Nonlinear semigroups for nonlocal conservation laws. (English) Zbl 1521.35083
Summary: We investigate a class of nonlocal conservation laws in several space dimensions, where the continuum average of weighted nonlocal interactions are considered over a finite horizon. We establish well-posedness for a broad class of flux functions and initial data via semigroup theory in Banach spaces and, in particular, via the celebrated Crandall-Liggett Theorem. We also show that the unique mild solution satisfies a Kružkov-type nonlocal entropy inequality. Similarly to the local case, we demonstrate an efficient way of proving various desirable qualitative properties of the unique solution.
MSC:
| 35F25 | Initial value problems for nonlinear first-order PDEs |
| 35Q49 | Transport equations |
| 45K05 | Integro-partial differential equations |
| 47H20 | Semigroups of nonlinear operators |
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