𝑒Λ^𝑠+𝛼𝜓²_𝐿2+𝐶𝑅_𝑒^{−(1+𝜃)/(1−𝜃)}Λ^𝑠𝜓_𝐿^{(2−𝜃)/(1−𝜃)}₂+𝐶𝑅_𝑒^{−(1+𝜉)/(1−𝜉)}Λ^𝑠𝜓_𝐿^{(2−𝜉)/(1−𝜉)}₂ (33) and we finally obtain the following estimate: 𝑑Λ𝑑𝑡^𝑠𝜓²_𝐿2+𝐹Λ^𝑠−1𝜓²_𝐿2+12𝑅_𝑒Λ^𝑠+𝛼𝜓²_𝐿2≤𝑅_𝑒Λ^{𝑠−𝛼−2}𝑓²_𝐿2+𝐶Λ^𝑠𝜓_𝐿^{2(2−𝜃)/(1−𝜃)}₂+Λ^𝑠𝜓_𝐿^{2(2−𝜉)/(1−𝜉)}₂. (34) Using Gronwall’s inequality, from estimate (34) we may deduce the existence of a positive time 𝑇=𝑓_{𝐿2̇𝐻(0,𝑇;𝑠−𝛼−2)},𝜓_0̇𝐻𝑠,𝑅_𝑒 (35) such that 𝜓∈𝐿^∞0,𝑇;𝐻̇^𝑠𝕋²∩𝐿²0,𝑇;𝐻̇^𝑠+𝛼𝕋². (36) 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 𝐻^𝑠 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 Δ𝜓_𝐿2≤𝐶Λ^𝑠𝜓^1−𝜃1_𝐿2Λ^𝑠+𝛼𝜓^𝜃1_𝐿2, (37) where 𝜃₁=(2−𝑠)/𝛼. From Sobolev embedding, we have Λ^𝑠−𝛼𝜓_𝐿2≤𝐶Λ^𝑠𝜓_𝐿2. (38) 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,∞;𝐻̇^{𝑠−𝛼−2}(𝕋²))∩𝐿²(0,∞;𝐿²(𝕋²)) and let 𝜓₀∈𝐻̇^𝑠(𝕋²) have zero mean on 𝕋², where 𝑠>3−2𝛼. There exists a small enough constant 𝜀>0 depending on 𝑅_𝑒, such that if 𝜓_{0̇𝐻𝑠𝑥}+𝑓_{𝐿2𝑡̇𝐻𝑥𝑠−𝛼−2}+𝜓₀^𝛾_𝐻1𝑥+𝑓^𝛾_{𝐿2𝑡𝐿2𝑥}𝜓_0̇𝐻^1−𝛾_𝑠𝑥+𝑓_𝐿^1−𝛾_{2𝑡̇𝐻𝑥𝑠−𝛼−2}<𝜀, (39) 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,∞;𝐻̇^𝑠(𝕋²)).

Proof. We proceed as in the proof of Theorem 6. The product term in (27) is now estimated by Λ^{𝑠−𝛼−1}∇^⟂𝜓⋅Δ𝜓_𝐿2≤𝐶Λ^𝑠−𝛼𝜓_𝐿2Δ𝜓_𝐿2+Λ^{𝑠−𝛼+1}𝜓_𝐿𝑝∇𝜓_{𝐿2𝑝/(𝑝−2)}, (40) where 𝑝=1/(1−𝛼), so that Λ^{𝑠−𝛼+1}𝜓_𝐿𝑝≤Λ^𝑠+𝛼𝜓_𝐿2. (41)
Similarly, in order to estimate ‖Λ^𝑠−𝛼𝜓‖_𝐿2‖Δ𝜓‖_𝐿2 in (40), we split it into two cases.
