Local-in-time strong solvability of Navier--Stokes type variational inequalities by Rothe's method

We consider parabolic variational inequalities in a Hilbert space $V$, which have a non-monotone nonlinearity of Navier–Stokes type represented by a bilinear operator $B:V\times V\to V^{\mathrm{\prime }}$ and a monotone type nonlinearity described by a convex, proper, and lower-semicontinuous functional $\phi :V\to (-\mathrm{\infty },+\mathrm{\infty }]$. Existence and uniqueness of a local-in-time strong solution in a maximal-$L^2$-regularity class and in a Kiselev–Ladyzhenskaya class are proved by discretization in time (also known as Rothe’s method), provided that a corresponding stationary Stokes problem admits a regularity structure better than $V$ (which is typically $H^2$-regularity in case of the Navier–Stokes equations). Since we do not assume the cancelation property $<B(u,v),v>=0$, in applications we may allow for broader boundary conditions than those treated by the existing literature.