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Large time behavior to the system of incompressible non-Newtonian fluids in \(\mathbb{R}^2\). (English) Zbl 1059.35104

From the text: We consider the optimal decay rate of global solutions to the Cauchy problem for two-dimensional non-Newtonian fluids \[ u_t-\Delta u+ (u\cdot\nabla)u- \nabla\cdot (|e(u)|^{p-2}e(u))+ \nabla\pi= 0, \]
\[ \nabla\cdot u=0, \qquad u(x,0)= u_0. \] It is proved that the weak solutions decay in \(L^2\) norm at \((1+t)^{-1/2}\) and the estimate for the decay rate is sharp in the sense that it coincides with the decay rate of a solution to the heat equation.

MSC:

35Q35 PDEs in connection with fluid mechanics
76A05 Non-Newtonian fluids
35B40 Asymptotic behavior of solutions to PDEs

Keywords:

weak solutions
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References:

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