2015/07/21 by Joep H. M. Evers, Evers, Joep H. M., Sander C. Hille +3
Engineering · Mathematics · #28A33 #34A12 #35F16 #45D05 #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions #Nonlinear Partial Differential Equations #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.1507.05730
openalex publication_date 2015/07/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we prove well-posedness for a measure-valued continuity\nequation with solution-dependent velocity and flux boundary conditions, posed\non a bounded one-dimensional domain. We generalize the results of [Evers, Hille\nand Muntean. Journal of Differential Equations, 259:1068-1097, 2015] to\nsettings where the dynamics are driven by interactions. In a forward-Euler-like\napproach, we construct a time-discretized version of the original problem and\nemploy the results of [Evers, Hille and Muntean. Journal of Differential\nEquations, 259:1068-1097, 2015] as a building block within each subinterval. A\nlimit solution is obtained as the mesh size of the time discretization goes to\nzero. Moreover, the limit is independent of the specific way of partitioning\nthe time interval [0,T]. This paper is partially based on results presented\nin Chapter 5 of [Evers. PhD thesis, Eindhoven University of Technology, 2015],\nwhile a number of issues that were still open there, are now resolved.\n