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On a class of forced active scalar equations with small diffusive parameters

Many equations that model fluid behaviour are derived from systems that encompass multiple physical forces. When the equations are written in non dimensional form appropriate to the physics of the situation, the resulting partial differential equations often contain several small parameters. We study a general class of such PDEs called active scalar equations which in specific parameter regimes produce certain well known models for fluid motion. We address various mathematical questions relating to well-posedness, regularity and long time behaviour of the solutions to this general class including vanishing limits of several diffusive parameters.

of considerable study in recent years, in part because they arise in many physical models and in part because they present challenging nonlinear PDEs. Various active scalar equations, such as surface quasi-geostrophic equation (SQG) and driftdiffusion equations, have received considerable attention in the past decade because of the challenging nature of the delicate balance between the nonlinear term and the dissipative term. The physics of an active scalar equation is encoded in the constitutive law that relates the transport velocity vector u with a scalar field θ. This law produces a differential operator that when applied to the scalar field determines the velocity. The singular or smoothing properties of this operator are closely connected with the mathematics of the nonlinear advection equation for θ.
(A3) For each ν > 0, there exists a constant C ν > 0 such that for all 1 ≤ i, j ≤ d, Our motivation for addressing such a class of active scalar equations mainly comes from several different physical systems, all of them take the form (1.1) under particular parameter regimes: 1. The first example comes from a model proposed by Moffatt and Loper [49], Moffatt [47] for magenetostrophic turbulence in the Earth's fluid core. 1 We point out that, most of the results given in our work hold for d ≥ 2.
2 Such mean zero assumption is common in many physical models which include SQG equation and MG equation; see [16] and [29] for example.
Under the postulates in [49], the governing equation becomes a 3D active scalar equation for a temperature field θ ∂ t θ + u · ∇θ = κ∆θ + S, u = M ν [θ], θ(x, 0) = θ 0 (x). (1.4) The expressions for the Fourier multiplier symbol M ν are explicitly given by where k = (k 1 , k 2 , k 3 ) ∈ Z 3 is the Fourier variable and D(k) = |k| 2 k 2 3 +(k 2 2 + ν|k| 4 ) 2 . The nonlinear equation (1.4) with u related to θ via the operator M ν is known as the magnetogeostrophic (MG ν ) equation (or simply MG equation) and its mathematical properties have been addressed in a series of papers which include [28], [29], [30], [31], [32], [33], [35], [36]. The behaviour of the MG ν equation is strikingly different when the parameters ν and κ are present (i.e. positive) or absent (i.e. zero). As the Fourier multiplier symbols M 0 given by (1.5) with ν = 0 are not bounded in all regions of Fourier space [32], when ν = 0 the relation between u and θ is given by a singular operator of order 1. The implications of such fact for the inviscid MG 0 equation are summarized in the survey article by Friedlander, Rusin and Vicol [35]. In particular, when κ > 0 the inviscid but thermally dissipative MG 0 equation is globally well-possed [32]; on the other hand when ν = 0 and κ = 0, the singular inviscid MG 0 equation is ill-possed in the sense of Hadamard in any Sobolev space [33]. In [28], Friedlander and Suen first addressed the system (1.4)-(1.5) for ν > 0 and obtained well-posedness results in Sobolev space. In a series of papers [29]- [30] the authors further examined the limit of vanishing viscosity as ν → 0 in the case when κ > 0 and κ = 0 and the long-time behaviour of solutions. They proved global existence of classical solutions to the forced MG ν equations and obtained convergences of solutions as ν vanishes. Moreover, it was shown in [31] that the equations (1.4)-(1.5) possess global attractors with various interesting properties. 2. The second example of a physical system comes from incompressible flow in a porous medium. It can be modeled by an active scalar equation where a small smoothing parameter enters into the constitutive law. Different from the usual incompressible porous media (IPM) equation, the incompressible porous media Brinkman (IPMB ν ) equation with an "effective viscosity" ν is derived via a modified Darcy's Law as suggested by Brinkman [2]. The IPM equation becomes the limiting case for IPMB ν equation when ν = 0. The 2D equation relating the velocity u, the density θ and the pressure P is given in non-dimensional form by which produces the constitutive law is the vector of Riesz transforms and e 2 = (0, 1). The corresponding active scalar equation is thus given by where the 2D components of the Fourier multiplier symbol corresponding to M ν as in (1.9) are Similar to the case of MG κ,ν equation, there is a noticeable difference in the operator M ν between the two cases ν > 0 and ν = 0: the operator is smoothing of order 2 when ν > 0, while for ν = 0 the operator is singular of order zero. The well known IPM equations, i.e. (1.8)-(1.10) without the effective viscosity ν, have been studied in a number of papers (see [24], [25] and the reference therein). As we pointed out before, when ν = 0 the operator in (1.8) becomes a singular integral operator of order zero, which is similar to the case for the SQG equations. Yet there is a crucial difference between the two operators: the SQG operator is odd while the IPM operator is even. Implications for well/ill posedness due to the odd/even structure of the operator in an active scalar equation are further explored in [26], [36], [42]. In a recent work, Friedlander and Suen [30] studied the system (1.8)-(1.10) in the limit of vanishing viscosity as ν → 0 and obtained convergence results in Sobolev spaces, which are not available for the case of MG equation when κ = 0. The foremost difference is that the MG 0 operator is singular of order 1 where as the IPM operator is singular of order zero. In view of such difference, for the "smoother" IPM case the convergence results are valid in Sobolev spaces rather than the Gevrey-class convergence results for the MG equation. 3. The third physical example comes from the modified surface quasi-geostrophic (SQG κ,ν ) equation. It relates the potential temperature θ κ,ν and the flow velocity u κ,ν as follows: where the Fourier multiplier symbol for M ν is given by (1.12) When ν = 0, the system (1.11)-(1.12) reduces to the well-known SQG equation which has been investigated by many researchers [5], [7], [8], [10], [14], [17], [22], [55]. For κ > 0 and γ = 1 2 , it can be used as a model for studying the temperature distribution θ on the 2D boundary of a rapidly rotating fluid with small Rossby and Ekman numbers [19], while for κ = 0, the inviscid model can be applied for studying frontogenesis in meteorology [43]. It is also worth mentioning that the 3D analog of (1.11)-(1.12) is widely used as a testing model for the vorticity evolution of the 3D Navier-Stokes equations [8]. We point out that, in view of the physical models as mentioned above, conditions (A1)-(A5) are both mathematically and physically important for studying the active scalar equations (1.1)-(1.3), which can be explained as follows: • Condition (A1) implies the drift velocity u in (1.1) is divergence-free, which is compatible with the incompressibility of the fluid described by those physical models. On the other hand, condition (A2) requires a uniform bound (independent of ν) on the Fourier multiplier symbol T ν . This conditions implies that the operator ∂ xi T ν ij is at most singular of order 1 for ν ≥ 0, which is consistent with the cases for M ν given in the MG κ,ν equation. We remark here that, however, for the cases of IPMB ν and SQG κ,ν equation, one can replace the condition (A2) by the following condition: (A2 * ) There exists a constant C 0 > 0 independent of ν, such that for all i, j ∈ {1, . . . , d}, .
Condition (A2 * ) requires that the operators ∂ xi T ν ij are less singular than those given by condition (A2), which allows us to obtain better regularity results on IPMB ν and SQG κ,ν equation.
• Condition (A4) imposes a minimal regularity requirement on the operators T ν ij , while condition (A3) describes the smoothing effect given by the parameter ν. Condition (A3) implies that the operators ∂ xi T ν ij are smoothing of degree two for ν > 0, which is crucial for proving well-posedness for the system (1.1) especially when κ = 0 (refer to [29] and [32] for the striking differences between the cases ν > 0 and ν = 0). • For condition (A5), we notice that the Fourier multiplier symbols M ν given in either (1.5), (1.10) or (1.12) do satisfy condition (A5), which implies that the drift velocity u converges to u κ,0 strongly in L 2 . Such behaviour of T ν ij allows strong convergence of solutions of (1.1) (refer to Theorem 7.4 in [29]), despite the fact that T ν ij are all matrices of zero-order pseudo-differential operators as required by condition (A4). In section 2, we state the results proved in [29] for the diffusive active scalar equations, which are the equations (1.1)-(1.3) for κ > 0, ν ≥ 0 and γ = 2. In this case, the system (1.1)-(1.3) is globally well-posed. To prove this fact, we control the term θ(t) L ∞ which can be done by using De Giorgi techniques. Having established the global well-posedness of (1.1)-(1.3), we proceed to address the convergence of solutions as ν → 0 which is based on some uniform estimates on θ which are independent of ν. Moreover, we define a weak solution to the MG 0 equation which we call a "vanishing viscosity"solution and prove the existence of a compact global attractor in L 2 (T 3 ) for the MG ν equations (1.4)-(1.5). We further obtain the upper semicontinuity of the global attractor as ν vanishes.
