期刊
SIAM JOURNAL ON MATHEMATICAL ANALYSIS
卷 48, 期 3, 页码 1929-1953出版社
SIAM PUBLICATIONS
DOI: 10.1137/15M1031655
关键词
measure-valued equations; nonlinearities; time discretization; flux boundary condition; mild solutions; particle systems
资金
- Netherlands Organisation for Scientific Research (NWO), Graduate Programme
In this paper we prove well-posedness for a measure-valued continuity equation with solution-dependent velocity and flux boundary conditions, posed on a bounded one-dimensional domain. We generalize the results of an earlier paper [J. Differential Equations, 259 (2015), pp. 10681097] to settings where the dynamics are driven by interactions. In a forward-Euler-like approach, we construct a time-discretized version of the original problem and employ those results as a building block within each subinterval. A limit solution is obtained as the mesh size of the time discretization goes to zero. Moreover, the limit is independent of the specific way of partitioning the time interval [0, T]. This paper is partially based on results presented in Chapter 5 of [Evolution Equations for Systems Governed by Social Interactions, Ph.D. thesis, Eindhoven University of Technology, 2015], while a number of issues that were still open there are now resolved.
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