Balance Laws with Integrable Unbounded Sources

Abstract

We consider the Cauchy problem for an $n\times n$ strictly hyperbolic system of balance laws $u_t+f(u)_x=g(x,u)$, $x\in\mathbb{R}$, $t>0$, $\|g(x,\cdot)\|_{\mathbf{C}^2}\leq\tilde{M}(x)\in\mathbf{L}^1$, endowed with the initial data $u(0,.)=u_o\in\mathbf{L}^1\cap\mathbf{BV}(\mathbb{R};\mathbb{R}^n)$. Each characteristic field is assumed to be genuinely nonlinear or linearly degenerate and nonresonant with the source, i.e., $|\lambda_i(u)|\geq c>0$ for all $i\in\{1,\dots,n\}$. Assuming that the $\mathbf{L}^1$ norms of $\|g(x,\cdot)\|_{\mathbf{C}^1}$ and $\|u_o\|_{\mathbf{BV}(\mathbb{R})}$ are small enough, we prove the existence and uniqueness of global entropy solutions of bounded total variation extending the result in [D. Amadori, L. Gosse, and G. Guerra, Arch. Ration. Mech. Anal., 162 (2002), pp. 327–366] to unbounded (in $\mathbf{L}^\infty$) 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.