Section: New Results

Microscopic description of moving interfaces

A large variety of models has been introduced to describe the evolution of a multiphase medium, e.g. the joint evolution of liquid and solid phases. These complex physical phenomena often feature absorbing phase transitions. For instance, the porous medium equation (PME)

where $m>1$ is a constant and $\text{div}$ and $\nabla$ are the divergence and gradient operators in ${ℝ}^d$, describes the evolution of the density $\rho :{ℝ}^d\times {ℝ}_{+}\to [0,1]$ of an ideal gas flowing in a homogeneous medium. It is known that, starting from an initial density ${\rho }_0$ with compact support, the solution $\rho (x,t)$ is nonnegative and has compact support in the space variable for each positive $t$. Thus there are interfaces separating the regions where $\rho$ is positive from those where it is zero.

In one submitted paper in collaboration with O. Blondel, C. Cancès, and M. Sasada, we have derived the PME (1) from a degenerate and conservative dynamics in [15], for any integer $m>1$. More precisely we improved the results previously obtained in [26], since we allow the solutions to feature moving interfaces, namely the initial condition may vanish. This moving boundary was not well apprehended at the microscopic level. Its rigorous definition is indeed very delicate, and its behavior (such that its speed, or fluctuation), as well as the relationship between the microscopic and macroscopic boundaries, are challenging questions that we aim to tackle in a near future.

When $m<1$, equation (1) is called fast diffusion equation. In a recent collaborative work (submitted) with O. Blondel, C. Erignoux and M. Sasada [16], we derive such a fast diffusion equation in dimension one from an interacting particle system belonging to the class of conserved lattice gases with active-absorbing phase transition [30]. The microscopic dynamics is very constrained: in a few words, a particle can jump to the right (resp. left) empty neighboring site if and only if it has a particle to its left (resp. right) neighboring site. This model is really complex: the state space is divided into transient states, absorbing states and ergodic states. Depending on the initial number of particles, the transient good configurations will lead to the ergodic component and the transient bad configurations will be absorbed to an inactive state. Because of the jump constraint, there are two distinct regimes for the macroscopic behavior. Either the macroscopic density is larger than $\frac{1}{2}$, in which case the system behaves diffusively, or the density is lower than $\frac{1}{2}$, in which case the system freezes rapidly.

The interfaces between these two phases propagate as particles from the supercritical phase ($\rho >\frac{1}{2}$) diffuse towards the subcritical phase ($\rho <\frac{1}{2}$). We expect that the macroscopic density profile evolves under the diffusive scaling according to the Stefan problem

The microscopic derivation of such Stefan problems is a well known difficult problem, only partially solved [27], [29]. In [16] we treat the liquid part of the problem (i.e. when the initial profiles ${\rho }_0$ are uniformly larger than the critical density $\frac{1}{2}$) and we provide a refined estimation of the time needed by the system to enter into the ergodic state. Then, we show that the macroscopic density profile evolves under the diffusive time scaling according to (1) with $m=-1$. The extension to more general initial profiles is our next goal.