A one-dimensional model of isentropic expansion of a compressible gas is the following. The gas initially occupies two disjoint intervals, \(a_ 0\leq x\leq b_ 0\) and \(c_ 0\leq x\leq d_ 0\) \((b_ 0<c_ 0)\), of the real \(x\)-axis, and the intermediate interval \(b_ 0<x<c_ 0\) is instead filled by a gas with very small density \(\varphi^ \delta\). Since the density and the velocity gradients are initially discontinuous at \(b_ 0\), and \(c_ 0\), one may expect that these discontinuities are preserved with time.
However, the solutions are more regular than what is predictable. In fact, if the instantaneous positions of the separation points are denoted by \(a^ \delta(t)\), \(b^ \delta(t)\), \(c^ \delta(t)\), \(d^ \delta(t)\), it is possible to prove that, for \(\delta>0\), there is a unique global weak solution, and the trajectories described by the points \(a^ \delta(t),\dots,d^ \delta(t)\) are uniquely defined; if instead \(\varrho^ \delta\) tends to zero there are still limiting trajectories, but their existence is restricted to bounded intervals of time. In addition, when \(\varrho^ \delta\) tends to zero, it may occur that, after a finite time, \(b^ \delta(t)\) coincides with \(c^ \delta(t)\), that is the two components collide and their individual momenta are not conserved.