Drift-diffusion equations with saturation on ℝd
Abstract
We study a class of nonlinear drift--diffusion equations on ℝd with saturation-type mobilities, not necessarily concave. Under structural assumptions on the mobility and potentials, bounded-domain approximations and local Caccioppoli estimates yield weak solutions for all positive times and an L1-contraction estimate. The same estimates provide local space--time compactness of time translates, whose subsequential limits are weak solutions. Under additional confinement assumptions, uniform mass-tail estimates yield a mass-preserving, free-energy dissipating C0-semigroup of L1(ℝd)-contractions. In this confining setting, every trajectory converges strongly in L1(ℝd) to a stationary state with the same mass, and the resulting time-limit operator is an L1-contraction. Under the same assumptions, we also identify the unique global and L1-local minimiser of the free energy at each fixed mass and prove that it is stationary under the semigroup.
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BibTeX
@misc{fernandezjimenez2026driftdiffusion,
title = {{Drift-diffusion equations with saturation on $\mathbbR^d$}},
author = {Alejandro Fernández-Jiménez},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.40294},
eprint = {2610.40294},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.40294},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Drift-diffusion equations with saturation on ℝ^d
AU - Alejandro Fernández-Jiménez
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.40294
UR - https://arxiv.org/abs/2610.40294
AB - We study a class of nonlinear drift--diffusion equations on ℝ^d with saturation-type mobilities, not necessarily concave. Under structural assumptions on the mobility and potentials, bounded-domain approximations and local Caccioppoli estimates yield weak solutions for all positive times and an L^1-contraction estimate. The same estimates provide local space--time compactness of time translates, whose subsequential limits are weak solutions. Under additional confinement assumptions, uniform mass-tail estimates yield a mass-preserving, free-energy dissipating C_0-semigroup of L^1(ℝ^d)-contractions. In this confining setting, every trajectory converges strongly in L^1(ℝ^d) to a stationary state with the same mass, and the resulting time-limit operator is an L^1-contraction. Under the same assumptions, we also identify the unique global and L^1-local minimiser of the free energy at each fixed mass and prove that it is stationary under the semigroup.
ER -