$L_p$-Theory for a Class of Non-Newtonian Fluids

Abstract

Local-in-time well-posedness of the initial-boundary value problem for a class of non-Newtonian Navier–Stokes problems on domains with compact $C^{\mbox{3-}}$-boundary is proven in an $L_p$-setting for any space dimension $n\geq2$. The stress tensor is assumed to be of the generalized Newtonian type, i.e., $\cS=2\mu(|{\mathcal E}|_2^2){\mathcal E} -\pi I$, ${\mathcal E}=\frac{1}{2}(\nabla u+\nabla u^{\sf T}),$ where $|{\mathcal E}|_2^2=\sum_{i,j=1}^n \varepsilon_{ij}^2$ denotes the Hilbert–Schmidt norm of the rate of strain tensor ${\mathcal E}$. The viscosity function $\mu\in C^{2-}({\mathbb R}_+)$ is subject only to the condition $\mu(s)>0$, $\mu(s)+2s\mu^\prime(s)>0$, $s\geq 0,$ which for the standard power-law–like function $\mu(s)=\mu_0(1+s)^{\frac{d-2}{2}}$ merely means $\mu_0>0$ and $d\geq 1$. This result is based on maximal regularity theory for a suitable linear problem and a contraction argument.