Energy estimates and convergence of weak solutions of the porous medium equation

Abstract

We study the convergence of the weak solution of the porous medium equation with a type of Robin boundary conditions, by tuning a parameter either to zero or to infinity. The convergence is in the strong sense, with respect to the L 2-norm, and the limiting function solves the same equation with Neumann (resp. Dirichlet) boundary conditions when the parameter is taken to zero (resp. infinity). Our approach is to consider an underlying microscopic dynamics whose space–time evolution of the density is ruled by the solution of those equations and from this, we derive sufficiently strong energy estimates which are the keystone to the proof of our convergence result.

From Exclusion to Slow and Fast Diffusion

“…where h (k) is given in Lemma 2.16. We highlight that although this gradient property was already known (see [6] for instance), the expression (2.18) for h (k) is new (we give the original expression of h (k) in the appendix, see (A.2)). Then, note that the expectation of c (k) (τ x η) under the invariant measure ν N ρ is…”

Hydrodynamic limits: The emergence of fractional boundary conditions

“…Note that, while above we arrived at the heat equation or the fractional heat equation, it is possible to obtain a nonlinear version of those equations of the form ∂ t 𝜚 = 𝒫𝜚 m , where m ∈ ℕ and 𝒫 = Δ or 𝒫 = −(−Δ) γ/2 , i.e., the porous medium equation and its fractional version. For details, we refer the reader to [13,15,21]. To arrive at these PDEs, one can simply start with an exclusion dynamics where the jump rate depends on the number of particles in the vicinity of the point where particles exchange positions; see [13,15,38].…”