scholarly journals Pollution Transfer as Optimal Mass Transport Problem

2016 ◽  
Vol 8 (6) ◽  
pp. 58
Author(s):  
L. Ndiaye ◽  
Mb. Ndiaye ◽  
A. Sy ◽  
D. Seck
Keyword(s):  
Porous Media ◽  
Optimal Transport ◽  
Existence Of Solution ◽  
Nonlinear Pde ◽  
Mathematical Framework ◽  
Sufficient Condition ◽  
Mass Transportation ◽  
Optimal Mass Transport ◽  
Transportation Theory ◽  
Optimal Transport Map

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.

scholarly journals On deterministic solutions for multi-marginal optimal transport with Coulomb cost

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Ugo Bindini ◽  
Luigi De Pascale ◽  
Anna Kausamo
Keyword(s):  
Positive Result ◽  
Transportation Problem ◽  
Optimal Transport ◽  
Sufficient Condition ◽  
Mass Transportation ◽  
Necessary And Sufficient Condition ◽  
Infinite Class ◽  
Necessary And Sufficient

<p style='text-indent:20px;'>In this paper we study the three-marginal optimal mass transportation problem for the Coulomb cost on the plane <inline-formula><tex-math id="M1">\begin{document}$ \mathbb R^2 $\end{document}</tex-math></inline-formula>. The key question is the optimality of the so-called Seidl map, first disproved by Colombo and Stra. We generalize the partial positive result obtained by Colombo and Stra and give a necessary and sufficient condition for the radial Coulomb cost to coincide with a much simpler cost that corresponds to the situation where all three particles are aligned. Moreover, we produce an infinite class of regular counterexamples to the optimality of this family of maps.</p>

MASS TRANSPORTATION IN GROUPS OF TYPE H

2005 ◽  
Vol 07 (04) ◽  
pp. 509-537 ◽  
Author(s):  
S. RIGOT
Keyword(s):  
Cost Function ◽  
Optimal Transport ◽  
Transport Problem ◽  
Mass Transportation ◽  
Heisenberg Groups ◽  
The One ◽  
Carathéodory Distance ◽  
The Cost ◽  
Special Case

We give a solution of the optimal transport problem in groups of type H when the cost function is the square of either the Carnot–Carathéodory or Korányi distance. This generalizes results previously proved for the Heisenberg groups. We use the same strategy that the one which was developed in that special case together with slightly refined technicalities that essentially reflect the fact that the center of the group can be of dimension larger than one. For each distance we prove existence, uniqueness and give a characterization of the optimal transport. In the case of the Carnot–Carathéodory distance we also prove that the optimal transport arises as the limit of the optimal transports in natural Riemannian approximations.

scholarly journals Optimal Mass Transport in Thin Domains

2014 ◽  
Vol 14 (1) ◽  
Author(s):  
José C. Navarro-Climent ◽  
Julio D. Rossi ◽  
Raúl C. Volpe
Keyword(s):  
Mass Transport ◽  
Euclidean Distance ◽  
Optimal Transport ◽  
Transport Problem ◽  
Thin Domains ◽  
Optimal Mass Transport

AbstractWe find the behavior of the solution of the optimal transport problem for the Euclidean distance (and its approximation by p−Laplacian problems) when the involved measures are supported in a domain that is contracted in one direction.

scholarly journals A homogenised model for flow, transport and sorption in a heterogeneous porous medium

2021 ◽  
Vol 932 ◽  
Author(s):  
L.C. Auton ◽  
S. Pramanik ◽  
M.P. Dalwadi ◽  
C.W. MacMinn ◽  
I.M. Griffiths
Keyword(s):  
Porous Medium ◽  
Porous Media ◽  
Multiple Scales ◽  
Effective Diffusivity ◽  
Rectangular Lattice ◽  
Mathematical Framework ◽  
Anisotropic Flow ◽  
Flow And Transport ◽  
Soft Porous Media ◽  
The Impact

A major challenge in flow through porous media is to better understand the link between microstructure and macroscale flow and transport. For idealised microstructures, the mathematical framework of homogenisation theory can be used for this purpose. Here, we consider a two-dimensional microstructure comprising an array of obstacles of smooth but arbitrary shape, the size and spacing of which can vary along the length of the porous medium. We use homogenisation via the method of multiple scales to systematically upscale a novel problem involving cells of varying area to obtain effective continuum equations for macroscale flow and transport. The equations are characterised by the local porosity, a local anisotropic flow permeability, an effective local anisotropic solute diffusivity and an effective local adsorption rate. These macroscale properties depend non-trivially on the two degrees of microstructural geometric freedom in our problem: obstacle size and obstacle spacing. We exploit this dependence to construct and compare scenarios where the same porosity profile results from different combinations of obstacle size and spacing. We focus on a simple example geometry comprising circular obstacles on a rectangular lattice, for which we numerically determine the macroscale permeability and effective diffusivity. We investigate scenarios where the porosity is spatially uniform but the permeability and diffusivity are not. Our results may be useful in the design of filters or for studying the impact of deformation on transport in soft porous media.

Freeform illumination optics construction following an optimal transport map

2016 ◽  
Vol 55 (16) ◽  
pp. 4301 ◽  
Author(s):  
Zexin Feng ◽  
Brittany D. Froese ◽  
Rongguang Liang
Keyword(s):  
Optimal Transport ◽  
Optimal Transport Map

scholarly journals Optimal transport with discrete long-range mean-field interactions

2020 ◽  
Vol 10 (02) ◽  
pp. 2050011
Author(s):  
Jiakun Liu ◽  
Grégoire Loeper
Keyword(s):  
Force Field ◽  
Long Range ◽  
Interaction Potential ◽  
Velocity Potential ◽  
Optimal Transport ◽  
Mean Field ◽  
External Force Field ◽  
Intermediate Time ◽  
Optimal Transport Map ◽  
Intermediate Density

We study an optimal transport problem where, at some intermediate time, the mass is either accelerated by an external force field or self-interacting. We obtain the regularity of the velocity potential, intermediate density, and optimal transport map, under the conditions on the interaction potential that are related to the so-called Ma–Trudinger–Wang condition from optimal transport [X.-N. Ma, N. S. Trudinger and X.-J. Wang, Regularity of potential functions of the optimal transportation problems, Arch. Ration. Mech. Anal. 177 (2005) 151–183.].

scholarly journals Incompressible flow in granulated porous media with null initial velocity

1998 ◽  
Vol 20 (20) ◽  
pp. 07
Author(s):  
José Luiz Boldrini ◽  
João Paulo Lukaszczyk
Keyword(s):  
Porous Media ◽  
Incompressible Fluid ◽  
Incompressible Flow ◽  
Initial Velocity ◽  
Type Equation ◽  
Existence Of Solution ◽  
Navier Stokes ◽  
Fixed Type

In this work we study a Navier-Stokes type equation which models the flow of a viscous, homogeneous and incompressible fluid in a isotropic granular (non consolidated) porous media. Using point fixed type arguments we obtain conditions for existence of solution for the equation in Hölder's spaces.

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