Existence and uniqueness of steady weak solutions to the non-Newtonian fluids in $\mathbb R^d$
Guillod and Wittwer (2018) investigated the existence and uniqueness of the weak solutions to the steady Navier–Stokes equations in the whole plane $\mathbb R ^2$. This problem is not trivial: due to the absence of boundaries the local behavior of the solutions cannot be controlled by the enstrophy in two dimensions. By introducing a prescribed mean velocity on some given bounded set, they obtained infinitely many weak solutions of the stationary Navier–Stokes equations in $\mathbb R ^2$ parameterized by this mean velocity. Furthermore, this explicit parameterization of the weak solutions allowed them to prove a weak-strong uniqueness theorem for small data. We are concerned with the steady equations for the non-Newtonian fluids in the whole space $\mathbb R ^d$ ($d=2\,,3$). For the non-Newtonian fluids, a similar problem (the local behavior of the solutions cannot be controlled by the enstrophy) arises when $p\geq d$. By following Guillod and Wittwer’s ideas, we succeed in establishing the existence of infinitely many weak solutions and a weak-strong uniqueness theorem for small data in $\mathbb R ^d$ with $p\geq d$.