Well-posedness of the Burgers' equation on the half-line through the Cole-Hopf transformation
Abstract
We rigorously establish the well-posedness of the initial-boundary value problem (IBVP) for the Burgers' equation on the half-line for data in Sobolev-type spaces. The formal solution to this problem was previously derived via the Cole-Hopf transformation and the Fokas unified transform method under the assumption of smooth, Schwartz-class data. In this work, we prove the validity of this formal representation for initial data u0∈ H̃s+1x (0,∞) and boundary data g0 ∈ Ht(2s+1)/4 where 1/2<s<3/2. Well-posedness is established under two complementary conditions: (1) sufficiently small initial and boundary data, and (2) non-positive boundary data of arbitrary size. The latter condition utilizes the strong minimum principle for the corresponding linear heat equation to ensure the global invertibility of the nonlinear Cole-Hopf map. The proofs combine linear estimates for the heat IBVP with nonlinear bounds obtained through Taylor series representations and the algebraic properties of Sobolev spaces.
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BibTeX
@misc{vo2026wellposedness,
title = {{Well-posedness of the Burgers' equation on the half-line through the Cole-Hopf transformation}},
author = {Huy K. Vo and Athanassios S. Fokas},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.33886},
eprint = {2610.33886},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.33886},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Well-posedness of the Burgers' equation on the half-line through the Cole-Hopf transformation
AU - Huy K. Vo
AU - Athanassios S. Fokas
PY - 2026
DA - 2026/10/24
PB - arXiv
DO - 10.48550/arxiv.2610.33886
UR - https://arxiv.org/abs/2610.33886
AB - We rigorously establish the well-posedness of the initial-boundary value problem (IBVP) for the Burgers' equation on the half-line for data in Sobolev-type spaces. The formal solution to this problem was previously derived via the Cole-Hopf transformation and the Fokas unified transform method under the assumption of smooth, Schwartz-class data. In this work, we prove the validity of this formal representation for initial data u_0∈ H̃^s+1_x (0,∞) and boundary data g_0 ∈ H_t^(2s+1)/4 where 1/2<s<3/2. Well-posedness is established under two complementary conditions: (1) sufficiently small initial and boundary data, and (2) non-positive boundary data of arbitrary size. The latter condition utilizes the strong minimum principle for the corresponding linear heat equation to ensure the global invertibility of the nonlinear Cole-Hopf map. The proofs combine linear estimates for the heat IBVP with nonlinear bounds obtained through Taylor series representations and the algebraic properties of Sobolev spaces.
ER -