General Reaction Systems. Invariant Manifolds and A Priori Bounds

General Reaction Systems. Invariant Manifolds and A Priori Bounds 

From this point and on it will be assumed that the transport coefficients, . are constant so that Eqs. (2.1.34), (2.1.35) simplify to 

The procedure of obtaining the invariant manifolds is similar to that of Section 1.2. By multiplying Eq.(2.4.1) by yn, summing over i, integrating twice, and using Eqs. (2.4.3), (2.4.4) there is obtained 

It follows from its definition that the manifold r uo) is closed and convex. It is also bounded as can be seen by setting Rp in Eq. (2.4.5) to obtain 

Due to the linearity of Eqs. (2.4.6), (2.4.7), the maximum and minimum of each state variable in r(wo) is attained at the intersection of r uo) with the surface dE +1, at points where one or more of the state variables vanish. These maxima and minima, which depend on the boundary condition Uq and the transport coefficients but are independent of the reaction rates and the size Tq of the region, are a priori bounds for the solutions of the boundary value problem (2.4.1) —(2.4.4). The independence of the a priori bounds of the reaction rates and the size of the region has very important mathematical implications and will be used extensively in the following sections. 

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