3D Navier-Stokes and Euler equations with initial data characterized by uniformly large vorticity (Q2767796)

The authors study the Cauchy problem for the three-dimensional Navier-Stokes equations, i.e. NEWLINE\[NEWLINE\begin{aligned} \partial_tV+(V\cdot\nabla) V-\nu\Delta V + \nabla p & =0\\ \nabla\cdot V & =0\\ V(0) & =V_0. \end{aligned}NEWLINE\]NEWLINE The initial velocity is assumed to be of the form NEWLINE\[NEWLINEV_0(y)=\widetilde V_0(y)+ {\Omega\over 2} (e_3\times y),NEWLINE\]NEWLINE where \(\widetilde V_0\in L^2 (T)\) for some periodic lattice \(T\) and \(\Omega\in\mathbb{R}\). The main result of the paper proves that any weak Leray solution of a certain form is regular after a finite time interval provided that \(\Omega\) is uniformly large. Important steps of the proof are the introduction of a suitably transformed system together with a careful study of the corresponding limit equation when sending \(\Omega\to\infty\).