On existence and uniqueness for transport equations with non-smooth velocity fields under inhomogeneous Dirichlet data
Abstract
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 $L^{p'}$ 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 $L^p$ initial and boundary data for $1/p+1/p'=1$. 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.
Fields of Study & Subjects
Advanced Mathematical Modeling in Engineering
Numerical methods in inverse problems
Nonlinear Partial Differential Equations
Publication details
- Journal / source
- Mathematische Zeitschrift
- Publisher
- Springer Science and Business Media LLC
- Volume
- 313
- Issue
- 1
- Publication date
- 9 April 2026
- Language
- EN