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Stopping a viscous fluid by a feedback dissipative field: thermal effects without phase changing. (English) Zbl 1082.35117

Rodrigues, José F. (ed.) et al., Trends in partial differential equations of mathematical physics. Selected papers of the international conference held on the occasion of the 70th birthday of V. A. Solonnikov, Óbidos, Portugal, June 7–10, 2003. Basel: Birkhäuser (ISBN 3-7643-7165-X/hbk). Progress in Nonlinear Differential Equations and their Applications 61, 1-14 (2005).
Authors’ abstract: We show how the action on two simultaneous effects (a suitable coupling about velocity and temperature and a low range of temperature but upper than the phase changing one) may be responsible of stopping a viscous fluid without any phase changing. Our model involves a system, on an unbounded pipe, given by the planar stationary Navier-Stokes equation perturbed with a sublinear term \(f(x,\theta,u)\) coupled with a stationary (and possibly nonlinear) advection diffusion equation for the temperature \(\theta\). After proving some results on the existence and uniqueness of weak solutions we apply an energy method to show that the velocity \(u\) vanishes for \(x\) large enough.
For the entire collection see [Zbl 1062.35005].

MSC:

35Q30 Navier-Stokes equations
76A05 Non-Newtonian fluids
35B37 PDE in connection with control problems (MSC2000)
76D07 Stokes and related (Oseen, etc.) flows
80A20 Heat and mass transfer, heat flow (MSC2010)
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