Time decay rates of non-Newtonian flows in \(\mathbf {R}^{n}_{+}\)
The authors study time decay rates for weak solutions to an incompressible non-Newtonian fluid model in \({\mathbb R}^{n}_{+}\) for \(n \geq 3\). The model is described by the so-called Ladyzhenskaya equations. Using the spectral decomposition of Stokes operator and \(L^{p}\)-\(L^{q}\) estimates, it can be shown that the weak solutions decay in \(L^{2}\) norm like \(t^{- {n \over 2} ({1 \over r} - {1 \over 2})}\) when the initial velocity \(u_{0} \in L^{2} \cap L^{r}\) for \(1 \leq r < 2\). If \(u_{0}\) satisfies additionally the momentum condition \[ \int_{{\mathbb R}^{n}_{+}} | x_{n} u_{0} (x)| ^{n} \,dx < \infty, \qquad 1 < r \leq 2, \] then the higher decay rates \(t^{- {n \over 2} ({1 \over r} - {1 \over 2}) - {1 \over 2}}\) are obtained. Furthermore, the difference between the non-Newtonian flow and Navier-Stokes flow is estimated.
scientific article; zbMATH DE number 5072987