In section 3, we examine the non-diffusive active scalar equations (1.1)-(1.3) by considering κ = 0 and ν ≥ 0 (cf [30]). For the case of ν > 0, the operators ∂ x T ν are smoothing order 2 which give rise to well-posedness for (1.1)-(1.3) with κ = 0, ν > 0 and θ 0 ∈ W s,d for s ≥ 0 and smooth forcing term S, and the results will be discussed in subsection 3.1. On the other hand, for the case when κ = 0 and ν = 0, in general the system (1.1)-(1.3) fails to be well-posed in Sobolev spaces. In [33], It was proved that the singular inviscid MG 0 equation is ill-possed in the sense of Hadamard in any Sobolev space. Yet it is possible to obtain the local existence and uniqueness of solutions to (1.1)-(1.3) with κ = ν = 0 in spaces of real-analytic functions, owing to the fact that the derivative loss in the nonlinearity u · ∇θ is of order at most one. We thus prove the local-in-time existence of Gevreyclass solutions which are summarised in subsection 3.2. We further address the convergence of Gevrey-class solutions as ν → 0 and apply the claimed results to inviscid MG ν equation (1.4)-(1.5) with κ = 0 and IPMB ν equation (1.8)-(1.10).
In section 4, we discuss the result obtained in [31] by investigating the properties of (1.1)-(1.3) in the full range γ ∈ [0, 2]. More precisely we prove the existence and convergence of solutions in various space for the cases ν > 0 and ν = 0, which are applicable for the critical SQG equation (1.11)-(1.12) with ν = 0. We then address the long-time behaviour for solutions when κ, ν > 0 and obtain global attractors in H 1 . We further prove some additional properties of the attractors which will be given in subsection 4.3. The results on the global attractors can be applied to MG ν equation (1.4)-(1.5) which are related to those discussed in section 2.

Diffusive active scalar equations
In this section, we discuss the global existence of classical solutions to (1.1)-(1.3) for κ > 0, ν ≥ 0 and γ = 2, and study the convergence of solutions as ν vanishes. Specifically, we consider the following abstract system: with ν ≥ 0, and S, θ 0 satisfy the zero mean assumptions (1.2)-(1.3). The results can then be applied to MG ν equations, which allows us to obtain the existence of a compact global attractor {A ν } ν≥0 in L 2 (T 3 ) including the critical equation where ν = 0.  [29]). Let θ 0 ∈ L 2 , S ∈ C ∞ and κ > 0 be given, and assume that Furthermore, for any ν ≥ 0, we prove that the smooth solutions θ ν obtained in Theorem 2.1 satisfy a uniform bound which is independent of ν: Theorem 2.2 (Friedlander and Suen [29]). Assume that the hypotheses and notations of Theorem 2.1 are in force. Then given 0 < t 1 < t 2 and s ≥ 0, there exists a positive constant C which depends on C 0 , t 1 , t 2 , s, d, κ, S, θ 0 L 2 but independent of ν such that where C 0 > 0 is the constant as stated in condition (A5).

The MG equations and existence of compact global attractor.
