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On Solutions of Three Quasi-Geostrophic Models

We consider the 2D quasi-geostrophic model and its two different regularizations. Global regularity results are established for the regularized models with subcritical or critical indices. The proof of Onsager's conjecture concerning weak solutions of the 3D Euler equations and the notion of dissipative solutions of Duchon and Robert are extended to weak solutions of the quasi-geostrophic equation.

Equations (1.1) and (1.2) are special cases of the general quasi-geostrophic approximations [17] for atmospheric and oceanic fluid flow with small Rossby and Ekman numbers. The variable θ represents potential temperature, u is the fluid velocity and ψ can be identified with the pressure. Equation (1.1) is an important example of a 2D active scalar with a specific structure most closely related to the 3D Euler equations while the equation in (1.2) with α = 1 2 is the dimensionally correct analogue of the 3D Navier-Stokes equations. These equations have recently been intensively investigated because of both their mathematical importance and their potential for applications in meteorology and oceanography ( [17], [12], [3], [18], [15], [4], [8], [5]).
In modeling long waves in nonlinear dispersive media, Benjamin,Bona and Mahony [1] introduced the BBM equation as an alternative to the KdV equation Equation (1.3) to (1.2) is like the BBM to the KdV equation and our motivation for proposing such a model for study comes from the effects of regularizations on the global regularity of weak solutions and the potential applications of this new model in geophysics.
These QG models appear to be simpler than the 3D hydrodynamics equations, but contain many of their difficult features. For instance, solutions of (1.1) and (1.2) exhibit strong nonlinear behavior, strikingly analogous to that of the potentially singular solutions of the 3D hydrodynamics equations [3]. Although progress has been made in the past several years ( [3], [18], [15], [4], [8], [5]), the theory remains fundamentally incomplete. In particular, it is not known whether or not weak solutions of (1.2) are regular for all time when α is equal to the critical index 1 2 . The critical case regularity issue turns out to be extremely difficult and was labeled by S. Klainerman [13] as one of the most challenging PDE problems of the 21st Century. In Section 2 we explore how far one can go toward a regularity proof in the critical case and what are the weakest assumptions needed to fill the gap.
In Section 3 we solve the global regularity problem for equation (1.3) with α ≥ 1 2 . We first construct a local solution θ in H s with s > 1 and then derive explicit bounds on the norms of all derivatives of the solution. From this we infer that for α > 1 2 the local solution θ remains bounded in H s for all time and thus no finite-time singularity can occur in this case. For the critical index α = 1 2 , global smoothness results are established under assumptions that are much weaker than those needed to guarantee regularity for equation (1.2). This leads us to conclude that solutions of (1.3) are better behaved and thus (1.3) constitutes a reasonable alternative to (1.2).
As is well-known, weak solutions of the 3D hydrodynamics equations in general only satisfy an energy inequality rather than equality. But Onsager conjectured in [16] that weak solutions of the 3D Euler equations in a Hölder space C γ with exponent γ > 1 3 should conserve energy. In [10] Eyink proved energy conservation for weak solutions in a strong form of Hölder space C γ * (γ > 1 3 ), in which the norm is defined in terms of absolute Fourier coefficients. Constantin, E and Titi provided a proof for the sharp version of Onsager's conjecture in [2]. Section 4 is concerned with solutions of the QG equation (1.1). First we verify Onsager's conjecture for weak solutions of the QG equation, extending the result of Constantin, E and Titi. Then the QG equation is shown to possess the dissipative weak solutions, a notion proposed by Duchon and Robert [9]. Finally the two models (1.1) and (1.3) are proven to be close by considering the limit of (1.3) as µ → 0. This provides further evidence for the validity of (1.3).
We now review the notations used throughout the sequel. The Fourier transform f of a tempered distribution f (x) on Ω is defined as We will denote the square root of the Laplacian (−∆) More generally, Λ β f for β ∈ R can be identified with the Fourier series The equality relating u to θ in (1.4) can be rewritten in terms of periodic Riesz transforms where R j , j = 1, 2 denotes the Riesz transforms defined by L p (Ω) denotes the space of the pth-power integrable functions normed by For any tempered distribution f on Ω and s ∈ R, we define and H s denotes the Sobolev space of all f for which f s is finite. For 1 ≤ p ≤ ∞ and s ∈ R, the space L p s () is a subspace of L p (Ω), consisting of all f which can be written in the form f = Λ −s g, g ∈ L p (Ω) and the L p s norm of f is defined to be the L p norm of g, i.e., f p,s = |g| p .

