Weak-BV stability of inflow and outflow problems for the isentropic Euler system
Abstract
We study the well-posedness of small BV solutions to the one-dimensional isentropic Euler system on the half-line under inflow and outflow boundary conditions. These boundary conditions are formulated in terms of admissible trace sets determined by Navier--Stokes boundary layers and zero-speed shocks. For both the inflow and outflow problems, we construct small BV solutions taking values in the subsonic region. These solutions are unique and satisfy the quantitative stability estimate |U(·, t)−V(·, t)|L2 ≲ √(|U(·, 0) − V(·, 0)|L2) , which holds for any such small BV solution V and any L∞ entropy solution U satisfying the strong trace property and the inflow/outflow boundary conditions. In particular, we emphasize that U may take values outside the subsonic region and need not have bounded variation or be constructed by the front tracking scheme. This establishes the first weak-BV stability and uniqueness theory for inflow and outflow initial-boundary value problems. A key difficulty lies in verifying that the boundary set is compatible with the a-contraction framework, and in proving that the front-tracking limit satisfies the prescribed inflow/outflow conditions.
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BibTeX
@misc{kang2026weakbv,
title = {{Weak-BV stability of inflow and outflow problems for the isentropic Euler system}},
author = {Moon-Jin Kang and Jiayun Meng and HyeonSeop Oh and Alexis F. Vasseur},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.03236},
eprint = {2610.03236},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.03236},
}
RIS
TY - UNPB
TI - Weak-BV stability of inflow and outflow problems for the isentropic Euler system
AU - Moon-Jin Kang
AU - Jiayun Meng
AU - HyeonSeop Oh
AU - Alexis F. Vasseur
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2610.03236
UR - https://arxiv.org/abs/2610.03236
AB - We study the well-posedness of small BV solutions to the one-dimensional isentropic Euler system on the half-line under inflow and outflow boundary conditions. These boundary conditions are formulated in terms of admissible trace sets determined by Navier--Stokes boundary layers and zero-speed shocks. For both the inflow and outflow problems, we construct small BV solutions taking values in the subsonic region. These solutions are unique and satisfy the quantitative stability estimate |U(·, t)−V(·, t)|_L^2 ≲ √(|U(·, 0) − V(·, 0)|_L^2) , which holds for any such small BV solution V and any L^∞ entropy solution U satisfying the strong trace property and the inflow/outflow boundary conditions. In particular, we emphasize that U may take values outside the subsonic region and need not have bounded variation or be constructed by the front tracking scheme. This establishes the first weak-BV stability and uniqueness theory for inflow and outflow initial-boundary value problems. A key difficulty lies in verifying that the boundary set is compatible with the a-contraction framework, and in proving that the front-tracking limit satisfies the prescribed inflow/outflow conditions.
ER -