I'm looking for some references about the theory of regular Lagrangian flows on a smooth domain $\Omega$ of $\mathbb{R}^d$ (say a smooth bounded open set of $\mathbb{R}^d$ or a half space).

Here, regular Lagrangian flows refer to the ones developped in the theory of Ambrosio which intends to explore the link between

• continuity equations : $\partial_t \rho +\mathrm{div}_x(b \rho)=0$ and
• ordinary differenrial equations : $\frac{d}{d t} X(t,x)=b(t,X(t,x))$ with $X(0,x)=x $,

for a vector field $b=b(t,x)$ which is not smooth (namely with Sobolev or BV regularity in space). See for instance Definition 13 p.13 in

http://php.math.unifi.it/users/cime/Courses/2005/02/CIME-2005-Ambrosio-Lecture_Notes.pdf

It actually extends the famous results of Di Perna and Lions of 1989 on transport and continuity equations. But all of these theories actually take place in $\mathbb{R}^d$, namely the fixed vector field $b$ (and then $\rho=\rho(t,x)$ and $X=X(t,x)$) are defined (for the space variable) on the whole space i.e. $b : \mathbb{R}^+ \times\mathbb{R}^d \rightarrow \mathbb{R}^d$.

However, the theory of Di Perna and Lions can be extended to domains included in $\mathbb{R}^d$ and which have a boundary , by constructing renormalized solutions to these equations where there is of course an additional boundary condition (see for instance https://hal.archives-ouvertes.fr/hal-00004420/document for results on a bounded domain with an absorption boundary condition)

I am therefore wondering if regular Lagrangian flows can be also defined and used on a domain with a boundary.

I'm looking for some references about the theory of regular Lagrangian flows on a smooth domain $\Omega$ of $\mathbb{R}^d$ (say a smooth bounded open set of $\mathbb{R}^d$ or a half space).

Here, regular Lagrangian flows refer to the ones developped in the theory of Ambrosio which intends to explore the link between

• continuity equations : $\partial_t \rho +\mathrm{div}_x(b \rho)=0$ and
• ordinary differential equations : $\frac{d}{d t} X(t,x)=b(t,X(t,x))$ with $X(0,x)=x $,

for a vector field $b=b(t,x)$ which is not smooth (namely with Sobolev or BV regularity in space). See for instance Definition 13 p.13 in

http://php.math.unifi.it/users/cime/Courses/2005/02/CIME-2005-Ambrosio-Lecture_Notes.pdf

It actually extends the famous results of Di Perna and Lions of 1989 on transport and continuity equations. But all of these theories actually take place in $\mathbb{R}^d$, namely the fixed vector field $b$ (and then $\rho=\rho(t,x)$ and $X=X(t,x)$) are defined (for the space variable) on the whole space i.e. $b : \mathbb{R}^+ \times\mathbb{R}^d \rightarrow \mathbb{R}^d$.

However, the theory of Di Perna and Lions can be extended to domains included in $\mathbb{R}^d$ and which have a boundary , by constructing renormalized solutions to these equations where there is of course an additional boundary condition (see for instance https://hal.archives-ouvertes.fr/hal-00004420/document for results on a bounded domain with an absorption boundary condition)

I am therefore wondering if regular Lagrangian flows can be also defined and used on a domain with a boundary.