Balance laws with integrable unbounded sources

Abstract

We consider the Cauchy problem for a $n\times n$ strictly hyperbolic system of balance laws $$ \{{array}{c} u_t+f(u)_x=g(x,u), x \in \mathbb{R}, t>0 u(0,.)=u_o \in L^1 \cap BV(\mathbb{R}; \mathbb{R}^n), | \lambda_i(u)| \geq c > 0 {for all} i\in \{1,...,n\}, \|g(x,\cdot)\|_{\mathbf{C}^2}\leq \tilde M(x) \in L1, {array}. $$ each characteristic field being genuinely nonlinear or linearly degenerate. Assuming that the $\mathbf{L}^1$ norm of $\|g(x,\cdot)\|_{\mathbf{C}^1}$ and $\|u_o\|_{BV(\reali)}$ are small enough, we prove the existence and uniqueness of global entropy solutions of bounded total variation extending the result in [1] to unbounded (in $L^\infty$) sources. Furthermore, we apply this result to the fluid flow in a pipe with discontinuous cross sectional area, showing existence and uniqueness of the underlying semigroup.