MEASURE-VALUED MASS EVOLUTION PROBLEMS WITH FLUX BOUNDARY CONDITIONS AND SOLUTION-DEPENDENT VELOCITIES

作者：Evers Joep H M^*; Hille Sander C; Muntean Adrian

来源：SIAM Journal on Mathematical Analysis, 2016, 48(3): 1929-1953.

DOI：10.1137/15M1031655

• measure-valued equations
• nonlinearities
• time discretization
• flux boundary condition
• mild solutions
• particle systems

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.

• 出版日期2016