We apply the results obtained in subsection 2.1 to the case for MG equations given by (1.4)-(1.5). We write for ν ≥ 0 and M ν is defined by the inverse Fourier transform of (1.5). Using [29, Lemma 5.1], one can verify that there are constants C 1 , C 2 > 0 independent of ν such that, for Hence conditions (A1)-(A5) are satisfied. Theorem 2.1 and Theorem 2.2 can therefore be applied to the MG equations in order to obtain the global-in-time existence and convergence of smooth solutions: Theorem 2.4 (Friedlander and Suen [29]). Let θ 0 ∈ L 2 , S ∈ C ∞ and κ > 0 be given. There exists a classical solution Theorem 2.5 (Friedlander and Suen [29]). Let θ 0 ∈ L 2 , S ∈ C ∞ and κ > 0 be given. Then if θ ν , θ are C ∞ smooth classical solutions of (1.4)-(1.5) for ν > 0 and ν = 0 respectively with initial data θ 0 , then given τ > 0, for all s ≥ 0, we have With the results of Theorems 2.4 and Theorem 2.5 in place, we define a weak solution to the MG 0 equation which we call a "vanishing viscosity"solution. We use this concept to prove the existence of a compact global attractor in L 2 (T 3 ) for the MG ν equations (1.4)-(1.5) including the critical equation where ν = 0, and we further obtain the upper semicontinuity of the global attractor as ν vanishes.
We prove that the system (1.4)-(1.5) driven by a force S possesses a compact global attractor in L 2 (T 3 ) which is upper semicontinuous at ν = 0. More precisely, we have Theorem 2.6 (Friedlander and Suen [29]). Assume S ∈ C ∞ . Then the system For any bounded set B ⊂ L 2 (T 3 ), and for any ε, T > 0, there exists t 0 such that for any t 1 > t 0 , every "vanishing viscosity" solution θ(t) with θ(0) ∈ B satisfies for some complete trajectory x(t) on the global attractor (x(t) ∈ A, ∀t ∈ (−∞, ∞)). Furthermore, for ν ∈ [0, 1], there exists a compact global attractor A ν ⊂ L 2 for (1.4)-(1.5) such that A 0 = A and A ν is upper semicontinuous at ν = 0, which means that We give a brief discussion of the proof of Theorem 2.6, and we refer the interested reader to [29] for full details. Roughly speaking, Theorem 2.6 can be divided into two parts, namely: 1. existence of global attractors A ν for (1.4)-(1.5) with ν ≥ 0; and 2. upper semicontinuity of A ν at ν = 0. For the existence of global attractors A ν , it can be proved in the following several steps: I. First let θ(t) be a "vanishing viscosity" solution of (1.4)-(1.5) on [0, ∞) with θ(0) ∈ L 2 . Then θ(t) satisfies the following energy equality: for all 0 ≤ t 0 ≤ t. The energy equality implies that: -Every "vanishing viscosity" solution to (1.4)-(1.5) is strongly continuous in time t.
-There exists an absorbing ball Y for (1.4)-(1.5) given by where R is any number larger than κ −1 S H −1 (T 3 ) . II. Next we define π ν : L 2 → L 2 as the map π ν θ 0 = θ ν , where θ ν is the solution to (1.4)-(1.5) given by Theorem 2.4. Then using [29, Lemma 6.7], for t > 0, π ν (t)θ 0 is continuous in ν, uniformly for θ 0 in compact subsets of L 2 . III. We denote the weak distance on L 2 (T 3 ) by whereφ k andψ k are the Fourier coefficients of φ and ψ. If then E is an evolutionary system (see [6] and [11] for the definition), so by [11,Theorem 4.5], there exists a weak global attractor A w to E with Applying the arguments given in [11], E satisfies all the following properties: ; -for any ε > 0, there exists δ > 0 such that for every θ ∈ E([0, ∞)) and t > 0, [11,Theorem 4.5], it implies that the strong global attractor A s for (1.4)-(1.5) with ν = 0 exists, it is strongly compact and A 0 := A s = A w . The case for (1.4)-(1.5) with ν > 0 is just similar. On the other hand, to prove the upper semicontinuity of A ν at ν = 0, we note that the absorbing ball Y as given by (2.18) has radius which is independent of ν.
Together with the continuity of π ν (t)θ 0 in ν, the result from [38] implies the weak upper semicontinuity, namely Moreover, for any φ νj ∈ A νj and ψ j ∈ A, if In other words, the weak upper semicontinuity implies the strong upper semicontinuity, and hence (2.19) further implies the strong upper semicontinuity of A ν at ν = 0.

