Short-time heat diffusion in compact domains with discontinuous transmission boundary conditions

Abstract

We consider a heat problem with discontinuous diffusion coefficients and discontinuous transmission boundary conditions with a resistance coefficient. For all compact (ǫ, δ) -domains Ω ⊂ R n with a d -set boundary (for instance, a self-similar fractal), we find the first term of the small-time asymptotic expansion of the heat content in the complement of Ω , and also the second-order term in the case of a regular boundary. The asymptotic expansion is different for the cases of finite and infinite resistance of the boundary. The derived formulas relate the heat content to the volume of the interior Minkowski sausage and present a mathematical justification to the de Gennes' approach. The accuracy of the analytical results is illustrated by solving the heat problem on prefractal domains by a finite elements method.

Semigroup-theoretic approach to diffusion in thin layers separated by semi-permeable membranes

“…We refer also to [40], where a compartment model with permeable walls (representing, e.g., cells, and axons in the white matter of the brain in particular) is analyzed, and to equation [42] there. Moreover, we acknowledge references [8,41,71], which we owe to an anonymous Referee, devoted to related models.…”

The Complex Airy Operator on the Line with a Semipermeable Barrier

“…The transmission boundary condition which is considered in this article appears in various exchange problems such as molecular diffusion across semi-permeable membranes [36,33,32], heat transfer between two materials [10,17,7], or transverse magnetization evolution in nuclear magnetic resonance (NMR) experiments [19]. In the simplest setting of the latter case, one considers the local transverse magnetization G(x, y ; t) produced by the nuclei that started from a fixed initial point y and diffused in a constant magnetic field gradient g up to time t. This magnetization is also called the propagator or the Green function of the Bloch-Torrey equation [38]:…”

An averaging principle for fast diffusions in domains separated by semi-permeable membranes

“…A systematic study of semigroups and cosine families related to such transmission conditions has been commenced in [15]. We note also the recent paper [9], where a heat problem for such transmission conditions is studied for quite irregular boundaries, and the monograph [1] in which related transmission conditions are analyzed.…”