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 \begin{align*} \|U(\cdot, t)-V(\cdot, t)\|_{L^2} \lesssim \sqrt{\|U(\cdot, 0) - V(\cdot, 0)\|_{L^2}} \, , \end{align*} which holds for any such small BV solution $V$ and any $L^\infty$ 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.