A note on positive solutions for conservation laws with singular source

Abstract:

We consider the Cauchy problem for the scalar conservation law \begin{equation*} \partial _t u+\partial _x f(u) =\displaystyle \frac {1}{g(u)},\qquad t>0,\ x\in \mathbb {R}, \end{equation*} with $g\in C^1(\mathbb {R})$, $g(0)=0$, $g(u)>0$ for $u>0$, and assume that the initial datum $u_0$ is nonnegative.

We show the existence of entropy solutions that are positive a.e. by means of an approximation of the equation that preserves positive solutions and by passing to the limit using a monotonicity argument. The difficulty lies in handling the singularity of the right-hand side (the source term) as $u$ possibly vanishes at the initial time. The source term is shown to be locally integrable.

Moreover, we prove a uniqueness and stability result for the above equation.