 Publication Date:
 NSFPAR ID:
 10231281
 Journal Name:
 Stochastic Systems
 Volume:
 10
 Issue:
 3
 Page Range or eLocationID:
 251 to 273
 ISSN:
 19465238
 Sponsoring Org:
 National Science Foundation
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Droplet formation happens in finite time due to the surface tension force. The linear stability analysis is useful to estimate the size of a droplet but fails to approximate the shape of the droplet. This is due to a highly nonlinear flow description near the point where the first pinchoff happens. A onedimensional axisymmetric mathematical model was first developed by Eggers and Dupont [“Drop formation in a onedimensional approximation of the Navier–Stokes equation,” J. Fluid Mech. 262, 205–221 (1994)] using asymptotic analysis. This asymptotic approach to the Navier–Stokes equations leads to a universal scaling explaining the selfsimilar nature of the solution. Numerical models for the onedimensional model were developed using the finite difference [Eggers and Dupont, “Drop formation in a onedimensional approximation of the Navier–Stokes equation,” J. Fluid Mech. 262, 205–221 (1994)] and finite element method [Ambravaneswaran et al., “Drop formation from a capillary tube: Comparison of onedimensional and twodimensional analyses and occurrence of satellite drops,” Phys. Fluids 14, 2606–2621 (2002)]. The focus of this study is to provide a robust computational model for onedimensional axisymmetric droplet formation using the Portable, Extensible Toolkit for Scientific Computation. The code is verified using the Method of Manufactured Solutions and validated using previousmore »

We consider the wellknown LiebLiniger (LL) model for
bosons interacting pairwise on the line via the\begin{document}$ N $\end{document} potential in the meanfield scaling regime. Assuming suitable asymptotic factorization of the initial wave functions and convergence of the microscopic energy per particle, we show that the timedependent reduced density matrices of the system converge in trace norm to the pure states given by the solution to the onedimensional cubic nonlinear Schrödinger equation (NLS) with an explict rate of convergence. In contrast to previous work [\begin{document}$ \delta $\end{document} 3 ] relying on the formalism of second quantization and coherent states and without an explicit rate, our proof is based on the counting method of Pickl [65 ,66 ,67 ] and Knowles and Pickl [44 ]. To overcome difficulties stemming from the singularity of the potential, we introduce a new shortrange approximation argument that exploits the Hölder continuity of the\begin{document}$ \delta $\end{document} body wave function in a single particle variable. By further exploiting the\begin{document}$ N $\end{document} subcritical wellposedness theory for the 1D cubic NLS, we can prove meanfield convergence when the limiting solution to the NLS has finitemore »\begin{document}$ L^2 $\end{document} 
The YBJ equation (Young & Ben Jelloul, J. Marine Res. , vol. 55, 1997, pp. 735–766) provides a phaseaveraged description of the propagation of nearinertial waves (NIWs) through a geostrophic flow. YBJ is obtained via an asymptotic expansion based on the limit $\mathit{Bu}\rightarrow 0$ , where $\mathit{Bu}$ is the Burger number of the NIWs. Here we develop an improved version, the YBJ + equation. In common with an earlier improvement proposed by Thomas, Smith & Bühler ( J. Fluid Mech. , vol. 817, 2017, pp. 406–438), YBJ + has a dispersion relation that is secondorder accurate in $\mathit{Bu}$ . (YBJ is firstorder accurate.) Thus both improvements have the same formal justification. But the dispersion relation of YBJ + is a Padé approximant to the exact dispersion relation and with $\mathit{Bu}$ of order unity this is significantly more accurate than the powerseries approximation of Thomas et al. (2017). Moreover, in the limit of high horizontal wavenumber $k\rightarrow \infty$ , the wave frequency of YBJ + asymptotes to twice the inertial frequency $2f$ . This enables solution of YBJ + with explicit timestepping schemes using a time step determined by stable integration of oscillations with frequency $2f$ . Other phaseaveraged equations have dispersion relations with frequency increasing as $k^{2}$more »

We study the focusing NLS equation in $R\mathbb{R}^N$ in the masssupercritical and energysubcritical (or intercritical ) regime, with $H^1$ data at the massenergy threshold $\mathcal{ME}(u_0)=\mathcal{ME}(Q)$, where Q is the ground state. Previously, Duyckaerts–Merle studied the behavior of threshold solutions in the $H^1$critical case, in dimensions $N = 3, 4, 5$, later generalized by Li–Zhang for higher dimensions. In the intercritical case, Duyckaerts–Roudenko studied the threshold problem for the 3d cubic NLS equation. In this paper, we generalize the results of Duyckaerts–Roudenko for any dimension and any power of the nonlinearity for the entire intercritical range. We show the existence of special solutions, $Q^\pm$, besides the standing wave $e^{it}Q$, which exponentially approach the standing wave in the positive time direction, but differ in its behavior for negative time. We classify solutions at the threshold level, showing either blowup occurs in finite (positive and negative) time, or scattering in both time directions, or the solution is equal to one of the three special solutions above, up to symmetries. Our proof extends to the $H^1$critical case, thus, giving an alternative proof of the Li–Zhang result and unifying the critical and intercritical cases. These results are obtained by studying the linearized equation around themore »

SUMMARY Tsunami generation by offshore earthquakes is a problem of scientific interest and practical relevance, and one that requires numerical modelling for data interpretation and hazard assessment. Most numerical models utilize twostep methods with oneway coupling between separate earthquake and tsunami models, based on approximations that might limit the applicability and accuracy of the resulting solution. In particular, standard methods focus exclusively on tsunami wave modelling, neglecting larger amplitude ocean acoustic and seismic waves that are superimposed on tsunami waves in the source region. In this study, we compare four earthquaketsunami modelling methods. We identify dimensionless parameters to quantitatively approximate dominant wave modes in the earthquaketsunami source region, highlighting how the method assumptions affect the results and discuss which methods are appropriate for various applications such as interpretation of data from offshore instruments in the source region. Most methods couple a 3D solid earth model, which provides the seismic wavefield or at least the static elastic displacements, with a 2D depthaveraged shallow water tsunami model. Assuming the ocean is incompressible and tsunami propagation is negligible over the earthquake duration leads to the instantaneous source method, which equates the static earthquake seafloor uplift with the initial tsunami sea surface height. Formore »