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Stabilizing effects of dissipation. (English) Zbl 0547.35014
Equadiff 82, Proc. int. Conf., Würzburg 1982, Lect. Notes Math. 1017, 140-147 (1983).
[For the entire collection see Zbl 0511.00014.]
The author investigates a problem that describes the evolution of an incompressible fluid (with the density \(\rho =1)\) which moves between two parallel plates occupying the planes \(x=0\) and \(x=1\) in the direction of the y-axis. The balance equations of momentum and energy are \((1)\quad v_ t=(\mu (\theta)v_ x)_ x,\quad\theta_ t=\mu (\theta)v^ 2_ x,\) 0\(\leq x\leq 1\), \(t\geq 0\), where v, \(\theta\) represent the y-component of velocity and respectively the temperature, while the viscosity \(\mu\) is assumed to be of the form \(\mu (\theta)=\theta^{-\gamma}\), \(0<\gamma <1\). The following boundary and initial conditions are associated to the system (1): \((2)\quad v(0,t)=0,\quad v(1,t)=1,\) \(t\geq 0\), \((3)\quad v(x,0)=v_ 0(x),\quad\theta (x,0)=\theta_ 0(x),\) 0\(\leq x\leq 1\). Assuming that \(v_ 0\in W^{2,2}(0,1)\), \(\theta_ 0\in W^{1,2}(0,1)\), \(v_ 0(0)=0\), \(v_ 0(1)=1\), \(\theta_ 0(x)>0\), the author proves the existence of a unique solution (v(x,t),\(\theta\) (x,t)) of the problem (1),(2),(3) which satisfies, as \(t\to\infty : v_ x(x,t)=1+O(t^{-(1-\gamma)/(1+\gamma)}),\quad v_ t(x,t)=O(t^{- 1}),\quad\int^{\theta (x,t)}_{\theta_ 0(x)}d\xi /\mu (\xi)=t+O(t^{2\gamma /(1+\gamma)}).\)
Reviewer: G.Moroşanu

35B40 Asymptotic behavior of solutions to PDEs
35B65 Smoothness and regularity of solutions to PDEs
35Q30 Navier-Stokes equations