Singularity formation in three-dimensional vortex sheets

[摘要] In this thesis, we investigate both theoretically and numerically the singularity formation and long time existence of three-dimensional vortex sheets.For the theoretical work, we divide it into two parts. In the first part, we study the early time singularity formation and the local form of the vortex sheet in the neighborhood of a singularity near the singularity time. We show that under a special set of coordinates, the three-dimensional vortex sheet can be viewed as a two-dimensional vortex sheet along certain space curves. As a result, the study of singularity formation of a three-dimensional vortex sheet can be related to that of the corresponding two-dimensional vortex sheet. And the singular behavior of these two problems is very similar. Moreover, by performing a transformation in the interface variables and deriving leading order asymptotic approximations for the evolution of these transformed variables, we show that the Kelvin-Helmholtz instability is a result of the coupling of two of these three variables to the leading order. This observation simplifies significantly our singularity analysis for three-dimensional vortex sheets and allows us to reveal clearly the nature of the curvature singularity in the three-dimensional vortex sheet equation. In the second part of our theoretical work, we prove the long time existence of the three-dimensional vortex sheet problem for analytic initial conditions near equilibrium. Moreover, the existence time is almost optimal if the initial perturbation over the equilibrium is sufficiently small.We have performed careful numerical study to validate our theoretical results. Well-resolved numerical study of the three-dimensional vortex sheet equation is difficult due to the complexity in evaluating the interface velocity. To alleviate this difficulty, we introduce two model equations. An important feature of these models equations is that they can be expressed in terms of convolution operators and consequently they can be computed efficiently by Fast Fourier Transform. Moreover, we show by asymptotic analysis that these model equations preserve the singularity type of the full equations. Our analysis also suggests that the model equations generate the same local form of curvature singularity near the physical singularity time as that of the full equations. Our detailed numerical computations on the two-dimensional problem show that the model equation captures all the essential singularity behavior of the full vortex sheet equation. Our calculations based on the three-dimensional model equation provide convincing evidences that a curvature singularity develops in finite time in the three-dimensional vortex sheet. And the type of the singularity is of order -1/2 in the mean curvature.

[发布机构] University:California Institute of Technology;Department:Engineering and Applied Science

[关键词] Applied Mathematics