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The geometry of dissipative evolution equations: The porous medium equation. (English) Zbl 0984.35089

The so-called porous medium equation (PME) \[ \frac{\partial \rho}{\partial t}- \Delta ^2 \rho^m = 0,\quad x\in\mathbb{R}^{N},\;t\in[0,\infty) \] models a fluid flow through a porous medium. An equation of parabolic type, in the case \( m > 1\) it is associated to small diffusion, while fast diffusion corresponds to \(m < 1\). Since in the latter situation no classical solution exists for nonzero initial data with compact support, weak solutions – with respect to both variables \(x,t\) – ought to be considered. The constraints \(m\geq 1 - \frac{1}{N}\) and \( m > \frac{N}{N+2}\) are assumed throughout this work.
This equation exhibits conservation properties, namely: if \(\rho\) is nonnegative at the initial state, it remains so; besides, the total mass \(\int \rho\) is constant on time. It is shown that PME presents a gradient flow structure which is richer than the classical, and more well known, structure. Such a structure is obtained through a change in the metric definition for the manifold of possible solutions.
This alternate approach, supported by physical and mathematical properties, allows a more clear way to deduce the asymptotical behavior of the solutions.

MSC:

35K65 Degenerate parabolic equations
76S05 Flows in porous media; filtration; seepage
35Q35 PDEs in connection with fluid mechanics
35K55 Nonlinear parabolic equations
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