We prove finite-time singularity formation for De Gregorio's model of the three-dimensional vorticity equation in the class of L p ∩ C α (R) vorticities for some α > 0 and p < ∞. We also prove finite-time singularity formation from smooth initial data for the Okamoto-Sakajo-Wunsch models in a new range of parameter values. As a consequence, we have finite-time singularity for certain infinite-energy solutions of the surface quasi-geostrophic equation which are C α -regular. One of the difficulties in the models we consider is that there are competing nonlocal stabilizing effects (advection) and destabilizing effects (vortex stretching) which are of the same size in terms of scaling. Hence, it is difficult to establish the domination of one effect over the other without having strong control of the solution.We conjecture that strong solutions to the De Gregorio model exhibit the following behavior: for each 0 < α < 1 there exists an initial ω 0 ∈ C α (R) which is compactly supported for which the solution becomes singular in finite-time; on the other hand, solutions to De Gregorio's equation are global whenever ω 0 ∈ L p ∩ C 1 (R) for some p < ∞. Such a dichotomy seems to be a genuinely non-linear effect which cannot be explained merely by scaling considerations since C α spaces are scaling subcritical for each α > 0.The absence of vortex stretching in two dimensions allows the two dimensional Euler equation to enjoy infinitely many coercive conserved quantities, notably all L p norms of the vorticity for 1 ≤ p ≤ ∞. The presence of vortex stretching makes the three-dimensional Euler equation unstable with respect to all L p norms which were conserved in the two dimensional case and leads to point-wise exponential or double-exponential growth of the vorticity in time (see, for example, [15], [11]). Despite the clear destabilizing properties of the vortex stretching term, the global regularity question of the three-dimensional Euler equation is wide open. In our view, there are three principle reasons why the regularity question is difficult:1. The quadratic and non-local nature of the vortex stretching term.2. The presence of both the vortex stretching and the transport terms.3. The 3D Euler equation is a system rather than a scalar equation.The importance of the second difficulty is discussed in great detail in [13]. In view of these many difficulties, experts in the field have devised simpler model equations that share some of the above properties for which more can be proven.

It has been known since work of Lichtenstein [42] and Gunther [29] in the 1920's that the 3D incompressible Euler equation is locally well-posed in the class of velocity fields with Hölder continuous gradient and suitable decay at infinity. It is shown here that these local solutions can develop singularities in finite time, even for some of the simplest three-dimensional flows.

We introduce a local-in-time existence and uniqueness class for solutions to the 2d Euler equation with unbounded vorticity. Furthermore, we show that solutions belonging to this class can develop stronger singularities in finite time, meaning that they experience finite time blow up and exit the wellposedness class. Such solutions may be continued as weak solutions (potentially non-uniquely) after the singularity. While the general dynamics of 2d Euler solutions beyond the Yudovich class will certainly not be so tame, studying such solutions gives a way to study singular phenomena in a more controlled setting.

For the 2D Euler equation in vorticity formulation, we construct localized smooth solutions whose critical Sobolev norms become large in a short period of time, and solutions which initially belong to L ∞ ∩ H 1 but escapes H 1 immediately for t > 0. Our main observation is that a localized chunk of vorticity bounded in L ∞ ∩ H 1 with odd-odd symmetry is able to generate a hyperbolic flow with large velocity gradient at least for a short period of time, which stretches the vorticity gradient.

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