Non-diffusive active scalar equations
We switch our attention to the non-diffusive active scalar equations (1.1)-(1.3) for κ = 0 and ν ≥ 0, namely 3.1. Well-posedness in various spaces for ν > 0. In [33], Friedlander and Vicol showed that the singular inviscid MG 0 equation is ill-possed in the sense of Hadamard in any Sobolev space. One of the main reasons for the ill-posedness is that, the Fourier multiplier symbols M 0 given by (1.5) is an even function in k.
When we perform energy estimates on (1.4), we may have trouble on controlling the following term Without the diffusive term κ∆θ, we loss control on the term R and the only hope to treat it would be to discover a commutator structure. If the operator ∂ xi T ν ij (k) is odd, then there is an extra cancellation which allows us to close the estimates (at the level of Sobolev spaces) by the Coifman-Meyer type commutator estimate. On the other hand, for the case when the operator ∂ xi T ν ij (k) is even, such argument fails and one has to seek for some other type of estimates. In the case of ν > 0, however, the operators ∂ x T ν are smoothing order 2 which give rise to well-posedness for (1.1) with θ 0 ∈ W s,d for s ≥ 0 and smooth forcing term S. The results are summarised as follows: Theorem 3.1 (Friedlander and Suen [30]). Let θ 0 ∈ W s,d for s ≥ 0 and S be a C ∞ -smooth source term. Then for each ν > 0, we have: • if s = 0, there exists unique global weak solution to (3.1) such that u ν ∈ C((0, ∞); W 2,d ).
In particular, θ ν (·, t) → θ 0 weakly in L d as t → 0 + . Here BC stands for bounded continuous functions. • if s > 0, there exists a unique global-in-time solution θ ν to (3.1) such that θ ν (·, t) ∈ W s,d for all t ≥ 0. Furthermore, for s = 1, we have the following single exponential growth in time on ∇θ ν (·, t) L d : where C > 0 is a constant which depend only on ν and the spatial dimension d.
The proof of Theorem 3.1 for the case of s = 0 relies on the existence and uniqueness of the flow map, which is essential for Euler system as well [4]. We briefly sketch here and the full details can be found in [30].
Next we study the Gevrey-class s solutions to (3.1) for ν > 0 when the initial datum θ 0 and forcing term S are in the same Gevrey-class. We prove that there exists a unique global-in-time Gevrey-class s solution θ ν with radius of convergence bounded below by some positive function τ (t) for all t ∈ [0, ∞). More precisely, we have: Here C > 0 is a constant which depends on ν but independent of t.
Using a Cauchy-Kowalewski-type argument and together with condition (A3), it can be showed in [30,Lemma 4 and we obtain that 1 2 By choosing τ > 0 such thatτ and τ satisfies the lower bound (3.7). We remark that, for the diffusive case given by the system (2.1), one can obtain global-in-time existence of solution Gevrey class s ≥ 1 with lower bound on τ (t) that does not vanish as t → ∞, refer to [30,Remark 4.3] for more details.

3.2.
