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We study the singularity formation of a quasi-exact 1D model proposed by Hou and Li (2008Commun. Pure Appl. Math.61661–97). This model is based on an approximation of the axisymmetric Navier–Stokes equations in therdirection. The solution of the 1D model can be used to construct an exact solution of the original 3D Euler and Navier–Stokes equations if the initial angular velocity, angular vorticity, and angular stream function are linear inr. This model shares many intrinsic properties similar to those of the 3D Euler and Navier–Stokes equations. It captures the competition between advection and vortex stretching as in the 1D De Gregorio (De Gregorio 1990J. Stat. Phys.591251–63; De Gregorio 1996Math. Methods Appl. Sci.191233–55) model. We show that the inviscid model with weakened advection and smooth initial data or the original 1D model with Hölder continuous data develops a self-similar blowup. We also show that the viscous model with weakened advection and smooth initial data develops a finite time blowup. To obtain sharp estimates for the nonlocal terms, we perform an exact computation for the low-frequency Fourier modes and extract damping in leading order estimates for the high-frequency modes using singularly weighted norms in the energy estimates. The analysis for the viscous case is more subtle since the viscous terms produce some instability if we just use singular weights. We establish the blowup analysis for the viscous model by carefully designing an energy norm that combines a singularly weighted energy norm and a sum of high-order Sobolev norms.

We present REGLO, a novel methodology for repairing pretrained neural networks to satisfy global robustness and individual fairness properties. A neural network is said to be globally robust with respect to a given input region if and only if all the input points in the region are locally robust. This notion of global robustness also captures the notion of individual fairness as a special case. We prove that any counterexample to a global robustness property must exhibit a corresponding large gradient. For ReLU networks, this result allows us to efficiently identify the linear regions that violate a given global robustness property. By formulating and solving a suitable robust convex optimization problem, REGLO then computes a minimal weight change that will provably repair these violating linear regions.