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Global well-posedness for the critical 2D dissipative quasi-geostrophic equation

We give an elementary proof of the global well-posedness for the critical 2D dissipative quasi-geostrophic equation. The argument is based on a non-local maximum principle involving appropriate moduli of continuity.


Introduction and main results
The 2D quasi-geostrophic equation attracted quite a lot of attention lately from various authors. Mainly it is due to the fact that it is the simplest evolutionary fluid dynamics equation for which the problem of existence of smooth global solutions remains unsolved. In this paper we will consider the so-called dissipative quasi-geostrophic equation where θ : R 2 → R is a scalar function, R 1 and R 2 are the usual Riesz transforms in R 2 and α > 0. It is well known (see [4,7]) that for α > 1 2 (the so-called subcritical case), the initial value problem θ(x, 0) = θ 0 (x) with C ∞ -smooth periodic initial data θ 0 has a global C ∞ solution.
For α = 1 2 , this equation arises in geophysical studies of strongly rotating fluid flows (see e.g. [1] for further references). Therefore, a significant amount of research focused specifically on the critical α = 1 2 case. In particular, Constantin, Cordoba, and Wu in [2] showed that the global smooth solution exists provided that θ 0 ∞ is small enough. Cordoba and Cordoba [5] proved that the viscosity solutions are smooth on time intervals t ≤ T 1 and t ≥ T 2 . The aim of this paper is to demonstrate that, in the critical case, smooth global solutions exist for any C ∞ periodic initial data θ 0 , with no additional qualifications or assumptions. What happens in the supercritical case 0 α < 1 2 remains an open question.
The main idea of our proof is quite simple: we will construct a special family of moduli of continuity that are preserved by the dissipative evolution, which will allow us to get an a priori estimate for ∇θ ∞ independent of time. More precisely, we will prove the following theorem.
Theorem. The quasi-geostrophic equation with periodic smooth initial data θ 0 (x) has a unique global smooth solution. Moreover, the following estimate holds: At this moment we do not know how sharp the upper bound (1) is. On the other hand, any a-priori bound for ∇θ ∞ is sufficient for the proof of well-posedness. Indeed, local existence and regularity results then allow to extend the unique smooth solution indefinitely. For the critical and supercritical quasi-geostrophic equation, such results can be found for example in [8] (Theorems 3.1 and 3.3). Hence, the rest of the paper is devoted to the proof of (1). This paper is built upon the ideas discovered in a related work on the dissipative Burgers equation [6].

Moduli of continuity
Let us remind the reader that a modulus of continuity is just an arbitrary increasing continuous concave function ω : [0, +∞) → [0, +∞) such that ω(0) = 0. Also, we say that a function f : Singular integral operators like Riesz transforms do not preserve moduli of continuity in general but they do not spoil them too much either. More precisely, we have Lemma. If the function θ has modulus of continuity ω, then u = (−R 2 θ, R 1 θ) has modulus of continuity The proof of this result is elementary but since we could not readily locate it in the literature, we provide a sketch in the appendix.
The flow term u · ∇θ in the dissipative quasi-geostrophic equation tends to make the modulus of continuity of θ worse while the dissipation term (−∆) α θ tends to make it better. Our aim is to construct some special moduli of continuity for which the dissipation term always prevails and such that every periodic C ∞ -function θ 0 has one of these special moduli of continuity.
Note that the critical (α = 1 2 ) equation has a simple scaling invariance: if θ(x, t) is a solution, then so is θ(Cx, Ct). This means that if we manage to find one modulus of continuity ω that is preserved by the dissipative evolution, then the whole family ω C (ξ) = ω(Cξ) of moduli of continuity will also be preserved.
Observe now that if ω is unbounded, then any C ∞ periodic function has modulus of continuity ω C if C > 0 is sufficiently large. Also, if the modulus of continuity ω has finite derivative at 0, it can be used to estimate ∇θ ∞ . Thus, our task reduces to constructing an unbounded modulus of continuity with finite derivative at 0 that is preserved by the dissipative evolution.
From now on, we will also assume that, in addition to unboundedness and the condition ω ′ (0) < +∞, we have lim ξ→0+ ω ′′ (ξ) = −∞. Then, if a C ∞ periodic function f has modulus of continuity ω, we have Indeed, take a point x ∈ R 2 at which max |∇f | is attained and consider the point ξ for all ξ > 0 and it remains to choose some ξ > 0 satisfying ρ(ξ) > C.

