An elementary proof of existence and uniqueness for the Euler flow in localized Yudovich spaces.

We revisit Yudovich's well-posedness result for the 2-dimensional Euler equations for an inviscid incompressible fluid on either a sufficiently regular (not necessarily bounded) open set or on the torus . We construct global-in-time weak solutions with vorticity in and in , where and are suitable uniformly-localized versions of the Lebesgue space and of the Yudovich space respectively, with no condition at infinity for the growth function  . We also provide an explicit modulus of continuity for the velocity depending on the growth function  .

A distributional approach to fractional Sobolev spaces and fractional variation: asymptotics I.

We continue the study of the space  of functions with bounded fractional variation in  of order introduced in our previous work (Comi and Stefani in J Funct Anal 277(10):3373-3435, 2019). After some technical improvements of certain results of Comi and Stefani (2019) which may be of some separated insterest, we deal with the asymptotic behavior of the fractional operators involved as . We prove that the -gradient of a -function converges in to the gradient for all as .