Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



On a nonlocal dispersive equation modeling particle suspensions

Author: Kevin Zumbrun
Journal: Quart. Appl. Math. 57 (1999), 573-600
MSC: Primary 35L65; Secondary 45K05, 76T99
DOI: https://doi.org/10.1090/qam/1704419
MathSciNet review: MR1704419
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Abstract: We study a nonlocal, scalar conservation law $ {u_t} + {\left( \left( {K_a} * u \right)u \right)_x} = 0$, modeling sedimentation of particles in a dilute fluid suspension, where $ {K_a}\left( x \right) = {a^{ - 1}}K\left( x/a \right)$ is a symmetric smoothing kernel, and $ \ast$ represents convolution. We show this to be a dispersive regularization of the Hopf equation, $ {u_t} + {\left( {u^2} \right)_x} = 0$, analogous to KdV and certain dispersive difference schemes. Using the smoothing property of convolution and the physical principle of conservation of mass, we establish the global existence of smooth solutions.

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DOI: https://doi.org/10.1090/qam/1704419
Article copyright: © Copyright 1999 American Mathematical Society

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