Abstract
In this paper, we use mass transportation theory to study pollution transfer in porous media. We show the existence of a $L^2-$regular vector field defined by a $W^{1, 1}-$ optimal transport map. A sufficient condition for solvability of our model, is given by a (non homogeneous) transport equation with a source defined by a measure. The mathematical framework used, allows us to show in some specifical cases, existence of solution for a nonlinear PDE deriving from the modelling. And we end by numerical simulations.
Highlights
Pollution problems are interesting and important topic in physics, in mathematical physics, in chemistry, in biology and even in complex sciences
We show the existence of a L2−regular vector field defined by a W1,1− optimal transport map
A sufficient condition for solvability of our model, is given by a transport equation with a source defined by a measure
Summary
Pollution problems are interesting and important topic in physics, in mathematical physics, in chemistry, in biology and even in complex sciences. After the modelling of the physical problem, we endeavour to use mass transportation and PDE theories to study pollution in porous media. The paper is organized as follows: The section 2 is devoted to the modelling of pollutant transfer in porous media. This mathematical framework used, allows us to solve the nonlinear system which derives from the modeling and in addition we give the optimality system. If we consi∫der Ω ∈ Rn any elementary domain of the porous domain D, the mass of the considered fluid is given by M(Ω, t) = Ω dm; where dm is a mass element of the fluid It is given by the following expression: dm = ρ(x, t)ε(x, t); where ρ(x, t) is the density.
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