On the existence of a global weak solution for the problem of sedimentation with compression (Q910582)

Two conservation laws that describe the sedimentation under gravity of solid particles in a fluid are given by \[ \partial \phi /\partial t+\partial f/\partial z=0,\quad \partial f/\partial z+\partial (f^ 2\phi^{-1}+p(\phi))/\partial z=0 \] in \(\Omega =\{(z,t):\) \(0<z\), \(0<t\}\), where the pressure p is of the form \(p(\phi)=a\cdot \exp (b\phi)\), b is a positive number and the following initial and boundary conditions \(\phi(z,0)=\phi_ 0(z)\), \(f(z,0)=f_ 0(z)\), \(f(0,t)=f_ 1(t)\) are supposed. The author's main result asserts that for bounded initial and boundary data \(\phi_ 0(z)\), \(f_ 0(z)\), \(f_ 1(t)\) having bounded total variation and for sufficiently large b, the above initial boundary value problem has a global weak solution which has bounded total variation on each line \(t=t_ 0\), \(t_ 0\) a positive constant.