Mathematical analysis of a three-dimensional eutrophication model

Abstract

In this paper we present and analyze a nutrient-phytoplankton-zooplankton-organic detritus-dissolved oxygen mathematical model simulating eutrophication processes into aquatic media. As a main result, we obtain existence and uniqueness results for the solution of the system, under realistic hypotheses of non-smooth coefficients (in particular, a non-regular water velocity). This lack or regularity prevent us from using the standard semigroup approach, forcing us towards the utilization of more refined techniques.

Analysis of a time optimal control problem related to the management of a bioreactor

“…Then, as was demonstrated by the authors in [3] (using similar arguments to previous paper [2] for a problem related to food technology), the eutrophication systems (2.4)-(2.5) admit a solution under non-smooth hypotheses. To be exact, if we assume that the initial conditions and the source terms are nonnegative and bounded, and that the fluid velocities and temperatures satisfy:…”

“…From the uniqueness of solution of state systems (2.4)-(2.5) we can assure that (u 1 n , u 2 n ) are defined for the whole time interval (0, T max ), for all n ∈ N. Then we have, thanks to the estimates obtained in [3] and to the boundedness of the set V ad , that the sequence of states {(u 1 n , u 2 n )} n∈N is bounded, i.e.…”

“…(In particular, for δ = 0 we have = 0, and for δ → ∞, we have → 1; see [3].) Moreover, this solution (u 1 , u 2 ) is nonnegative and bounded (in previous space norm) by a value depending on the time bound T max , the fluid velocities (w 1 , w 2 ), the controls (ρ 1 , ρ 2 ), and the initial condition u 1 0 .…”

“…The most remarkable results of existence of solution for a complete nutrient-phytoplankton-zooplankton-oxygen model can be found in the paper of Allegretto et al [1] (for a particular periodic case) or in the recent work of the authors [3] (presenting also uniqueness and regularity results for a realistic situation with non-smooth water velocity).…”

Optimal control of a bioreactor

“…For the fully three-dimensional case, several numerical models have been proposed, for instance, by Drago et al [13], Yamashiki et al [19] or Park et al [16], but no one gives theoretical results of existence or uniqueness of solution, but only numerical simulations for particular areas. The most remarkable results of existence of solution for a nutrient-phytoplankton-zooplankton-oxygen model can be found in the paper of Allegretto et al [3] (for a particular periodic case), and in the recent work of the authors [6] (presenting also 0096-3003/$ -see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2010.03.097 uniqueness and regularity results for a more general non-smooth situation modelling nutrient-phytoplankton-zooplankton-oxygen-detritus concentrations).…”