Dissipative QG Equation
In this section we focus on the nonlinear behavior of weak solutions of the initial-value problem (IVP) for the dissipative QG equation where 0 ≤ α ≤ 1 and κ > 0 are real numbers. We establish nonlinear estimates which characterize the regularity of weak solutions of the IVP (2.1) with α greater than or equal to the critical index 1 2 .
For 0 ≤ α ≤ 1, weak solutions of the IVP (2.1) are known to exist globally in time [18]. More precisely, for any holds for any t ≤ T .
Let s ≥ 0. We now estimate θ s = |Λ s θ| 2 . If we take the inner product of Λ 2s θ with the first equation in (2.1), we obtain The first term on the right hand side is bounded above by For the second term, we have where β ≤ α remains to be determined. To proceed, we need the calculus inequality where γ > 0, 1 < r ≤ p ≤ ∞ and 1/r = 1/p + 1/q. The use of (2.5) gives where 2 < q ≤ ∞ and 1/p + 1/q = 1/2. Taking into account of the second equation in (2.1) and the inclusion H s+2− 2 It then follows from (2.6) that In the above, β is essentially arbitrary and we may choose for any s ≥ 0, 2 < q ≤ ∞ and β = 1/2 + 1/q ≤ α.
We now prove that weak solutions of the IVP (2.1) with α > 1 2 are actually regular. More precisely, we have the following theorem.
Theorem 2.1 Let α > 1 2 and assume for T > 0, s > 0 and 2 < q < 2 Then any weak solution θ of the IVP (2.1) are regular in the sense that Proof. The idea of the proof is to obtain from (2.8) a closed differential equality for |Λ s θ| 2 . Since u is essentially the Riesz transform of θ from the second equation in (2.1), we have for To eliminate the occurrence of |Λ s+β θ| 2 , we use one of the inequalities of Gagliardo and Nirenberg We use Hölder's inequality to find Inserting (2.9) and (2.10) into (2.8) and canceling a factor of |Λ s+α θ| 2 where C only depends on |θ 0 | q and T 0 |f (·, τ )| q dτ . The proof of Theorem 2.1 is then concluded after we apply Gronwall's lemma to (2.11). Now We turn our attention to the regularity issue of weak solutions of the IVP (2.1) with α equal to the critical index 1 2 . The purpose of the next several theorems is to show how far one can go toward a regularity proof and what assumptions are needed to fill the gap.
. Consider a weak solution θ of the IVP (2.1) and assume that where C 0 is a constant as in (2.8). Then θ is regular in the sense that In particular, if θ 0 , f and u satisfy then (2.13) holds.
Proof. The situation is different when α = 1 2 . The differential inequality (2.8) only holds for β = α = 1 2 and q = ∞. We have after replacing β with Using the assumption (2.12), we have which gives (2.13). In view of the maximum principle it then follows that (2.14) implies (2.12) and thus (2.13).
For the critical index α = 1 2 , the terms |θ| ∞ and |u| ∞ have so far stood in the way of finding a regularity proof. The problem of how to deal with |θ| ∞ and |u| ∞ has to be solved. In order to estimate |θ| ∞ and |u| ∞ , we first prove the following lemma.
In the above, we choose R as and we have after some manipulation the inequality (2.15). Now we can prove the following theorem, which can be reviewed as a ladder theorem for the QG equation.
If we further assume that θ(·, t) σ is bounded and small, say, for some constant C then θ is regular in the sense that

Regularized QG Equation
In this section we are concerned with the IVP for the regularized QG equation where 0 ≤ α ≤ 1 and µ > 0 are real numbers. The central issue is still whether or not weak solutions are regular for all time. Because of the insignificant role of f and for the sake of clarity of our presentation, we will set f = 0 in the rest of this section.
The exposition below is organized as follows. We first construct a local solution in H s for s > 1 and then produce explicit bounds on the norms of all derivatives. From this we infer that for α > 1 2 the solutions are smooth and unique for all time. For α = 1 2 , global regularity is established under a weak assumption.
We reformulate the problem as an integral equation and then apply the Banach contraction mapping principle to prove local existence. Now rewrite the first equation in (3.1) in the form and invert the operator (1 + µΛ 2α ) subject to periodic boundary condition to obtain where the convolution kernel G(x) is defined through its Fourier transform Now notice that for α ≥ 1 2 and any s ≥ 0 We now state and prove a local existence result for smooth solutions of the IVP (3.1).
Theorem 3.1 Let α ≥ 1 2 and assume that θ 0 ∈ H s for some s > 1. Let b = θ 0 s , and set R = 2b. Define B R to be the ball with radius R centered at the origin in E. We now show that if T is sufficiently small, then A is a contraction map on B R . Let θ andθ be any two elements of B R . Then we have Then applying the calculus inequality (2.5), we find that and Since s > 1, we have the Sobolev inequality |F | ∞ ≤ C F s , for some constant C. Since u and θ, as well asū andθ, are related by the second equation in (3.1), one has ū s ≤ θ s , u −ū s ≤ θ −θ s It then follows from (3.9) that It is indeed that T depends only on θ 0 s (since R = 2 θ 0 s ). For this choice of T , we have The conclusion of part (a) then follows from contraction mapping principle.
To prove part (b), suppose on the contrary that T * < ∞ and that there exists a number M > 0 and a sequence t n approaching T * from below such that θ(·, t n ) s ≤ M, for n = 1, 2, 3, · · ·  Proof. Let θ ∈ L ∞ ([0, T ]; H s ) be the solution of the IVP (3.1) we constructed in Theorem 3.1. Take the inner product of Λ 2s−2α θ with the first equation in (3.1) The estimates for the term on the right hand side are similarly to those in the previous section, so we only give the most important lines here. Instead of (2.8), we have for any 2 < q ≤ ∞ and β = 1/2 + 1/q ≤ α. Inserting (2.9) and the following modified version of (2.10) in Theorem 2.1 into (3.10), we obtain Gronwall's lemma then implies that for any t > 0 This estimate indicates that the solution of the IVP (3.1) with α > 1 2 remains bounded at later time. Therefore, by part (b) of Theorem 3.1, Now we turn our attention to α = 1 2 . The following theorem asserts that if the solution θ loses its regularity at a later time, then the maximum of θ or the maximum of u necessarily grows without a bound. Proof. The differential inequality (3.10) is valid for all α > 0, therefore we have by choosing α = 1 2 , q = ∞ and β = 1 which implies The conclusion of the theorem is then inferred from this inequality.

