Chemically reacting fluid flows: Weak solutions and global attractors (Q1283887)

The three-dimensional Navier-Stokes equations of incompressible fluid are coupled with the reaction-diffusion equations to describe chemically reacting flows. These equations were earlier considered in the two-dimensional case by \textit{O. Manley, M. Marion} and \textit{R. Temam} [Indiana Univ. Math. J. 42, No. 3, 941-967 (1993; Zbl 0796.35164)]. The boundary conditions for the chemistry and temperature equations are required to be inhomogeneous mixed Robin and Neumann boundary conditions depending in a specific way on the fluid flow at the boundary \(u(x,t)= v(x,t)+ w(x)\) for the velocity \(u\), where \(v\) satisfies homogeneous boundary conditions and \(w\) satisfies inhomogeneous boundary conditions and does not depend on time. The same procedure is applied for temperature and chemicals. A priori estimates are derived through the proper choice of homogeneous terms like \(w\). The solution is proved to be physically reasonable, i.e., temperature is bounded from below, and volume concentrations remain nonnegative and their total sum is equal to one at any point and at any moment. The existence of global attractors is proved in a manner of \textit{G. R. Sell} [J. Dyn. Differ. Equ. 8, No. 1, 1-33 (1996; Zbl 0855.35100)].