Existence of Periodic Solutions to Steady Viscous Burgers Equation with a General Force
Abstract
In this paper, we will construct periodic solutions to the viscous steady Burgers equation with an external force f(x), based on the following formal expansion uε(x)=u0(x)+ε u1(x)+⋯+εn un(x)+⋯, where ε ≥ 0 represents the viscosity and u0(x) is a solution to non-viscous steady Burgers equation with the external force f(x). We will focus on the solutions which are uniformly bounded with respect to the viscosity. In our previous work, starting from u0=−(2+cos x), the authors constructed the periodic solutions to the viscous steady Burgers equation with the external force f=u0(x)u0x=−2sin x−sin xcos x. In this paper, we will extend the main result obtained in our previous work and construct the solutions starting from general u0 and f satisfying the non-viscous steady Burgers equation u0(x)u0x=f. It will be shown that there exists a ε0>0, which depends on n, such that for any 0<ε<ε0, the viscous steady Burgers equation with the external force f has a periodic solution uε(x) ∈ C2([0,2π]), satisfying |uε(x)−u0(x)−ε u1(x)−⋯−εn un(x)| ≤ Cεn+1, where C>0 is a constant which depends on n, but is independent of ε. Compared with Jauslin-Kreiss-Moser's result, we present a new approach to construct periodic solutions to the viscous steady Burgers with a general external force. The constructed solutions will tend to the ones of the non-viscous Burgers equation with a sharper convergence rate (up to higher order) when the viscosity vanishes.
Related
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BibTeX
@misc{cao2026existence,
title = {{Existence of Periodic Solutions to Steady Viscous Burgers Equation with a General Force}},
author = {Yuhan Cao and Quansen Jiu},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.29704},
eprint = {2608.29704},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.29704},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Existence of Periodic Solutions to Steady Viscous Burgers Equation with a General Force
AU - Yuhan Cao
AU - Quansen Jiu
PY - 2026
DA - 2026/08/30
PB - arXiv
DO - 10.48550/arxiv.2608.29704
UR - https://arxiv.org/abs/2608.29704
AB - In this paper, we will construct periodic solutions to the viscous steady Burgers equation with an external force f(x), based on the following formal expansion u^ε(x)=u_0(x)+ε u_1(x)+⋯+ε^n u_n(x)+⋯, where ε ≥ 0 represents the viscosity and u_0(x) is a solution to non-viscous steady Burgers equation with the external force f(x). We will focus on the solutions which are uniformly bounded with respect to the viscosity. In our previous work, starting from u_0=−(2+cos x), the authors constructed the periodic solutions to the viscous steady Burgers equation with the external force f=u_0(x)u_0x=−2sin x−sin xcos x. In this paper, we will extend the main result obtained in our previous work and construct the solutions starting from general u_0 and f satisfying the non-viscous steady Burgers equation u_0(x)u_0x=f. It will be shown that there exists a ε_0>0, which depends on n, such that for any 0<ε<ε_0, the viscous steady Burgers equation with the external force f has a periodic solution u^ε(x) ∈ C^2([0,2π]), satisfying |u^ε(x)−u_0(x)−ε u_1(x)−⋯−ε^n u_n(x)| ≤ Cε^n+1, where C>0 is a constant which depends on n, but is independent of ε. Compared with Jauslin-Kreiss-Moser's result, we present a new approach to construct periodic solutions to the viscous steady Burgers with a general external force. The constructed solutions will tend to the ones of the non-viscous Burgers equation with a sharper convergence rate (up to higher order) when the viscosity vanishes.
ER -