The initial-value problem for the Navier-Stokes equations with a free surface in $L^q$-Sobolev spaces

By Helmut Abels. Advances in Differential Equations, 2005

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On some free boundary problem of the Navier-Stokes equations in the maximal $L_p$-$L_q$ regularity class (2015, unknown): “lt; η(x′),x′ = (x1,...,xN−1) ∈ RN−1}. A local in time unique existence theorem was proved by Beale [5], Allain [2] and Tani [36] in the L2 Sobolev-Slobodetskii space with surface tension and by Abels [1] in the Lp Sobolev-Slobodetskii space without surface tension. A global in time unique existence ”
On a free boundary problem for the Navier-Stokes equations (2007, background): “Abels [1] proved a local in time unique existence theorem.”
Decay of viscous surface waves without surface tension (2010, unknown): “22”