zbMATH — the first resource for mathematics

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.

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