The Geometry of Dissipative Evolution Equations: The Porous Medium Equation

We show that the porous medium equation has a gradient flow structure which is both physically and mathematically natural. In order to convince the reader that it is mathematically natural, we show the time asymptotic behavior can be easily understood in this framework. We use the intuition and the calculus of Riemannian geometry to quantify this asymptotic behavior. Contents 1 The porous medium equation as a gradient flow 2 1.1 The porous medium equation . . . . . . . . . . . . . . . . . . 2 1.2 Abstract gradient flow . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Two interpretations of the porous medium equation as gradient flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2 A physical argument in favor of the new gradient flow 6 3 A mathematical argument in favor of new gradient flow 9 3.1 Self similar solutions and asymptotic behaviour . . . . . . . . 9 3.2 A new asymptotic result . . . . . . . . . . . . . . . . . . . . . 10 3.3 The asymptotic result express...