This talk concerns the Benjamin-Ono (BO) equation of internal wave theory, and properties of the solution of the Cauchy initial-value problem in the situation that the initial data is fixed but the coefficient of the nonlocal dispersive term in the equation is allowed to tend to zero (i.e., the zero-dispersion limit). It is well-known that existence of a limit requires the weak topology because high-frequency oscillations appear even though they are not present in the initial data. Physically, this phenomenon corresponds to the generation of a dispersive shock wave. In the setting of the Korteweg-de Vries (KdV) equation, it has been shown that dispersive shock waves exhibit a universal form independent of initial data near the two edges of the dispersive shock wave, and also near the gradient catastrophe point for the inviscid Burgers equation from which the shock wave forms. In this talk, we will present corresponding universality results for the BO equation. These have quite a different character than in the KdV case; while for KdV one has universal wave profiles expressed in terms of solutions of Painlevé-type equations, for BO one instead has expressions in terms of classical Airy functions and Pearcey integrals. These results are proved for general rational initial data using a new approach based on an explicit formula for the solution of the Cauchy problem for BO. This is joint work with Elliot Blackstone, Louise Gassot, Patrick Gérard, and Matthew Mitchell.
Abstract

We present an accurate and efficient boundary integral (BI) method to simulate the deformation of drops and bubbles in Stokes flow with soluble surfactant. Soluble surfactant advects and diffuses in bulk fluids while adsorbing and desorbing from interfaces. Since the fluid velocity depends on the bulk surfactant concentration C, the advection-diffusion equation governing C is nonlinear, which precludes the Green’s function formulation necessary for a BI method. However, in the physically representative large Peclet number limit an analytical reduction of the surfactant dynamics surprisingly permits a Green’s function formulation. Despite this, existing fast algorithms for similar BI formulations, such as those developed for the heat equation, do not readily apply. To address this challenge, we present a new fast algorithm for our formulation which gives a mesh-free solution to the fully coupled moving interface problem, including soluble surfactant effects. The method extends to other problems involving advection-diffusion in the large Peclet number limit. This is joint work with Michael Booty (NJIT), Samantha Evans (NJIT), and Johannes Tausch (SMU). Zoom: https://tamu.zoom.us/j/94220070032

Viscous multilayer shear flows of immiscible liquids can become unstable due to the presence of interfaces. The nonlinear patterns that emerge are complex and extreme events such as interface pinching and/or wall touching are possible. It is of interest, therefore, to develop reduced dimension and reduced parameter nonlinear models that can be used to study the dynamics and quantify solutions and allied phenomena. I will describe how asymptotic analysis can be used to derive classes of equations that couple thin and thick regions and produce nonlocal PDEs. The physical origins can be free-stream shear, electric or magnetic fields etc. The particular case of two-layer viscous flows in channels will be considered in detail and comparisons of the nonlocal equations with experiments will be shown that confirm their usefulness. Finally I will discuss models that describe multilayer extrusion processes that in many cases support slip at the interface between viscous fluids. Such slip is the viscous analogue of classical inviscid slip that leads to Kelvin-Helmholtz instability. It will be shown that the viscous case also supports short-wave instabilities, albeit not catastrophic. Nonlinear aspects will also be discussed. Seminar will be via Zoom: https://tamu.zoom.us/j/94220070032
Abstract

We present a discontinuity aware quadrature (DAQ) rule, and use it to develop implicit self-adaptive theta (SATh) schemes for the approximation of scalar hyperbolic conservation laws. Our SATh schemes require the solution of a system of two equations, one controlling the cell averages of the solution at the time levels, and the other controlling the space-time averages of the solution. These quantities are used within the DAQ rule to approximate the time integral of the hyperbolic flux function accurately, even when the solution may be discontinuous somewhere over the time interval. We prove that DAQ is accurate to second order when there is a discontinuity in the solution and third order when it is smooth. The resulting scheme is a finite volume theta time stepping method, with theta defined implicitly (or self-adaptively). When upstream weighted, SATh-up is unconditionally stable, satisfies the maximum principle, and is total variation diminishing under appropriate monotonicity and boundary conditions, provided that theta is restricted to be at least 1/2, or even 0 with additional restrictions. We present numerical results showing the performance of the SATh schemes, sometimes using the more general Lax-Friedrichs numerical flux. Compared to solutions of finite volume schemes using Crank-Nicolson and backward Euler time stepping, SATh solutions often approach the accuracy of the former but without oscillation, and they are numerically less diffuse than the later.