On continuity equations in space-time domains

Yuming Zhang

doi:10.3934/dcds.2018212

Article Contents

2018, Volume 38, Issue 10: 4837-4873. Doi: 10.3934/dcds.2018212

This issue Previous Article Periodic solutions for indefinite singular equations with singularities in the spatial variable and non-monotone nonlinearity Next Article Emergent dynamics of the Kuramoto ensemble under the effect of inertia

On continuity equations in space-time domains

Yuming Zhang^,

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

^* Corresponding author: Yuming Zhang
^* Corresponding author: Yuming Zhang

Received: May 2017

Revised: May 2018

Published: October 2018

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Abstract

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.

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Mathematics Subject Classification: Primary: 35D30, 45K05; Secondary: 49K20.

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Figure 1. the particle system and the P_x,t operator

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