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

TL;DR

This work addresses existence and uniqueness for the transport equation $u_t + \mathbf{b}\cdot\nabla u = 0$ in domains with inhomogeneous Dirichlet data, where the velocity field $\mathbf{b}$ is non-smooth but belongs to $L^1(0,T;W^{1,p'}(\Omega))$ with bounded $\operatorname{div}\mathbf{b}$. The authors introduce a boundary‑adapted mollification and prove a DiPerna–Lions–type renormalization property via a relabeling lemma, yielding uniqueness in $L^\infty(0,T;L^p(\Omega))$ for $1/p+1/p'=1$. Existence is established through two routes: a Grönwall-type argument for essentially bounded $\mathbf{b}$ and a renormalization/compactness approach that handles more general data; they also extend the theory to non-divergence-free fields by refining the commutator interchange. Overall, the paper extends DiPerna–Lions theory to ambient spaces with boundary and unbounded domains, providing well-posedness results for transport with non-smooth velocities and inhomogeneous boundary 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.

Figures (5)

• Figure 1: The standard mollifier
• Figure 2: The tailored mollifier
• Figure 3: The cone and the support of $\rho^{(d)}_{\eta(\varepsilon)}(\mathbf{0})$
• Figure 4: The graphs of $\psi_\delta(x_d)$ and $\rho^+_{\eta}(x_d)$. Here, $\delta = 0.3$ and $\eta = 0.25$.
• Figure 5: The graphs of $\theta$ and $\widetilde{\theta}$ for $\theta(\sigma) = \tanh{\sigma}$.

Theorems & Definitions (35)

• Definition 1.1
• Theorem 1.1: Uniqueness
• Theorem 1.2: Existence
• Lemma 1.1: Relabeling lemma
• Remark 2.1
• Lemma 2.1
• Remark 2.2
• proof : Proof of Lemma \ref{['lem:approx']}
• Remark 2.3
• Lemma 2.2