Seiro Omata

Convergence to diffusion waves of the solutions for benjamin-bona-mahony-burgers equations

This paper is concerned with the large-time behavior of solution of the Cauchy problem for the Benjamin-Bona-Mahony-Burgers equation. We prow that the solution unique globally exists and time-asymptotically tends to its corresponding diffusion wave, when the initial perturbation is small enough. The corresponding diffusion wave is constructed by the heat equation or the Burgers equation. In particular, we obtain the convergence rates in Lq-spaces (2≤q≤∞). The mathematical proof is based on the Fourier transform method and the energy method.Furthermore, we take the numerical computations on such a problem. The numerical simulations show that the convergence rates obtained theoretically seem to be sharp

A variational method for multiphase volume-preserving interface motions

We develop a numerical method for realizing mean curvature motion of interfaces separating multiple phases, whose volumes are preserved throughout time. The foundation of the method is a thresholding algorithm of the Bence-Merriman-Osher type. The original algorithm is reformulated in a vector setting, which allows for a natural inclusion of constraints, even in the multiphase case. Moreover, a new method for overcoming the inaccuracy of thresholding methods on non-adaptive grids is designed, since this inaccuracy becomes especially prominent in volume-preserving motions. Formal analysis of the method and numerical tests are presented.

A numerical approach to the asymptotic behavior of solutions of a one-dimensional free boundary problem of hyperbolic type

A free boundary problem which arises from the physical model called “peeling” will be analyzed numerically. To obtain an equation which describes the motion of the free boundary, the fixed domain method is applied. By using the equation, numerical computations are carried out. Our numerical computations suggest that the peeling speed plays an important role in the existence of global solutions.

Mathematical model for self-propelled droplets driven by interfacial tension

We propose a model for the spontaneous motion of a droplet induced by inhomogeneity in interfacial tension. The model is derived from a variation of the Lagrangian of the system and we use a time-discretized Morse flow scheme to perform its numerical simulations. Our model can naturally simulate the dynamics of a single droplet, as well as that of multiple droplets, where the volume of each droplet is conserved. We reproduced the ballistic motion and fission of a droplet, and the collision of two droplets was also examined numerically.

A Hyperbolic Obstacle Problem with an Adhesion Force

We will treat free boundary problems of a wave type in this section. Examples of the physical phenomena that we have in mind are a motion of a soap film attached to a water surface or a droplet motion on a planner surface. The surface acts as an obstacle and there may exist adhesion forces when the film or the droplet detach from the obstacle. We consider the case with a positive contact angle in an equilibrium state. We also calculate the moving contact angle according to a dynamical action functional.