Dynamic growth estimates of maximum vorticity for 3Dincompressible Euler equations and the SQG model

By performing estimates on the integral of the absolute value ofvorticity along a local vortex line segment, we establish a relativelysharp dynamic growth estimate of maximum vorticity under someassumptions on the local geometric regularity of the vorticity vector.Our analysis applies to both the 3D incompressible Euler equationsand the surface quasi-geostrophic model (SQG).As an application of our vorticity growth estimate, we apply ourresult to the 3D Euler equation with the two anti-parallel vortex tubesinitial data considered by Hou-Li [12]. Under someadditional assumption on the vorticity field, which seems to beconsistent with the computational results of [12], we showthat the maximum vorticity can not grow faster than double exponentialin time. Our analysis extends the earlier results byCordoba-Fefferman [6, 7] and Deng-Hou-Yu [8, 9].