Well-posedness in various spaces for ν = 0. In this subsection we study the non-diffusive equations (3.1) for ν = 0: (3.8) When ν = 0 and condition (A2) is imposed, as it was proved in [33], the equation (3.8) is ill-posed in the sense of Hadamard, which means that the solution map associated to the Cauchy problem for (3.8) is not Lipschitz continuous with respect to perturbations in the initial datum around a specific steady profile θ 0 , in the topology of a certain Sobolev space X. Nevertheless, as pointed out in [33], it is possible to obtain the local existence and uniqueness of solutions to (3.8) in spaces of real-analytic functions, owing to the fact that the derivative loss in the nonlinearity u 0 · ∇θ 0 is of order at most one (both in u 0 and in ∇θ 0 ). In [30], we extended the results of [33] to the case of Gevrey-class solutions which are summarised as follows:
In contrast, when ν = 0 and condition (A2 * ) is in force, the operator ∂ x T 0 becomes a zero order operator with ∂ x T 0 : L 2 → L 2 being bounded. Following the idea given in [35], we show that the equation (3.8) is locally well-posed in Sobolev space H s for s > d 2 + 1: Theorem 3.4 (Friedlander and Suen [30]). For d ≥ 2, we fix s > d 2 + 1. Assume that θ 0 , S ∈ H s (T d ) have zero-mean on T d . Then for ν = 0, under the condition (A2 * ), there exists a T > 0 and a unique smooth solution θ 0 to (3.8) such that The proof of Theorem 3.4 consists of three steps, which can be briefly outline as follows (details can be found in [30]): I. We first construct a sequence of approximations {θ n } n≥1 given by the solutions of and for n > 1, Then by applying [35, Theorem A1], one can show that θ n ∈ L ∞ (0, T ; H s ) for all n ∈ N. II. Next, by induction on n, we prove that Λ s θ n (·, t) L 2 is bounded on [0, T ] for some T > 0. Assume that Λ s θ j L ∞ (0,T ;L 2 ) ≤ C Λ s θ 0 L 2 , for 1 ≤ j ≤ n − 1. We apply Λ s on (3.11) and take inner product with Λ s θ n to obtain 1 2 Using commutator estimates (see [35] for example), the term involving u n−1 can be bounded in terms of θ n and θ n−1 : Hence by integrating the identity (3.12) over t, choosing T small enough and applying the induction hypothesis, Λ s θ n (·, t) L 2 is bounded on [0, T ] as well. III. Finally, we show that {θ n } n≥0 is a Cauchy sequence. This can be done by considering the differenceθ n = θ n − θ n−1 and one can prove that Integrating the above over t and choosing T small enough, we obtain Thus θ n is Cauchy in L ∞ (0, T ; H s−1 ) with θ n converges strongly to θ 0 in L ∞ (0, T, H s−1 ). Since we assume that s > d 2 + 1, this also implies that the strong convergence occurs in a Hölder space relative to x as n → ∞, hence the limiting function θ 0 is a solution of (3.8). Uniqueness of θ 0 follows by the same argument given in [35] and we omit the details.
3.3. Convergence of solutions as ν → 0. In this subsection, we address the convergence of solutions to (3.1) as ν → 0. Depending on the conditions (A2) and (A2 * ), we can address the convergence of solutions in two cases respectively: 3.3.1. Gevrey-class solutions: We focus on the case for Gevrey-class solutions θ ν to (3.1) when (A2) is in force. By Theorem 3.3, given Gevrey-class s initial datum θ 0 and forcing S, there existsT ,τ > 0 and a unique Gevrey-class solution θ ν to (3.1) defined on [0,T ] with radius of convergence at leastτ for all ν ≥ 0. In particular, the Gevrey-class solutions θ ν converges to θ 0 in some Gevrey-class norm as ν → 0 and the results are summarised in the following theorem: Theorem 3.5 (Friedlander and Suen [30]). Under the condition (A2), if θ ν and θ 0 are Gevrey-class s solutions to (3.1) for ν > 0 and ν = 0 respectively with initial datum θ 0 on T d × [0,T ] with radius of convergence at leastτ as described in Theorem 3.3, then there exists T <T and τ = τ (t) <τ such that, for t ∈ [0, T ], we have The proof of Theorem 3.5 relies on the estimates of the difference φ ν := θ ν − θ 0 , and it can be shown that φ ν satisfies where the terms R 1 and R 2 can be bounded as follows: By choosing τ = τ (t) ≤τ such that τ + C θ ν τ,r + C Λ 1 2s θ 0 2 τ,r < 0, τ <τ , and applying the bound (3.10) to conclude that Integrating the above with respect to t and using the condition (A2), we have lim ν→0 φ ν τ,r = 0 and (3.13) follows.
