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arXiv · 2501.12575

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

Tokuhiro Eto·Yoshikazu Giga
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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.

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Tokuhiro Eto, Yoshikazu Giga. 2025-01-22. On existence and uniqueness for transport equations with non-smooth velocity fields under inhomogeneous Dirichlet data. https://arxiv.org/abs/2501.12575

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