Porous Medium Equation Notes - Adam Oberman Math

Porous Medium Equation Notes

Contents

The porous medium equation

Introduction

  • The porous medium equation (PME)
    • A nonlinear heat equation:
\partial_t u = \triangle(u^m), \quad m>1.
  • The signed PME
    • The antisymmetric extension of the nonlinearity
    • For brevity, often write um instead of | u | m − 1u
\partial_t u = \triangle (|u|^{m-1} u).
  • The heat equation (HE)
    • When m = 1, the PME is the HE
\partial_t u = \triangle u.
  • The fast diffusion equation (FDE)
    • When m < 1, the FDE has to be written in the "modified form"
\partial_t u = \triangle (u^m/m) = \text{div} (u^{m-1} \nabla u).

PME as a nonlinear parabolic equation

Demonstration of the change of character of the PME at the level u = 0

  • m = 2, \quad u \geq 0
    • \partial_t u = 2u \triangle u + 2|\nabla u|^2
    • In the region where u \neq 0, the leading term in the right-hand side is the Laplacian modified by the variable coefficient 2u
    • for u\rightarrow 0, the equation simplifies into the eikonal equation, (a first-order equation of Hamilton--Jacobi type, that propagates along characteristics)
\partial_t u \sim 2|\nabla u|^2
  • General m \neq 1
    • Pressure variable: v = cu^{m-1}, \quad c \geq 0
    • \partial_t v = av \triangle v + b|\nabla v|^2, \quad \text{with} \quad a = m/c, \quad b = m/(c(m-1)).
    • This is a fundamental transformation in the theory of the PME that allows us to get similar conclusions about the behavior of the equation for u, v \sim 0 \quad \text{when} \quad m \neq 2.
    • The standard choice for c in the literature is c = m / (m − 1), because it simplifies the formulas (a = m − 1,b = 1) and makes sense for dynamical considerations, but c = 1 is also used.

Special solutions

  • A fundamental example of solution was obtained around 1950 in Moscow by Zel'dovich and Kompaneets and Barenblatt, who found and analysed a solution representing heat release from a point source.
  • Explicit formula
\mathcal{U}(x,t) = t^{-\alpha} (C - k|x|^2 t^{-2\beta})_{+}^{\frac{1}{m-1}}
where (s) + = max{s,0},
\alpha = \frac{d}{d(m-1) + 2}, \quad \beta = \frac{\alpha}{d}, \quad k = \frac{\alpha(m-1)}{2md} .
and C > 0 is an arbitrary constant.
  • The name source-type solution is due to the fact that it takes as initial data a Dirac mass: as t \rightarrow 0 we have \mathcal{U}(x,t) \rightarrow M \delta (x), where M is a function of the free constant C (and m and d).
  • We will use the shorter term source solution, and very often the name ZKB solution.
  • Recall that the solution was subsequently found by Pattle in 1959. The names Barenblatt solution and Barenblatt--Pattle solution are found in the literature.
  • The source solution has compact support in space for every fixed time, since the free boundary is the surface given by the equation
t = c | x | d(m − 1) + 2,
where c = c(C,m,d).
In physical terms, the disturbance propagates with a precise finite speed. This is to be compared with the properties of the Gaussian kernel,
E(x,t) = M(4πt) − d / 2exp( − x2 / 4t),
which is the source solution for the HE.

Nonlinear diffusion. Related equations

A quite general form of nonlinear diffusion equation, as it appears in the specialized literature, is

\partial_t H(x,t,u) = \sum_{i=1}^{d} \partial_{x_i}(A_i(x,t,u,Du)).

FAST DIFFUSION

  • In the range of exponents m < 1
  • Since the diffusion coefficient D(u) = | u | m − 1 goes now to infinity as u \rightarrow 0, the equation is called in this new range the fast diffusion equation, FDE.
  • In this terminology, the PME becomes a slow diffusion equation.

FILTRATION EQUATIONS

  • The generalized porous medium equation, GPME,
\partial_t u = \triangle \Phi(u) + f.
Also called the filtration equation.
  • The Stefan problem
    • An important role in the development of the topic of the filtration equation has been played by the Stefan problem, a simple but powerful model of phase transition developed in the study of the evolution of a medium composed of water and ice.
    • It can written as a filtration equation with \Phi(u) = (u-1)_{+} \quad \text{for} \quad u \geq 0, \quad \Phi(u) = u \quad \text{for} \quad u<0.
    • More generally, we can put Φ(u) = c1(u − L) + for u \geq 0, and Φ(u) = c2u for u < 0, where c1,c2 and L are positive constants.
    • The Stefan problem and the PME have had a somewhat parallel history.

p-LAPLACIAN EVOLUTIONS

  • Another popular nonlinear degenerate parabolic equation
\partial_t u = \text{div} (|\nabla u|^{p-2} \nabla u)
  • Called the p-Laplacian evolution equation, PLE.

References

  1. Juan Luis Vazquez. The Porous Medium Equation. Oxford University Press, 2006. [1]
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