In particular, θ ν (·, t) → θ 0 weakly in L 2 as t → 0 + . • if s > 0, there exists a unique global-in-time solution θ ν to (1.8)- (1.10) such that θ ν (·, t) ∈ W s,2 for all t ≥ 0. Furthermore, for s = 1, we have the following single exponential growth in time on ∇θ ν (·, t) L 2 : where C > 0 is a constant which depends only on some dimensional constants. Here C > 0 is a constant which depends on ν but independent of t. 4.1. Existence and convergence of H s -solutions when ν > 0. When the parameter ν is taken to be positive, the ensuing smoothing properties of T ν ij permits existence and convergence in Sobolev space H s as κ goes to zero. The global-in-time existence theorem is given as follows: Theorem 4.1 (Friedlander and Suen [31]). Fix ν > 0, s ≥ 0 and γ ∈ (0, 2], and let θ 0 ∈ H s and S ∈ H s ∩ L ∞ be given. • For any κ > 0, there exists a global-in-time solution to (1.1) such that . • For κ = 0, if we further assume that θ 0 ∈ L ∞ , then exists a global-in-time solution to (1.1)-(1.3) such that θ 0 (·, t) ∈ H s for all t ≥ 0.
In view of the case when κ > 0, the most subtle part for proving Theorem 4.1 is to estimate the L ∞ -norm of θ κ (·, t) when θ 0 is not necessarily in L ∞ . In achieving our goal, we apply De Giorgi iteration method (see [31,Lemma 4.5]) and obtain

4.2.
Existence and convergence of Gevrey-class solutions when ν = 0. In contrast to the case for ν > 0, when the parameter ν is set to zero, condition (A2) implies that ∂ xi T ν ij is a singular operator. In this case the existence and convergence results for (1.1)-(1.3) are restricted to Gevrey-class solutions which are summarised in the following theorem: Theorem 4.3 (Friedlander and Suen [31]). Let κ ≥ 0 and γ ∈ (0, 2] be fixed, and let θ 0 and S be the initial datum and forcing term respectively. Fix s ≥ 1 and K 0 > 0. Suppose θ 0 and S both belong to Gevrey-class s with radius of convergence τ 0 > 0 and where r > max{ d 2 + 3 2 , d 2 + 1 + 3 2s }. Then there exists T * = T * (τ 0 , K 0 ) > 0 and a unique Gevrey-class s solution on [0, T * ) to the initial value problem associated to (1.1)-(1.3) with ν = 0. Furthermore, if θ κ , θ 0 are Gevrey-class s solutions to (1.1)-(1.3) with ν = 0 for κ > 0 and κ = 0 respectively with initial datum θ 0 on T d ×[0,T ] with radius of convergence at leastτ , then there exists T ≤T and τ = τ (t) <τ such that, for t ∈ [0, T ], we have: For κ > 0 and γ ∈ [1,2], under a smallness assumption on the initial data, it can be proved that the Gevrey-class solutions obtained in Theorem 4.3 exist for all time: Theorem 4.4 (Friedlander and Suen [31]). Let κ > 0 and γ ∈ [1,2], and suppose that both θ 0 and S belong to some Gevrey-class s with s ≥ 1. There exists ε > 0 depending on κ such that, if θ 0 and S satisfy and The proofs of Theorem 4.3-4.5 can be found in [31]. We point out that all the abstract results obtained in Theorem 4.3-4.5 can be applied to the critical SQG equation, which is a special example of (1.1)-(1.3) with ν = 0 and γ = 1.

4.3.
Long time behaviour for solutions when ν > 0 and κ > 0. In this subsection, we study the long time behaviour for solutions to the active scalar equations (1.1) when ν > 0 and κ > 0. Based on the global-in-time existence results established in Theorem 4.1, for fixed ν > 0 and κ > 0, we can define a solution operator π ν (t) for the initial value problem (1.1) via π ν (t) : We study the long-time dynamics of π ν (t) on the phase space H 1 . Specifically, we establish the existence of global attractors for π ν (t) in H 1 and address some properties for the attractors. The following theorem first gives the existence of global attractors: Theorem 4.6 (Friedlander and Suen [31]). Let S ∈ L ∞ ∩ H 1 . For ν, κ > 0 and γ ∈ (0, 2], the solution map π ν (t) : H 1 → H 1 associated to (1.1) possesses a unique global attractor G ν . Moreover, there exists M G ν which depends only on ν, κ, γ, S L ∞ ∩H 1 and universal constants, such that if θ 0 ∈ G ν , we have that where θ(·, t) = π ν (t)θ 0 .
