Quantitative enstrophy bounds for measure vorticities

where ω ν 0 := curl u ν 0 for a given divergence-free initial velocity u ν 0 . In (NS-Vort) it is tacitly assumed that u ν is the incompressible vector field such that curl u ν = ω ν , unique among the ones with a fixed spatial average. By compatibility, the spatial average of u ν is set to be the same of u ν 0 for all positive times. For any ν > 0 and any divergence-free initial condition u ν 0 ∈ L 2 (T 2 ), global-in-time weak solutions u ν ∈ L ∞ ([0, ∞); L 2 (T 2 )) ∩ L 2 ([0, ∞); Ḣ1 (T 2 )) are known to exist since the seminal work of Leray [23], and also Hopf [16]. In two space dimensions they are unique [1,29], they instantaneously become smooth, and they satisfy the energy identity

At least on the whole space R 2 , the two-dimensional Navier-Stokes equations are well-posed for any ω ν 0 ∈ M(R 2 ) as well [15]. This is more delicate with respect to the more classical setting of Leray since it also allows initial velocities with infinite kinetic energy 1 .

1.1. Main results. We are mainly interested in establishing estimates for the enstrophy ∥ω ν (t)∥ 2 L 2 when the initial vorticities are finite Borel measures. When {ω ν 0 } ν ⊂ M(T 2 ) is bounded, the estimate

is well-known. The same holds under the orthogonal assumption that {u ν 0 } ν ⊂ L 2 (T 2 ) is bounded. We refer, for instance, to [6] for more details.

Our goal is to improve on (1.1), possibly in a sharp way, under the additional assumption that the absolute vorticity on balls, i.e.

M ω (r) := sup

|ω ν (y, t)| dy ∀r > 0, (1.2) vanishes in the limit as r → 0, uniformly in time and viscosity. Building on the strategy introduced in [2, 13,22], in Section 3 we present how enstrophy bounds can be obtained from the knowledge of M ω (r). Although the precise expression might be implicit in general, here we state the main implication of the approach in a couple of specific cases.

Theorem 1.1. Let {u ν 0 } ν ⊂ L 2 (T 2 ) be such that {ω ν 0 } ν ⊂ M(T 2 ) is bounded. Let {ω ν } ν be the corresponding solutions to (NS-Vort) and let M ω be defined as in (1.2).

(a) If M ω (r) ≲ r α for some α ∈ (0, 2), then

for all times t, T > 0 such that νt < 1 and νT < 1, with the implicit constants independent of ν, t, T .

and for all times t > 0 such that νt < 1, with the implicit constant that does not depend on ν, t. In particular, for any δ > 0, there holds

and for any T > δ such that νT < 1, with the implicit constant independent of ν, δ, T .

The restriction to time scales below ν -1 is natural since otherwise the sharp estimate would be provided by the trivial bound (1.1). We emphasize that Theorem 1.1 does not require the sequence of initial velocities to stay bounded in L 2 (T 2 ). In fact, the assumption {u ν 0 } ν ⊂ L 2 (T 2 ) is most likely useless (see Remark 3.1). As an immediate corollary, we deduce lower bounds on the dissipation timescale.

) is bounded. Let {ω ν } ν be the corresponding solutions to (NS-Vort) and let M ω be defined as in (1.2). Let {T ν } ν be a sequence of positive times.

(a ′ ) If M ω (r) ≲ r α for some α ∈ (0, 2), then

and, in addition, {u ν 0 } ν ⊂ L 2 (T 2 ) is strongly compact, for any κ ∈ (0, 1 2 ) it holds

In fact, the above results are not specific to the Navier-Stokes equations, since they apply to any advection diffusion equation with a divergence-free drift (see Remark 3.2). The rest of the introduction is devoted to describe the main context in which these results fit, together with the main improvements over the existing literature. We also discuss, and in some cases prove, their optimality.

1.2. Main context and related literature. Vorticity concentration relates to the global existence of weak solutions to the two-dimensional Euler equations with measure initial vorticity. This was first noted by Delort [7] who proved global existence for a finite energy initial velocity as soon as the singular2 part of the vorticity has distinguished sign. After more than 30 years, and several contributions [3,14,17,21,24,26,30,32,33] by many authors, the type of initial data considered by Delort essentially remains the largest class for which global existence of weak solutions is known. Previous results were obtained in the seminal works by DiPerna and Majda [8][9][10], in which several tools for the study of “loss of compactness” issues in nonlinear PDEs have been developed.

The main observation by Delort [7] is that the L 2 loc (T 2 × [0, ∞)) strong compactness of the approximating velocity fields is not essential to obtain the global existence of a weak solution to the Euler equations. This is due to the special structure of the nonlinearity. He established an abstract convergence result for time-dependent divergence-free vector fields as soon as the sequence of vorticities does not display spatial concentrations, uniformly in time. In our notation (1.2), this re