Behaviour of the velocity of one-dimensional flows in porous media

Abstract:

The one-dimensional flow of gas of density $u$ through a porous medium obeys the equation ${u_t} = {({u^m})_{xx}}$, where $m > 1,x \in {\mathbf {R}}$ and $t > 0$. We prove that the local velocity of the gas, given by $\upsilon = - m{u^{m - 2}}{u_x}$, not only is bounded for $t \geqslant \tau > 0$ but approaches an $N$-wave profile as $t \to \infty$. $N$-waves are the typical asymptotic profiles for some first-order conservation laws, a class of nonlinear hyperbolic equations. The case $m \leqslant 1$ is also studied: there are solutions with unbounded velocity while others have bounded velocity.