Linear transport equations

We develop an existence uniqueness theory for linear transport equations with discontinuous coefficients. We study the case of existence and uniqueness of continuous solutions to linear transport equations. We impose minimal regularity on the transport velocity to guarantee that weak solutions are well defined. We use the Filippov ODE approach to give sufficient conditions for existence of a continuous weak solution. In this generality, we can have many continuous weak solutions. We show that there is a unique weak solution which is stable under smoothings of the transport velocity and the initial data. We also derive analytic properties of the stable solution which in the one-dimensional case can serve as entropy conditions. For example, in the case of initial data with bounded variation, the stable solution is the only weak solution which preserves the variation of the initial data for all times. In the two-dimensional case, we derive a new property of the flow solution of the associated ODE which is the analogue of upper and lower solutions in the one-dimensional case. Although the Toneli variation of the initial data is not preserved, we show that the stable solution of linear transport equation preserves the Banach indicatrix of the initial data for all times. We also show that this property is not enough to separate the stable solution from the set of all weak solutions. We give an example of a transport equations with many flow solutions and each flow solution preserves the Banach indicatrix of the initial data for all times. We point out the reason for this phenomenon—the structure of connected sets in the multidimensional case is much more complicated than the one in the one-dimensional case. Finally, we use the DiPerna-Lions regularization approach to give sufficient conditions on the transport velocity for uniqueness of a continuous weak solution if exists. Our result for uniqueness of continuous weak solutions is the natural extension of DiPerna-Lions theory for $Lp$ solutions, $1\le p\le \infty$.