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Periodic solutions for indefinite singular equations with singularities in the spatial variable and non-monotone nonlinearity
October  2018, 38(10): 4837-4873. doi: 10.3934/dcds.2018212

## On continuity equations in space-time domains

 Department of Mathematics, University of California, Los Angeles, Los Angeles, CA 90025, USA

* Corresponding author: Yuming Zhang

Received  May 2017 Revised  May 2018 Published  July 2018

In this paper we consider a class of continuity equations that are conditioned to stay in general space-time domains, which is formulated as a continuum limit of interacting particle systems. Firstly, we study the well-posedness of the solutions and provide examples illustrating that the stability of solutions is strongly related to the decay of initial data at infinity. In the second part, we consider the vanishing viscosity approximation of the system, given with the co-normal boundary data. If the domain is spatially convex, the limit coincides with the solution of our original system, giving another interpretation to the equation.

Citation: Yuming Zhang. On continuity equations in space-time domains. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 4837-4873. doi: 10.3934/dcds.2018212
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##### References:
the particle system and the Px,t operator
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