On Singular Vortex Patches, I: Well-posedness Issues

Preservation of log-Hölder coefficients of the vorticity in the transport equation

“…In turn, this forces the Lagrangian deformation to be divergent; |∂ x 1 Φ 1 (0, t)| = +∞ for t > 0. This is very interesting, since a formal analysis given in [4] suggests that ∇u(•, t) L ∞ ≃ ct −1 for some absolute constant c > 0, which is barely non-integrable in time. To clarify, in general (without the odd-odd assumption) it is possible that |ω(x, t)| ≃ log( 1 |x| ) −γ and u(x, t) ∈ L 1 t Lip hold at the same time for any γ > 0.…”

Vortex stretching and anomalous dissipation for the incompressible 3D Navier-Stokes equations

“…However, using the fact that ∂ 2 u 1 is uniformly bounded when ω is given exactly by the Bahouri-Chemin stationary solution (this can be shown using either Fourier series with Poisson summation formula or radial-angular decomposition; cf. [14,9,10]), we just need to estimate the part where ω L n (t) is different from the Bahouri-Chemin solution. Moreover, without loss of generality we take x = (x 1 , x 2 ) with 0 ≤ x 1 ≤ x 2 ≤ L 2 and we need to show a bound on the following:…”

The incompressible Euler equations under octahedral symmetry: singularity formation in a fundamental domain

“…• One may extend the solutions provided by Theorem A to R 3 and obtain ω ∈ L ∞ ([0, T ); L ∞ (R 3 )) which can be considered as a 3D vortex patch. In 2D, a global well-posedness result for vortex patches with corner singularity has been established in [22] under the assumption that the vorticity is m-fold rotationally symmetric for some m ≥ 3. Hence Theorem A can be considered as an extension of this result to the 3D case, although in the current setting the patch boundary near the origin is fixed in time by reflection symmetries.…”