Global well-posedness for the incompressible Euler equations in an endpoint Sobolev space

2026/07/19 by Raphaël Danchin, In-Jee Jeong
#math.AP

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Abstract

We consider the initial value problem for the vorticity equation in the endpoint critical Sobolev space W^d,1(ℝ^d) for d = 2, 3. In two dimensions, we prove global propagation of the W^2,1(ℝ²) regularity of the vorticity. In three dimensions, for axisymmetric flows without swirl, we propagate W^3,1(ℝ³) regularity of the vorticity for all times. These are in stark contrast to existing strong ill-posedness results in critical Sobolev spaces W^d/p,p(ℝ^d) for all 1 < p < ∞, which were based on axisymmetric flows without swirl when d = 3.

Citations

• On the Well-posedness of the Ideal MHD Equations in the Triebel–Lizorkin Spaces

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