Details of the proof of Theorem 4.6 can be found in [31, Subsection 6.1], we also refer to [18] for the case of SQG equations. The steps of proof can be outlined as follows: I. By standard energy method (see for example [16] for the case γ = 1), one can show that θ κ satisfies where c 0 > 0 is a universal constant which depends only on the dimension d. Moreover, by [31, Lemma 6.2], the set is an absorbing set for π ν (t) and II. Next by [31, Lemma 6.3], we obtain the necessary a priori estimate in C α -space with some appropriate exponent α ∈ (0, 1). Furthermore, as pointed out in [16], we can see that the solutions to (1.1) emerging from data in a bounded subset of H 1 are absorbed in finite time by B ∞ . Hence if θ 0 ∈ H 1 ∩ L ∞ and fix ν, κ > 0, then there exists α ∈ (0, γ 2γ+2 ] which depends on θ 0 L ∞ , S L ∞ , ν, κ, γ such that where C > 0 is a positive constant, K ∞ andK ∞ are given respectively by With the help of (4.7), we obtain the following result which can be regarded as an improvement of the regularity of the absorbing set B ∞ , namely there exists α ∈ (0, γ 2γ+2 ] and a constant C α = C α ( S L ∞ , α, ν, κ, γ, K ∞ ,K ∞ ) ≥ 1 such that the set is an absorbing set for π ν (t).
III. As in [31,Lemma 6.7], by establishing an a priori estimate for initial data in H 1 ∩ C α , we can show that there exists a bounded absorbing set for π ν (t) in H 1 . More precisely, there exists α ∈ (0, γ 2γ+2 ] and a constant R 1 = R 1 ( S L ∞ ∩H 1 , α, ν, κ, γ) ≥ 1 such that the set The bound (4.8) turns out to be crucial for improving the regularity of the absorbing set B 1 to H 1+ γ 2 , which allows us to obtain an absorbing set B 1+ γ 2 for π ν (t) given by for some constant R 1+ γ 2 ≥ 1 which depends on S L∞∩H 1 , ν, κ, γ. The existence and regularity of the global attractor claimed by Theorem 4.6 now follows by applying the argument given in [9,Proposition 8] and the bound (4.8). After we have obtained the global attractors as described in Theorem 4.6, we prove some additional properties on the attractors under the assumption that γ ∈ [1,2] (also refer to [31, Subsection 6.2] for details): Theorem 4.7 (Friedlander and Suen [31]). Let S ∈ L ∞ ∩ H 1 . For ν, κ > 0, assume that the exponent γ ∈ [1,2]. Then the global attractor G ν of π ν (t) further enjoys the following properties: • G ν is fully invariant, namely π ν (t)G ν = G ν , ∀t ≥ 0.
• G ν is maximal in the class of H 1 -bounded invariant sets.
To prove the invariance and the maximality of the attractor G ν , we observe that the solution map π ν (t) is indeed continuous in the H 1 -topology, in other words for every t > 0, the solution map π ν (t) : B 1+ γ 2 → H 1 is continuous in the topology of H 1 . The key ingredient for the proof of continuity is the bound where C ν > 0 is a constant depending on ν > 0, and such bound comes from the condition (A3) and the assumption that d = 2 or 3. Following the argument given in [16,Proposition 5.5] and using the log-convexity method introduced by [1], we can also prove that the solution map π ν is injective on the absorbing set B 1+ γ 2 . Hence by applying [18, Proposition 6.4], we can obtain the invariance and the maximality of the attractor G ν stated in Theorem 4.7.
It remains to address the fractal dimensions for the global attractors G ν . In order to prove that dim f (G ν ) is finite, we need to show that the solution map π ν is uniform differentiable (refer to [31,Definition 6.18] for the definition for being uniform differentiable). And by [31,Lemma 6.21], the large-dimensional volume elements which are carried by the flow of π ν (t)θ 0 , with θ 0 ∈ G ν , actually have exponential decay in time. The argument in [13,and Chapter 14] can then be applied which shows that dim f (G ν ) is finite.
• G ν is maximal in the class of H 1 -bounded invariant sets.

Department of Mathematics, University of Southern California
Email address: susanfri@usc.edu

Department of Mathematics and Information Technology, The Education University of Hong Kong
Email address: acksuen@eduhk.hk