For 2D Navier-Stokes with only horizontal viscosity in a channel, unique global weak solutions exist for initial velocities in L^2 with one y-derivative in L^2.
Elgindi, Finite-time singularity formation for C 1,α solutions to the incompressible Euler equations on R3, Ann
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On the two-dimensional Navier-Stokes equations with horizontal viscosity
For 2D Navier-Stokes with only horizontal viscosity in a channel, unique global weak solutions exist for initial velocities in L^2 with one y-derivative in L^2.