Case 3 (3−2𝛼<2<𝑠). From Sobolev imbedding, we have Λ^𝑠−𝛼𝜓_𝐿2≤𝐶Λ^𝑠+𝛼𝜓_𝐿2,Δ𝜓_𝐿2≤𝐶Λ^𝑠𝜓_𝐿2. (42)
Case 4 (3−2𝛼<𝑠<2). Using Sobolev imbedding, we have Λ^𝑠−𝛼𝜓_𝐿2≤𝐶Λ^𝑠𝜓_𝐿2,Δ𝜓_𝐿2≤𝐶Λ^𝑠+𝛼𝜓_𝐿2. (43)
So, we can always obtain the following estimate Λ^𝑠−𝛼𝜓_𝐿2Δ𝜓_𝐿2≤𝐶Λ^𝑠𝜓_𝐿2Λ^𝑠+𝛼𝜓_𝐿2. (44)
With this choice of 𝑝 and the above embedding, the product estimate gives us Λ^{𝑠−𝛼−1}∇^⟂𝜓⋅Δ𝑢_𝐿2≤𝐶Λ^𝑠+𝛼𝜓_𝐿2Λ^𝑠𝜓_𝐿2+Λ^𝑠+𝛼𝜓_𝐿2∇𝜓_{𝐿2𝑝/(𝑝−2)}≤𝐶Λ^𝑠+𝛼𝜓_𝐿2Λ^𝑠𝜓_𝐿2+∇𝜓_{𝐿2𝑝/(𝑝−2)}. (45) Combining (24) with (45) and proceeding as in (34) we obtain 12𝑑Λ𝑑𝑡^𝑠𝜓²_𝐿2+𝐹Λ^𝑠−1𝜓²_𝐿2+1𝑅_𝑒Λ^𝑠+𝛼𝜓²_𝐿2≤Λ^{𝑠−𝛼−2}𝑓_𝐿2Λ^𝑠+𝛼𝜓_𝐿2+𝐶Λ^𝑠+𝛼𝜓²_𝐿2Λ^𝑠𝜓_𝐿2+∇𝜓_{𝐿2𝑝/(𝑝−2)} (46) which in turn implies 12𝑑Λ𝑑𝑡^𝑠𝜓²_𝐿2+𝐹Λ^𝑠−1𝜓²_𝐿2+12𝑅_𝑒Λ^𝑠+𝛼𝜓²_𝐿2≤𝑅_𝑒2Λ^{𝑠−𝛼−2}𝑓²_𝐿2+𝐶Λ^𝑠+𝛼𝜓²_𝐿2Λ^𝑠𝜓_𝐿2+∇𝜓_{𝐿2𝑝/(𝑝−2)}. (47)
Observe that ∇𝜓_{𝐿2𝑝/(𝑝−2)}≤𝐶𝜓^𝛾_𝐿2Λ^𝑠𝜓_𝐿^1−𝛾₂, (48) where 𝛾=1−(3−2𝛼)/𝑠. Therefore, if 𝐶Λ^𝑠𝜓_𝐿2+𝜓^𝛾_𝐿2Λ^𝑠𝜓_𝐿^1−𝛾₂≤14𝐶𝑅_𝑒 (49) estimate (47) combined with Sobolev imbedding inequality ‖Λ^𝑠+𝛼𝜓‖_𝐿2≥‖Λ^𝑠𝜓‖_𝐿2 shows that 𝑑Λ𝑑𝑡^𝑠𝜓²_𝐿2+𝐹Λ^𝑠−1𝜓²_𝐿2+12𝑅_𝑒Λ^𝑠𝜓²_𝐿2≤𝑅_𝑒𝑓²_{̇𝐻𝑥𝑠−𝛼−2} (50) and hence sup_0≤𝑡<∞Λ^𝑠𝜓²_𝐿2+𝐹Λ^𝑠−1𝜓²_𝐿2≤Λ^𝑠𝜓₀²_𝐿2+𝐹Λ^𝑠−1𝜓₀²_𝐿2+𝑅_𝑒𝑓²_{𝐿2𝑡̇𝐻𝑥𝑠−𝛼−2}. (51) By Sobolev imbedding, we have Λ^𝑠−1𝜓_0𝐿2≤𝐶Λ^𝑠𝜓_0𝐿2. (52) Combining (51) and (52), we get Λ^𝑠𝜓²_𝐿2≤𝐶Λ^𝑠𝜓₀²_𝐿2+𝑅_𝑒𝑓²_{𝐿2𝑡̇𝐻𝑥𝑠−𝛼−2}. (53) Note that taking the 𝐿²-product of (3) with 𝜓 gives for any 𝑡>0𝑑𝑑𝑡∇𝜓²_𝐿2+𝐹Λ^𝑠−1𝜓²_𝐿2+1𝑅_𝑒^𝑡₀Λ^1+𝛼𝜓²_𝐿2≤𝑅_𝑒𝑓²_𝐿2𝑥. (54) Thus, there exists some constant 𝐾 (dependent on 𝐹) such that sup_0≤𝑡<∞∇𝜓²_𝐿2+𝐹𝜓²_𝐿2≤𝐾𝜓₀²_𝐻1+𝐾𝑓_{𝐿2𝑡𝐿2𝑥} (55) which gives us a basic uniform estimate of 𝜓 in 𝐿^∞_𝑡𝐻¹_𝑥.
