On a nonhomogeneous system of pressureless flow

In this paper, a nonhomogeneous system of pressureless flow \[ {\rho _t} + {\left ( \rho u \right )_x} = 0, \qquad {\left ( \rho u \right )_t} + {\left ( \rho {u^2} \right )_x} = \rho x\] 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., \[ \rho \left ( x, t \right ) = - \frac {\partial }{{\partial {x^2}}}\min \limits _y F\left ( y; x, t \right ), \qquad \rho \left ( x, t \right )u\left ( x, t \right ) = \frac {{{\partial ^2}}}{{\partial x\partial t}}\min \limits _y F\left ( y; x, t \right )\] 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].