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Global BV entropy solutions and uniqueness for hyperbolic systems of balance laws. (English) Zbl 1024.35066
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.

##### MSC:
 35L60 First-order nonlinear hyperbolic equations 35L65 Hyperbolic conservation laws 35A35 Theoretical approximation in context of PDEs
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