associate professor
karel (please add @math.kyoto-u.ac.jp)
Research Area
Applied Mathematics, Partial Differential Equations, Numerical analysis

My research concerns investigating properties of solutions to partial differential equations that describe motion of interfaces such as soap films in soap bubbles or grain boundaries in crystals, and developing mathematical models and numerical methods for reproducing such phenomena in computer simulations.

For instance, a soap film tries to minimize its surface area, settling in the shape of a so-called minimal surface. If the soap bubble does not have this shape, it deforms so as to decrease its surface energy as fast as possible. This deformation is known to produce the interfacial motion called "mean curvature flow", where each point on the soap film moves with normal velocity proportional to the curvature of the film at that point. Namely,
$$ \boldsymbol{v}= - \gamma \kappa \boldsymbol{n} . $$

Mathematically interesting things happen when the soap film pinches off, or when two soap bubbles merge or attach to each other. In such cases, the topology changes and singularities appear. In order to deal with such singularities, a mathematical approach called "level set method" has been developed. The interface is expressed as a level set of a function $u$ and in the case of mean curvature flow the corresponding evolution for this functions becomes
$$ \frac{\partial u}{\partial t}-| \nabla u| \text{div} \left( \frac{\nabla u}{| \nabla u|} \right) = 0, $$
for which it is possible to construct a weak solution that can include singularities.

In my research, I am interested in the analysis of motions of several attached bubbles or bubble motions including oscillations due to inertia. Moreover, I am also trying to analyze motions with apriori unknown free boundaries, which are the sets of points where the soap film meets the soapy liquid (for example, when a soap bubble floats on soap water surface or when a soap film is pulled up from a soapy solution using a frame).

I am working not only on mathematical analysis but also on developing suitable mathematical models for such interfacial phenomena and effective numerical schemes for solving nonlinear equations thus obtained.