On the Possibility of Kinetic Flow through Fractal Holes in a Surface

The paper considers initial boundary value problems for the non-linear Boltzmann equation with large L 1 initial data and for the lin-earised Bolzmann equation with L 2 initial data. Particular problems under analysis describe ows in a domain consisting of two subdomains connected through a joint part ? of the boundary which can have a complicated fractal geometry. The presence of interaction between these two parts of such a ow and the possibility to reach a uniform stationary state when time goes to innnity depends on initial data and on the Hausdorr dimension of the contact set ?. Geometrical conditions for the existence of such a global stabilisation for the nonlinear Boltzmann equation and for the linearised Boltzmann equation are studied and found to be diierent. Renormalised L 1 solutions can in general penetrate ? only if its 2-dimensional Hausdorr measure is positive. Solutions to the linearised Boltzmann equation in L 2 can go through much thinner "holes" that have Hausdorr dimension larger than 1.