On existence and uniqueness for transport equations with non-smooth velocity fields under inhomogeneous Dirichlet data

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2026年4月9日

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Tokuhiro Eto,
Yoshikazu Giga

摘要

A transport equation with a non-smooth velocity field is considered under inhomogeneous Dirichlet boundary conditions. The spatial gradient of the velocity field is assumed in $L^{p'}$ $L^{p^{'}}$ in space and the divergence of the velocity field is assumed to be bounded. By introducing a suitable notion of solutions, it is shown that there exists a unique renormalized weak solution for $L^p$ $L^{p}$ initial and boundary data for $1/p+1/p'=1$ $1 / p + 1 / p^{'} = 1$ . Our theory is considered as a natural extension of the theory due to DiPerna and Lions (1989), where there is no boundary. Although a smooth domain is considered, it is allowed to be unbounded. A key step is a mollification of a solution. In our theory, mollification in the direction normal to the boundary is tailored to approximate the boundary data.