Uniqueness of the solution of the continuity equation with discontinuous vector field

Question

I'm trying to prove the uniqueness of the following continuity equation with discontinuous vector field:

$ \begin{cases} m_t(t,x) + [(-\alpha x + \sigma u^{*}(x) + c) \ m(t,x)]_x = 0 \qquad (t,x) \in [0,T] \times [-X_{max}, X_{max}] \\ m(0,x) = m_0 \end{cases} $

where

\begin{equation} u^{*}(x):= \begin{cases} 1 &\text{ if } x>x_0 \\ 0 &\text{ if } x<x_0 \\ \frac{1}{2} &\text{ if } x=x_0 \end{cases} \end{equation}

and where $T >0, X_{max}>0, \alpha>0, c>0, x_0 >0, \sigma < 0$.

This problem corresponds to a distribution that accumulates into the value $x_0$. In fact, we see immediately that a (weak) solution is given by a certain measure in which it appears a delta dirac centered in $x_0$. I planned to use the result of DiPerna-Lions 'Ordinary differential equations, transport theory and Sobolev spaces', but I realized that here the assumptions are not completely fulfilled. Anyone have any suggestions?

Thanks a lot

Answer 1

Use the method of characteristics, and check how far the go before two different ones intersect, or whether there are regions not accessible by characteristics starting from the $x-$axis.