Balance laws with integrable unbounded sources

TL;DR

This paper proves the existence and uniqueness of global entropy solutions for hyperbolic systems of balance laws with unbounded sources, extending previous results to cases with unbounded in $L^\infty$ sources, and applies it to fluid flow in pipes.

Contribution

It extends the theory of hyperbolic balance laws to include unbounded sources, establishing existence and uniqueness of solutions in this more general setting.

Findings

Existence and uniqueness of solutions for unbounded sources.

Application to fluid flow in pipes with discontinuous cross sections.

Extension of previous results to $L^\infty$ 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.