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Remark on the \(L^2\) decay for weak solution to equations of non-Newtonian incompressible fluids in the whole space. II. (English) Zbl 1107.76311
Salvi, Rodolfo (ed.), The Navier-Stokes equations: theory and numerical methods. Proceedings of the international conference, Varenna, Lecco, Italy, 2000. New York, NY: Marcel Dekker (ISBN 0-8247-0672-2/pbk). Lect. Notes Pure Appl. Math. 223, 221-232 (2002).
Introduction: The Cauchy problem of power-law incompressible fluids in \(\mathbb R^3\) is described by the system of equations \[ \begin{gathered} \text{div}\,u=0,\,u_t+(u\cdot\nabla)u=\text{div}\,T,\,u(\cdot,0)=u_0,\end{gathered} \] where \(u=(u_1,\dots,u_3)\) represents the velocity field, \(T\) is the stress tensor and \(u_0\) is the initial value of the velocity. The stress tensor is decomposed as \(T_{ij}=-\pi\delta_{ij}+\tau_{ij}^v\), where \(\pi\) is the pressure, \(\delta_{ij}\) is the Kronecker delta and \(\tau^v\) is the viscous part of the stress. We assume the stress tensor \(\tau^v\) of the form \(\tau^v=\tau(Du)\) with \(\tau\) a nonlinear tensor function, where the components of the symmetric deformation velocity tensor are given by \(Du_{ij}=\frac12(\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i})\). We consider the following growth conditions: \[ \begin{gathered} |\tau_{ij}(Du)|\leq C_1(|Du|+|Du|^{p-1}),\quad C_1>0, p\geq2,\\|\tau_{ij}(Du)|\leq C_1|Du|^{p-1},\quad 1<p<2,\end{gathered} \] as well as the strong coercivity condition \[ \begin{gathered} \tau_{ij}(Du)Du_{ij}\geq C_2(|Du|^p+| Du|^2),\𝟙<p<\infty, C_2>0, (|D|=(Du_{ij} Du_{ij})^{1/2}).\end{gathered} \]
The aim of this paper is to investigate the problem of shear thickening \((p>2)\) when the stress tensor has the following form: \[ \tau_{ij}^v=(\mu_0+\mu_1|Du|^{p-2})Du_{ij}. \]
We do not assume \(u_0\) to be small or to belong to some \(L_p\) space, but \(u_0\) is also the initial data of the linear heat equation. We see that the decay results depend solely on the decay properties of the solution of the heat equation. Finally, we deal with lower bounds of rates of decay of power law fluids. We consider the case when the average of \(\int_{\mathbb R^n}u_0\,dx\) is nonzero.
For Part I see [Š. Matušů-Nečasová and P. Penel, Ann. Univ. Ferrara, Nuova Ser., Sez. VII 46, 187-207 (2000; Zbl 1009.76005)].
For the entire collection see [Zbl 0972.00046].
76A05 Non-Newtonian fluids
35B40 Asymptotic behavior of solutions to PDEs
35Q35 PDEs in connection with fluid mechanics