The breakthrough scenario
Now assume that θ has modulus of continuity ω for all times t < T . Then θ remains C ∞ smooth up to T and, according to the local regularity theorem, for a short time beyond T . By continuity, we see that θ must also have modulus of continuity ω at the moment T . Suppose that |θ(x, T )−θ(y, T )| < ω(|x−y|) for all x = y. We claim that then θ has modulus of continuity ω for all t > T sufficiently close to T . Indeed, by the remark above, at the moment T we have ∇θ ∞ < ω ′ (0). By continuity of derivatives, this also holds for t > T close to T , which immediately takes care of the inequality |θ(x, t) − θ(y, t)| < ω(|x − y|) for small |x−y|. Also, since ω is unbounded and θ ∞ doesn't grow with time, we automatically have |θ(x, t) − θ(y, t)| < ω(|x − y|) for large |x − y|. The last observation is that, due to periodicity of θ, it suffices to check the inequality |θ(x, t) − θ(y, t)| < ω(|x − y|) for x belonging to some compact set K ⊂ R 2 . Thus, we are left with the task to show that, if |θ(x, T ) − θ(y, T )| < ω(|x − y|) for all x ∈ K, δ |x − y| δ −1 with some fixed δ > 0, then the same inequality remains true for a short time beyond T . But this immediately follows from the uniform continuity of θ.
This implies that the only scenario in which the modulus of continuity ω may be lost by θ is the one in which there exists a moment T > 0 such that ω has modulus of continuity ω for all t ∈ [0, T ] and there are two points x = y such that θ(x, T )−θ(y, T ) = ω(|x−y|). We shall rule this scenario out by showing that, in such case, the derivative ∂ ∂t (θ(x, t) − θ(y, t)) t=T must be negative, which, clearly, contradicts the assumption that the modulus of continuity ω is preserved up to the time T .

Estimate of the derivative: the dissipation term
Recall that the dissipative term can be written as d dh P h * θ h=0 where P h is the 2dimensional Poisson kernel. Thus, our task is to estimate (P h * θ)(x) − (P h * θ)(y) under the assumption that θ has modulus of continuity ω. Since everything is translation and rotation invariant, we may assume that x = ( ξ 2 , 0) and y = (− ξ 2 , 0). Write where P h is the 1-dimensional Poisson kernel. Here we used symmetry and monotonicity of the Poisson kernels together with the observation that R P h (η, ν) dν = P h (η). The last formula can also be rewritten as Recalling that ∞ 0 P h (η) dη = 1 2 , we see that the difference (P h * θ)(x) − (P h * θ)(y) − ω(ξ) can be estimated from above by Recalling the explicit formula for P h , dividing by h and passing to the limit as h → 0+, we finally conclude that the contribution of the dissipative term to our derivative is bounded from above by Note that due to concavity of ω, both terms are strictly negative.
Note that, for small δ, the left derivative of ω at δ is about 1 while the right derivative equals γ 4δ < 1 4 . So ω is concave if δ is small enough. It is clear that ω ′ (0) = 1, lim ξ→0+ ω ′′ (ξ) = −∞ and that ω is unbounded (it grows at infinity like double logarithm). The hard part, of course, is to show that, for this ω, the negative contribution to the time derivative coming from the dissipative term prevails over the positive contribution coming from the flow term. More precisely, we have to check the inequality Observing that ω ′ (ξ) 1, we conclude that the positive part of the left hand side is bounded by Aξ(3 + log δ ξ ). To estimate the negative part, we just use the second order Taylor formula and monotonicity of ω ′′ on [0, ξ] to get the bound 1 π 8. Checking the inequality: case ξ δ In this case, we have ω(η) η for 0 η δ and ω(η) ω(ξ) for δ η ξ. Hence if γ < δ 2 and δ is small enough. Thus, the positive term on the left hand side is bounded from above by the expression Aω(ξ) 4 + log ξ δ ω ′ (ξ) = Aγ ω(ξ) ξ . To estimate the negative term, note that, for ξ δ, we have under the same assumptions on γ and δ as above. Also, due to concavity, we have ω(2η But ω(ξ) ξ (Aγ − 1 π ) < 0 if γ is small enough.

Concluding remarks
Here we just want to quote (with necessary minor modifications) a paragraph from [5]. Note that it was written just 2 years ago.
The case α = 1 2 is specially relevant because the viscous term (−∆) 1 2 θ models the so-called Eckmann's pumping, which has been observed in quasi-geostrophic flows. On the other hand, several authors have emphasized the deep analogy existing between the dissipative quasi-geostrophic equation with α = 1 2 and the 3D incompressible Navier-Stokes equations. This paper provides an elementary treatment of the α = 1 2 case. Unfortunately, the argument does not seem to extend to the Navier-Stokes equations due to the different structure of nonlinearity. So, while our paper resolves the global existence and regularity question in a physically relevant model, it also suggests that there is a significant structural difference between the critical 2D quasi-geostrophic equation and 3D Navier-Stokes equations.

Appendix
Here we provide a sketch of the proof of the Lemma.
A similar estimate holds for the second integral in (2). Next, letx = x+y 2 . Then when |x − t| ≥ 3ξ, the first integral is estimated by Cξ ∞ 3ξ ω(r) r 2 dr. The second integral is estimated by Cω(3ξ), and hence is controlled by 3C ξ 0 ω(r) r dr.