On Existence, Uniqueness and Qualitative Properties of a system of Partial Dierential Equations involved in a Statistical Learning Tool intending livestock Data Assimilation. Working-Paper Version

In the article Flourent et al. (Submitted), a Biomimetic Statistical Learning tool was developed. This tool is based on a system of Partial Dierential Equations. Some applications showed that this tool provides accurate predictions and has an extrapolation capability. Hence, the mathematical analysis of the integrated PDE system becomes mandatory. Yet, the two rst equations of this system contain a diusion operator depending on the sum of two unknown densities. This fact, coupled with the presence of a convection term, makes impossible the direct application of the classical results about parabolic equations. Therefore, the mathematical analysis of this model constitutes a new challenge for PDE analysis. Some properties typically veried in the framework of the parabolic PDEs are not obtainable for the studied PDE system. Hence, we worked on two toy models to put in evidence the lost properties and the ones which can be expected despite the presence of an atypical diusion term.