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 Ω âŠ‚ R 2 or on the torus Ω = T 2 . We construct global-in-time weak solutions with vorticity in L 1 ∩ L ul p and in L 1 ∩ Y ul Θ , where L ul p and Y ul Θ are suitable uniformly-localized versions of the Lebesgue space L p and of the Yudovich space Y Θ respectively, with no condition at infinity for the growth function  Θ . We also provide an explicit modulus of continuity for the velocity depending on the growth function  Θ . We prove uniqueness of weak solutions in L 1 ∩ Y ul Θ under the assumption that  Θ grows moderately at infinity. In contrast to Yudovich's energy method, we employ a Lagrangian strategy to show uniqueness. Our entire argument relies on elementary real-variable techniques, with no use of either Sobolev spaces, Calderón-Zygmund theory or Littlewood-Paley decomposition, and actually applies not only to the Biot-Savart law, but also to more general operators whose kernels obey some natural structural assumptions.

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

We continue the study of the space  B V α ( R n ) of functions with bounded fractional variation in  R n of order α ∈ ( 0 , 1 ) 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 α → 1 - . We prove that the α -gradient of a W 1 , p -function converges in L p to the gradient for all p ∈ [ 1 , + ∞ ) as α → 1 - . Moreover, we prove that the fractional α -variation converges to the standard De Giorgi's variation both pointwise and in the Γ -limit sense as α → 1 - . Finally, we prove that the fractional ß -variation converges to the fractional α -variation both pointwise and in the Γ -limit sense as ß â†’ α - for any given α ∈ ( 0 , 1 ) .