On a nonhomogeneous system of pressureless flow
In this paper, a nonhomogeneous system of pressureless flow Pt + {pu)x = 0, (pu)t + (pu2)x = px 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., d2 <92 p(x,t) = --r-^minF(y;x,t), p(x,t)u{x,t) = minF(y;x,t) ox y oxot y hold in the sense of distributions, where F(y; x, t) 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(x, t) 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],