arXiv subjects

Ian Tice

Publications and source records attributed to Ian Tice.

46 records · Page 3Linked to original sources

arXiv2011

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.

math.AP↗

arXiv2011

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 $6/5 < γ< 4/3$.

math.AP↗

arXiv2011

Lorentz space estimates for the Coulombian renormalized energy

In this paper we obtain optimal estimates for the "currents" associated to point masses in the plane, in terms of the Coulombian renormalized energy of Sandier-Serfaty \cite{ss1,ss3}. To derive the estimates, we use a technique that we introduced in \cite{st}, which couples the "ball construction method" to estimates in the Lorentz space $L^{2,\infty}$.

math.AP↗

arXiv2011

Compressible, inviscid Rayleigh-Taylor instability

We consider the Rayleigh-Taylor problem for two compressible, immiscible, inviscid, barotropic fluids evolving with a free interface in the presence of a uniform gravitational field. After constructing Rayleigh-Taylor steady-state solutions with a denser fluid lying above the free interface with the second fluid, we turn to an analysis of the equations obtained from linearizing around such a steady state. By a natural variational approach, we construct normal mode solutions that grow exponentially in time with rate like $e^{t \sqrt{\absξ}}$, where $ξ$ is the spatial frequency of the normal mode. A Fourier synthesis of these normal mode solutions allows us to construct solutions that grow arbitrarily quickly in the Sobolev space $H^k$, which leads to an ill-posedness result for the linearized problem. Using these pathological solutions, we then demonstrate ill-posedness for the original non-linear problem in an appropriate sense. More precisely, we use a contradiction argument to show that the non-linear problem does not admit reasonable estimates of solutions for small time in terms of the initial data.

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arXiv2011

Decay of viscous surface waves without surface tension

Consider a viscous fluid of finite depth below the air. In the absence of the surface tension effect at the air-fluid interface, the long time behavior of a free surface with small amplitude has been an intriguing question since the work of Beale \cite{beale_1}. In this monograph, we develop a new mathematical framework to resolve this question. If the free interface is horizontally infinite, we establish that it decays to a flat surface at an algebraic rate. On the other hand, if the free interface is periodic, we establish that it decays at an almost exponential rate, i.e. at an arbitrarily fast algebraic rate determined by the smallness of the data. Our framework contains several novel techniques, which include: (1) a local well-posedness theory of the Navier-Stokes equations in the presence of a moving boundary; (2) a two-tier energy method that couples the boundedness of high-order energy to the decay of low-order energy, the latter of which is necessary to balance out the growth of the highest derivatives of the free interface; (3) control of both negative and positive Sobolev norms, which enhances interpolation estimates and allows for the decay of infinite surface waves; (4) a localization procedure that is compatible with the energy method and allows for curved lower surface geometry in the periodic case. Our decay results lead to the construction of global-in-time solutions to the surface wave problem.

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arXiv2010

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 $\ep \to 0$, when the intensity of the electric current and applied magnetic field on the boundary scale like $\lep$. We show that the limiting velocity of the vortices is the sum of a Lorentz force, due to the current, and a pinning force. We state an analogous result for a model Ginzburg-Landau equation without magnetic field but with forcing terms. Our proof provides a unified approach to various proofs of dynamics of Ginzburg-Landau vortices.

math.AP↗

arXiv2010

Ginzburg-Landau vortex dynamics driven by an applied boundary current

In this paper we study the time-dependent Ginzburg-Landau equations on a smooth, bounded domain $Ω\subset \Rn{2}$, subject to an electrical current applied on the boundary. The dynamics with an applied current are non-dissipative, but via the identification of a special structure in an interaction energy, we are able to derive a precise upper bound for the energy growth. We then turn to the study of the dynamics of the vortices of the solutions in the limit $\ep \to 0$. We first consider the original time scale, in which the vortices do not move and the solutions undergo a "phase relaxation." Then we study an accelerated time scale in which the vortices move according to a derived dynamical law. In the dynamical law, we identify a novel Lorentz force term induced by the applied boundary current.

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arXiv2009

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, and in both periodic and non-periodic 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 small periodicity avoids instability in the presence of surface tension.

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arXiv2007

Lorentz Space Estimates for the Ginzburg-Landau Energy

In this paper we prove novel lower bounds for the Ginzburg-Landau energy with or without magnetic field. These bounds rely on an improvement of the "vortex balls construction" estimates by extracting a new positive term in the energy lower bounds. This extra term can be conveniently estimated through a Lorentz space norm, on which it thus provides an upper bound. The Lorentz space $L^{2,\infty}$ we use is critical with respect to the expected vortex profiles and can serve to estimate the total number of vortices and get improved convergence results.

math.AP↗

arXiv2007

Lorentz Space Estimates and Jacobian Convergence for the Ginzburg-Landau Energy with Applied Magnetic Field

In this paper we continue the study of Lorentz space estimates for the Ginzburg-Landau energy started in our previous paper, \cite{p1}. We focus on getting estimates for the Ginzburg-Landau energy with external magnetic field $h_{ex}$ in certain interesting regimes of $h_{ex}$. This allows us to show that for configurations close to minimizers or local minimizers of the energy, the vorticity mass of the configuration $(u,A)$ is comparable to the $L^{2,\infty}$ Lorentz space norm of $\nabla_A u$. We also establish convergence of the gauge-invariant Jacobians (vorticity measures) in the dual of a function space defined in terms of Lorentz spaces.

math.AP↗