Large self-similar solutions of a convection--diffusion equation
Abstract
We construct self-similar solutions to a nonlinear convection--diffusion equation posed on the whole Euclidean space, for a class of homogeneous initial data. The corresponding self-similar profiles solve a nonlinear elliptic equation on ℝN. We obtain these profiles as limits of solutions to Dirichlet problems on bounded domains with truncated nonlinearities, using a priori estimates that are uniform with respect to both the domain size and the truncation level. Our approach is inspired by recent constructions of large self-similar solutions for the Navier--Stokes system. The convection--diffusion equation considered here shares some structural features with the Navier--Stokes equations, but its scalar parabolic nature allows us to exploit tools such as the maximum principle and thereby obtain a more direct construction of self-similar solutions.
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BibTeX
@misc{zheng2026large,
title = {{Large self-similar solutions of a convection--diffusion equation}},
author = {Xiaoxin Zheng and Dominika Pilarczyk and Grzegorz Karch and Marco Cannone},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.07259},
eprint = {2610.07259},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.07259},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Large self-similar solutions of a convection--diffusion equation
AU - Xiaoxin Zheng
AU - Dominika Pilarczyk
AU - Grzegorz Karch
AU - Marco Cannone
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.07259
UR - https://arxiv.org/abs/2610.07259
AB - We construct self-similar solutions to a nonlinear convection--diffusion equation posed on the whole Euclidean space, for a class of homogeneous initial data. The corresponding self-similar profiles solve a nonlinear elliptic equation on ℝ^N. We obtain these profiles as limits of solutions to Dirichlet problems on bounded domains with truncated nonlinearities, using a priori estimates that are uniform with respect to both the domain size and the truncation level. Our approach is inspired by recent constructions of large self-similar solutions for the Navier--Stokes system. The convection--diffusion equation considered here shares some structural features with the Navier--Stokes equations, but its scalar parabolic nature allows us to exploit tools such as the maximum principle and thereby obtain a more direct construction of self-similar solutions.
ER -