Unique solvability of the free-boundary Navier-Stokes equations with surface tension

We prove the existence and uniqueness of solutions to the time-dependent incompressible Navier-Stokes equations with a free-boundary governed by surface tension. The solution is found using a topological fixed-point theorem for a nonlinear iteration scheme, requiring at each step, the solution of a model linear problem consisting of the time-dependent Stokes equation with linearized mean-curvature forcing on the boundary. We use energy methods to establish new types of spacetime inequalities that allow us to find a unique weak solution to this problem. We then prove regularity of the weak solution, and establish the a priori estimates required by the nonlinear iteration process.Comment: 73 pages; typos corrected; minor details adde

On the splash singularity for the free-surface of a Navier-Stokes fluid

In fluid dynamics, an interface splash singularity occurs when a locally smooth interface self-intersects in finite time. We prove that for $d$-dimensional flows, $d=2$ or $3$, the free-surface of a viscous water wave, modeled by the incompressible Navier-Stokes equations with moving free-boundary, has a finite-time splash singularity. In particular, we prove that given a sufficiently smooth initial boundary and divergence-free velocity field, the interface will self-intersect in finite time.Comment: 21 pages, 5 figure

Well-posedness of the free-surface incompressible Euler equations with or without surface tension

We provide a new method for treating free boundary problems in perfect fluids, and prove local-in-time well-posedness in Sobolev spaces for the free-surface incompressible 3D Euler equations with or without surface tension for arbitrary initial data, and without any irrotationality assumption on the fluid. This is a free boundary problem for the motion of an incompressible perfect liquid in vacuum, wherein the motion of the fluid interacts with the motion of the free-surface at highest-order.Comment: To appear in J. Amer. Math. Soc., 96 page

On the impossibility of finite-time splash singularities for vortex sheets

In fluid dynamics, an interface splash singularity occurs when a locally smooth interface self-intersects in finite time. By means of elementary arguments, we prove that such a singularity cannot occur in finite time for vortex sheet evolution, i.e. for the two-phase incompressible Euler equations. We prove this by contradiction; we assume that a splash singularity does indeed occur in finite time. Based on this assumption, we find precise blow-up rates for the components of the velocity gradient which, in turn, allow us to characterize the geometry of the evolving interface just prior to self-intersection. The constraints on the geometry then lead to an impossible outcome, showing that our assumption of a finite-time splash singularity was false.Comment: 39 pages, 8 figures, details added to proofs in Sections 5 and

Regularity of the velocity field for Euler vortex patch evolution

We consider the vortex patch problem for both the 2-D and 3-D incompressible Euler equations. In 2-D, we prove that for vortex patches with $H^{k-0.5}$ Sobolev-class contour regularity, $k \ge 4$, the velocity field on both sides of the vortex patch boundary has $H^k$ regularity for all time. In 3-D, we establish existence of solutions to the vortex patch problem on a finite-time interval $[0,T]$, and we simultaneously establish the $H^{k-0.5}$ regularity of the two-dimensional vortex patch boundary, as well as the $H^k$ regularity of the velocity fields on both sides of vortex patch boundary, for $k \ge 3$.Comment: 30 pages, added references and some details to Section

A versatile all-channel stimulator for electrode arrays, with real-time control

Over the last few decades, technology to record through ever increasing numbers of electrodes has become available to electrophysiologists. For the study of distributed neural processing, however, the ability to stimulate through equal numbers of electrodes, and thus to attain bidirectional communication, is of paramount importance. Here, we present a stimulation system for multi-electrode arrays which interfaces with existing commercial recording hardware, and allows stimulation through any electrode in the array, with rapid switching between channels. The system is controlled through real-time Linux, making it extremely flexible: stimulation sequences can be constructed on-the-fly, and arbitrary stimulus waveforms can be used if desired. A key feature of this design is that it can be readily and inexpensively reproduced in other labs, since it interfaces to standard PC parallel ports and uses only off-the-shelf components. Moreover, adaptation for use with in vivo multi-electrode probes would be straightforward. In combination with our freely available data-acquisition software, MeaBench, this system can provide feedback stimulation in response to recorded action potentials within 15 ms

Layout of Multiple Views for Volume Visualization: A User Study

Abstract. Volume visualizations can have drastically different appearances when viewed using a variety of transfer functions. A problem then occurs in trying to organize many different views on one screen. We conducted a user study of four layout techniques for these multiple views. We timed participants as they separated different aspects of volume data for both time-invariant and time-variant data using one of four different layout schemes. The layout technique had no impact on performance when used with time-invariant data. With time-variant data, however, the multiple view layouts all resulted in better times than did a single view interface. Surprisingly, different layout techniques for multiple views resulted in no noticeable difference in user performance. In this paper, we describe our study and present the results, which could be used in the design of future volume visualization software to improve the productivity of the scientists who use it

On the Motion of Vortex Sheets with Surface Tension in the 3D Euler Equations with Vorticity

We prove well-posedness of vortex sheets with surface tension in the 3D incompressible Euler equations with vorticity.Comment: 28 page