Global \textbf{BV} entropy solutions and uniqueness for hyperbolic systems of balance laws
The authors consider the Cauchy problem for \(n \times n\) strictly hyperbolic systems of nonresonant balance laws \[ \begin{aligned} u_t+f(u)_x &= g(x,u), \qquad x \in \mathbb R, t>0, \\ u(0,.) &= u_o \in L^1 \cap \text{BV} (\mathbb Ri; {\mathbb R}^n), \\ |\lambda_i(u)|&\geq c > 0 \text{ for all } i\in \{1,\ldots,n\}, \\ |g(.,u)|+ \|\nabla_u g(.,u) \|&\leq \omega \in L^1 \cap L^\infty (\mathbb Ri), \end{aligned} \] each characteristic field being genuinely nonlinear or linearly degenerate. Assuming that \(\|\omega \|_{L^1(\mathbb R)}\) and \(\|u_0 \|_{\text{BV}(\mathbb R)}\) are small enough, the authors prove the existence and uniqueness of global entropy solutions of bounded total variation as limits of special wave-front tracking approximations for which the source term is localized by means of Dirac masses. Moreover, they give a characterization of the resulting semigroup trajectories in terms of integral estimates.
scientific article; zbMATH DE number 1773416