Ian Tice

Linear Rayleigh–Taylor Instability for Viscous, Compressible Fluids

We study the equations obtained from linearizing the compressible Navier–Stokes equations around a steady-state profile with a heavier fluid lying above a lighter fluid along a planar interface, i.e., a Rayleigh–Taylor instability. We consider the equations with or without surface tension, with the viscosity allowed to depend on the density, in both periodic and nonperiodic settings. In the presence of viscosity there is no natural variational framework for constructing growing mode solutions to the linearized problem. We develop a general method of studying a family of modified variational problems in order to produce maximal growing modes. Using these growing modes, we construct smooth (when restricted to each fluid domain) solutions to the linear equations that grow exponentially in time in Sobolev spaces. We then prove an estimate for arbitrary solutions to the linearized equations in terms of the fastest possible growth rate for the growing modes. In the periodic setting, we show that sufficiently sm...

Instability theory of the Navier-Stokes-Poisson equations

The stability question of the Lane‐Emden stationary gaseous star configurations is an interesting problem arising in astrophysics. We establish both linear and nonlinear dynamical instability results for the Lane‐Emden solutions in the framework of the Navier‐Stokes‐Poisson system with adiabatic exponent 6 5 < < 4 3 .

The Viscous Surface-Internal Wave Problem: Nonlinear Rayleigh–Taylor Instability

We consider the free boundary problem for two layers of immiscible, viscous, incompressible fluid in a uniform gravitational field, lying above a rigid bottom in a three-dimensional horizontally periodic setting. The effect of surface tension is either taken into account at both free boundaries or neglected at both. We are concerned with the Rayleigh–Taylor instability, so we assume that the upper fluid is heavier than the lower fluid. When the surface tension at the free internal interface is below a critical value, which we identify, we establish that the problem under consideration is nonlinearly unstable.

The Viscous Surface-Internal Wave Problem: Global Well-Posedness and Decay

We consider the free boundary problem for two layers of immiscible, viscous, incompressible fluid in a uniform gravitational field, lying above a general rigid bottom in a three-dimensional horizontally periodic setting. We establish the global well-posedness of the problem both with and without surface tension. We prove that without surface tension the solution decays to the equilibrium state at an almost exponential rate; with surface tension, we show that the solution decays at an exponential rate. Our results include the case in which a heavier fluid lies above a lighter one, provided that the surface tension at the free internal interface is above a critical value, which we identify. This means that sufficiently large surface tension stabilizes the Rayleigh–Taylor instability in the nonlinear setting. As a part of our analysis, we establish elliptic estimates for the two-phase stationary Stokes problem.

Ginzburg–Landau Vortex Dynamics with Pinning and Strong Applied Currents

We study a mixed heat and Schrödinger Ginzburg–Landau evolution equation on a bounded two-dimensional domain with an electric current applied on the boundary and a pinning potential term. This is meant to model a superconductor subjected to an applied electric current and electromagnetic field and containing impurities. Such a current is expected to set the vortices in motion, while the pinning term drives them toward minima of the pinning potential and “pins” them there. We derive the limiting dynamics of a finite number of vortices in the limit of a large Ginzburg–Landau parameter, or

The Compressible Viscous Surface-Internal Wave Problem: Nonlinear Rayleigh–Taylor Instability

{\varepsilon \to 0}

Stability of Contact Lines in Fluids: 2D Stokes Flow

, when the intensity of the electric current and applied magnetic field on the boundary scale like