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

Abstract:

We revisit Yudovich’s well-posedness result for the 2-dimensional Euler equations for an inviscid incompressible fluid. Existence of global-in-time weak solutions holds in suitable uniformly-localized versions of the Lebesgue space and of the Yudovich space, respectively, for the vorticity, with explicit modulus of continuity for the associated velocity. Uniqueness of weak solutions, in contrast to Yudovich’s energy method, follows from a Lagrangian comparison. 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. This is a joint work in collaboration with Gianluca Crippa.

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