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

TL;DR

This paper proves local-in-time strong solvability of Navier--Stokes type variational inequalities in Hilbert spaces using Rothe's method, accommodating broader boundary conditions without the cancelation property.

Contribution

It establishes existence and uniqueness of solutions for Navier--Stokes type inequalities under less restrictive boundary conditions by leveraging time discretization and regularity assumptions.

Findings

Proves local-in-time strong solutions exist and are unique.

Uses Rothe's method for discretization in time.

Allows broader boundary conditions than previous studies.

Abstract

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 $B: V \times V \to V'$ and a monotone type nonlinearity described by a convex, proper, and lower-semicontinuous functional $\varphi : V \to (-\infty, +\infty]$. 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 $\left< B(u, v), v \right> = 0$, in applications we may allow for broader boundary conditions than those treated by the existing literature.