The Cauchy problem for the 1-D Gurevich-Zybin system. (English) Zbl 1415.35186
Summary: In this paper, we prove well-posedness of the one dimensional Gurevich-Zybin system in Besov spaces \(B_{p, r}^s\). This will imply the existence and uniqueness of solutions in the aforementioned spaces along with the continuity of the data-to-solution map, provided that the initial data belong to \(B_{p, r}^s\). We also establish persistence properties of solutions by showing that strong solutions to the Gurevich-Zybin system decay at infinity in the spatial variable, provided that the initial data do. Furthermore, it is shown that the system exhibits finite speed of propagation in the sense that if the initial data are endowed with compact support, then the solution carries this support along particle trajectories. As a concluding note, we discuss blow-up criterion for solutions.
©2019 American Institute of Physics
©2019 American Institute of Physics
MSC:
| 35L60 | First-order nonlinear hyperbolic equations |
| 35F40 | Initial value problems for systems of linear first-order PDEs |
| 30H25 | Besov spaces and \(Q_p\)-spaces |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 35A02 | Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness |
| 49K40 | Sensitivity, stability, well-posedness |
| 35B44 | Blow-up in context of PDEs |
| 35L45 | Initial value problems for first-order hyperbolic systems |
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