Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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



On a nonhomogeneous system of pressureless flow

Authors: Yi Ding and Feimin Huang
Journal: Quart. Appl. Math. 62 (2004), 509-528
MSC: Primary 35L60; Secondary 35D05, 35L65, 76N10
DOI: https://doi.org/10.1090/qam/2086043
MathSciNet review: MR2086043
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Abstract: In this paper, a nonhomogeneous system of pressureless flow

$\displaystyle {\rho _t} + {\left( \rho u \right)_x} = 0, \qquad {\left( \rho u \right)_t} + {\left( \rho {u^2} \right)_x} = \rho x$

is investigated. It is found that there exists a generalized variational principle from which the weak solution is explicitly constructed by using the initial data; i.e.,

$\displaystyle \rho \left( x, t \right) = - \frac{\partial }{{\partial {x^2}}}\m... ...l ^2}}}{{\partial x\partial t}}\mathop {\min }\limits_y F\left( y; x, t \right)$

hold in the sense of distributions, where $ F\left( {y; x, t} \right)$ is a functional depending on the initial data. The weak solution is unique under an Oleinik-type entropy condition when the initial data is of measurable function. It is further shown that the solution $ u\left( x, t \right)$ converges to $ x$ as $ t$ tends to infinity. The proofs are based on the generalized variational principle and careful studies on the generalized characteristics introduced by Dafermos [5].

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DOI: https://doi.org/10.1090/qam/2086043
Article copyright: © Copyright 2004 American Mathematical Society

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