Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2605.01523

Optimal transport of signed measures: existence, uniqueness and fractal structure part I: the separated-support case research announcement, with a numerical validation report

Bwo'Nyahre Baidi Barthelemy·Kouakep Tchaptche Yannick·Houpa Danga Duplex Elvies
View original source ↗Read source document ↗Export metadata ↓

Abstract

We announce a rigorous theory of optimal transport for signed (real) measures on Rd in the separated-support regime: the supports of the positive and negative parts of measures are assumed spatially disjoint. Under this structural hypothesis, together with regularity and non-degeneracy conditions on the absolutely continuous, discrete, and fractal singular components of the underlying Jordan and Lebesgue decompositions, we obtain existence and uniqueness of an optimal transport map for a cost that distinguishes same-sign and opposite-sign transports via a positional penalty, a coupled Monge-Ampere system together with a double Legendre transform characterising the solution, and exact preservation of Hausdorff dimension on the fractal singular components, the latter via a novel adaptive regularisation kernel and a direct measure-theoretic argument requiring no continuity of the transport map. This note states the results and outlines the two main proof techniques, full proofs, uniform estimates, and technical lemmas appear in the complete version of this work. We further report, in full and exhaustive summary, a complete numerical validation of this theory in ambient dimension one, two, and three: an entropic Sinkhorn solver for the single combined transport problem underlying the existence proof, applied to an explicit test case in which inter-sign transport is forced by mass imbalance and calibrated to occur with a large, robust cost margin, recovers exactly the predicted four-region structure with zero measured crossing of fractal mass between signs, and an exact-assignment refinement confirms pointwise-exact preservation of Hausdorff dimension under transport, with box-counting estimates converging to the theoretical dimension as resolution increases. The case of overlapping supports and the fully general unbalanced setting are the subject of Part II, in preparation.

Explore related subjects

math.AP

Keep this discovery

BibTeXRIS

Bwo'Nyahre Baidi Barthelemy, Kouakep Tchaptche Yannick, Houpa Danga Duplex Elvies. 2026-05-02. Optimal transport of signed measures: existence, uniqueness and fractal structure part I: the separated-support case research announcement, with a numerical validation report. https://arxiv.org/abs/2605.01523

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Well-posedness of the two-dimensional unsteady Prandtl system in Sobolev space with degenerate critical points

This paper is devoted to the well-posedness of classical Prandtl equations in a finite order Sobolev space. For a initial data with degenerate critical points and general outflow, we obtain the local-in-time existence and uniqueness of the solution to the Prandtl equations in a Sobolev space, by introducing a new iteration scheme and linear cancelation. This result shows that Oleinik's monotonicity condition is not a necessary condition for the Prandtl equations to be well-posed in Sobolev spaces and provides evidence to demonstrate that zero shear stress does not necessarily lead to boundary layer separation in two-dimensional unsteady boundary layers.

math.AP↗
arXiv2026

Global existence and time decay for a bipolar Euler-Poisson system with one pressureless and undamped fluid

We study the Cauchy problem for a three-dimensional bipolar Euler--Poisson system in which one fluid is pressureless and undamped, while the other is subject to momentum relaxation. For sufficiently small smooth perturbations of a constant equilibrium, we prove the global existence and uniqueness of smooth solutions under an irrotationality assumption on the initial velocity of the pressureless fluid, together with algebraic time-decay estimates. The main difficulty is that the velocity of the pressureless fluid is dissipated only indirectly through the Poisson coupling, and this mechanism degenerates strongly at high frequencies, leading to a regularity-loss structure. We overcome this difficulty by combining refined Green-function estimates, a low--middle--high frequency decomposition, and high-order nonlinear energy estimates adapted to the asymmetric regularity hierarchy. The result establishes a global small-data theory for this asymmetric regime, in which pressure and damping are simultaneously absent from the same fluid.

math.AP↗
arXiv2026

Boundary layer of 2D Chemotaxis Navier-Stokes equations with logarithmic Sensitivity. II. viscous vanishing limit

This is the second part of a two-part work concerning boundary layer solutions to the coupled Chemotaxis-Navier-Stokes system in the two-dimensional half-space. In the present work, we address the convergence of boundary layer solutions to singular chemotaxis-fluid equations under slip boundary conditions with respect to the chemical diffusion-viscosity parameter $\varepsilon$ in the two-dimensional half-plane. More precisely, we show that the boundary layer for $\varepsilon>0$ (viscous convection coefficient) converges to the superposition of the outer layer (solution with $\varepsilon=0$) and the inner layer as $\varepsilon\rightarrow0$. The outer and inner profiles are explicitly derived as in the first part\cite{WWZ}. Furthermore, the well-posedness results of the coupled Chemotaxis-Navier-Stokes system in conormal Sobolev spaces will be presented in Appendix. They answer the question mentioned in the first part of the two-part work. This study could help the understanding of the chemotactic movement of aerobic bacteria to the water-air surface observed experimentally in fluids, and enrich the theoretical results of boundary layer in chemotactic fluid models.

math.AP↗