Hence, from (53) and (55) we obtain that condition (49) is satisfied for all 𝑡>0 as long as we have 𝜓_{0̇𝐻𝑠𝑥}+𝑓_{𝐿2𝑡̇𝐻𝑥𝑠−𝛼−2}+𝜓₀^𝛾_𝐻1𝑥+𝑓^𝛾_{𝐿2𝑡𝐿2𝑥}×𝜓_0̇𝐻^1−𝛾_𝑠𝑥+𝑓_𝐿^1−𝛾_{2𝑡̇𝐻𝑥𝑠−𝛼−2}<𝜀, (56) 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∈𝐻̇^𝑠(𝕋²) and 𝑓∈𝐿²(0,∞;𝐻̇^𝑠−(5/2)(𝕋²))∩𝐿²(0,∞;𝐿²(𝕋²)) have zero mean on 𝕋². There exists a sufficiently small constant 𝜀>0 depending on 𝑅_𝑒, such that if 𝜓₀^𝛾1_𝐻1𝑥+𝑓^𝛾1_{𝐿2𝑡̇𝐻𝑥𝑠−(5/2)}𝜓₀^1−𝛾1_{̇𝐻𝑠𝑥}+𝑓^1−𝛾1_{𝐿2𝑡̇𝐻𝑥𝑠−(5/2)}+𝜓₀^𝛾2_𝐻1𝑥+𝑓^𝛾2_{𝐿2𝑡𝐿2𝑥}𝜓₀^1−𝛾2_{̇𝐻𝑠𝑥}+𝑓^1−𝛾2_{𝐿2𝑡̇𝐻𝑥𝑠−(5/2)}<𝜀, (57) where 𝛾₁=1−(2/𝑠) and 𝛾₂=1−(3/𝑠), then the unique smooth solution 𝜓∈𝐿^∞0,∞;𝐻̇^𝑠𝕋² (58) 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 12𝑑Λ𝑑𝑡^𝑠𝜓²_𝐿2+Λ^𝑠−1𝜓²_𝐿2+1𝑅_𝑒Λ^𝑠+(1/2)𝜓²_𝐿2≤Λ^𝑠−(5/2)𝑓_𝐿2Λ^𝑠+(1/2)𝜓_𝐿2+Λ^𝑠−(3/2)∇^⟂𝜓⋅Δ𝜓_𝐿2Λ^𝑠+(1/2)𝜓_𝐿2. (59)
The second term on the right side is estimated using the product estimate in Proposition 2. Thus we obtain, similar to (45), Λ^𝑠−(3/2)∇^⟂𝜓⋅Δ𝜓_𝐿2≤𝐶∇𝜓_𝐿∞Λ^𝑠+(1/2)𝜓_𝐿2+Λ^𝑠−(1/2)𝜓_𝐿2Δ𝜓_𝐿∞≤𝐶∇𝜓_𝐿∞Λ^𝑠+(1/2)𝜓_𝐿2+Λ^𝑠+(1/2)𝜓_𝐿2Δ𝜓_𝐿∞. (60) By interpolation inequality, we have ∇𝜓_𝐿∞≤Λ^𝑠𝜓^1−𝛾1_𝐿2𝜓^𝛾1_𝐿2,Δ𝜓_𝐿∞≤Λ^𝑠𝜓^1−𝛾2_𝐿2𝜓^𝛾2_𝐿2, (61) where 𝛾₁=1−(2/𝑠) and 𝛾₂=1−(3/𝑠). Combining estimates (61) gives 12𝑑Λ𝑑𝑡^𝑠𝜓²_𝐿2+Λ^𝑠−1𝜓²_𝐿2+1𝑅_𝑒Λ^𝑠+(1/2)𝜓²_𝐿2≤12𝑅_𝑒Λ^𝑠−(5/2)𝑓²_𝐿2+𝐶𝑅_𝑒2Λ^𝑠+(1/2)𝜓²_𝐿2×Λ^𝑠𝜓^1−𝛾1_𝐿2𝜓^𝛾1_𝐿2+Λ^𝑠𝜓^1−𝛾2_𝐿2𝜓^𝛾2_𝐿2. (62)
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.

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).