Zero width limit of the heat equation on moving thin domains
Tatsu‐Hiko Miura · Interfaces and Free Boundaries Mathematical Analysis Computation and Applications · 2017
We study the behavior of a variational solution to the Neumann type problem of the heat equation on a moving thin domain \Omega_{\varepsilon}(t) that converges to an evolving surface \Gamma (t) as the width of \Omega_\varepsilon(t) goes to zero. We show that, under suitable assumptions, the average in the normal direction of \Gamma(t) of a variational solution to the heat equation converges weakly in a function space on \Gamma(t) as the width of \Omega_\varepsilon(t) goes to zero, and that the limit is a unique variational solution to a limit equation on \Gamma(t) , which is a new type of linear diffusion equation involving the mean curvature and the normal velocity of \Gamma(t) . We also estimate the difference between variational solutions to the heat equation on \Omega_\varepsilon(t) and the limit equation on \Gamma(t) .