Weak solutions of the QG equation
Onsager conjectured [16] that weak solutions of the 3D Euler equations in a Hölder space C γ with exponent γ > 1 3 should conserve energy. In [10] Eyink proved energy conservation for weak solutions in a strong form of Hölder space C γ * (γ > 1 3 ), in which the norm is defined in terms of absolute Fourier coefficients. Constantin, E and Titi [2] proved a sharp version of Onsager's conjecture in Besov space B s,∞ 3 with s > 1 3 . In [9] Duchon and Robert explored possible sources for energy losses and proved the existence of the so-called dissipative weak solutions for the 3D Euler equations. Their notion of dissipative weak solutions can be regarded as a special type of dissipative solutions proposed by Lions [14]. For the 2D Euler equations, DiPerna and Majda constructed weak solutions of the vorticity-velocity formulation with data in L p (p > 4 3 ) and Eyink [11] validated such weak solutions as dissipative weak solutions in the sense of Duchon and Robert and argued for their relevance to the enstrophy cascade of 2D turbulence.
The goal of this section is to prove Onsager's conjecture and extend the notion of dissipative weak solutions to the IVP for the QG equation (4.1) Understanding the zero-dissipation limits of the Navier-Stokes equations ( [5], [6]) is crucial in hydrodynamics turbulence theory and we believe that similar limits for the regularizations of the QG equation will be equally important in the turbulence theory for quasi-geostrophic flows. At the end of this section we show that the smooth solution of the regularized QG equation converges to that of the QG equation as µ → 0.
For θ 0 ∈ L 2 and any T > 0, the IVP (4.1) has been shown to possess global weak solutions (in the distributional sense) in L ∞ ([0, T ]; L 2 ) [18]. In the rest of this section we will use φ to denote the standard mollifier in R 2 , φ ǫ (x) = 1 ǫ 2 φ x ǫ and F ǫ = φ ǫ * F for any tempered distribution F .
First we show that if the weak solution θ is also in the Besov space B s,∞ 3 with s > 1 3 , then θ conserves the L 2 -norm. More details on Besov spaces can be found in [19], but here we only list some of the basic facts we will use. If F ∈ B s,∞ p for p ≥ 1, then Proof. The idea of the proof is similar to the one in [2]. If θ solves the IVP (4.1), then θ ǫ satisfies the following equation where σ ǫ = u ǫ θ ǫ − (uθ) ǫ . It is easy to check that σ ǫ can be represented by the formula It then follows (4.4) that We now estimate the term on the right hand side.
Since θ and u are related by the second equation in (4.1), we have from (4.2) , and and it approaches zero as ǫ → 0. The proof is then completed after letting ǫ → 0 in (4.5).
The following theorem extends the notion of dissipative weak solutions of the inviscid hydrodynamics equations ( [9], [11]) to the QG equation.
be a weak solution of the IVP (4.1) corresponding to θ 0 ∈ L 2 and arbitrary T > 0. If a function G : R → R is C 2 , strictly convex and has bounded derivative, then the equation holds in the sense of distribution, where F (G, θ) is the limit of in the sense of distribution.
Proof. Let G ∈ C 2 (R, R). Multiplying (4.4) by G ′ (θ ǫ ), we have Now we start showing that (4.7) converges to (4.6) in the distributional sense. Since G has bounded derivative as ǫ → 0. Now we show that u ǫ G(θ ǫ ) converges to uG(θ) in L 1 , which can be deduced from the following estimate and the fact that θ ǫ → θ in L 2 . It now remains to show that G ′ (θ ǫ )σ ǫ → 0 in the distributional sense.
which approaches zero as ǫ → 0. Therefore the limit of the term on the right hand side should also exist in the distributional sense. This completes the proof of the theorem.
uniformly for 0 ≤ t ≤ T , where C is a pure constant and T is any fixed time less than the existence time for θ.