arXiv subjects

Anthony Suen

Publications and source records attributed to Anthony Suen.

25 records · Page 2Linked to original sources

arXiv2019

Wellposedness and convergence of solutions to a class of forced non-diffusive equations with applications

This paper considers a family of non-diffusive active scalar equations where a viscosity type parameter enters the equations via the constitutive law that relates the drift velocity with the scalar field. The resulting operator is smooth when the viscosity is present but singular when the viscosity is zero. We obtain Gevrey-class local well-posedness results and convergence of solutions as the viscosity vanishes. We apply our results to two examples that are derived from physical systems: firstly a model for magnetostrophic turbulence in the Earth's fluid core and secondly flow in a porous media with an "effective viscosity".

math.AP↗

arXiv2018

Solutions to a class of forced drift-diffusion equations with applications to the magneto-geostrophic equations

We prove the global existence of classical solutions to a class of forced drift-diffusion equations with $L^2$ initial data and divergence free drift velocity $\{u^ν\}_{ν_\ge0}\subset L^\infty_t BMO^{-1}_x$, and we obtain strong convergence of solutions as the viscosity $ν$ vanishes. We then apply our results to a family of active scalar equations which includes the three dimensional magneto-geostrophic $\{$MG$^ν\}_{ν\ge0}$ equation that has been proposed by Moffatt in the context of magnetostrophic turbulence in the Earth's fluid core. We prove the existence of a compact global attractor $\{\mathcal{A}^ν\}_{ν\ge0}$ in $L^2(\mathbb{T}^3)$ for the MG$^ν$ equations including the critical equation where $ν=0$. Furthermore, we obtain the upper semicontinuity of the global attractor as $ν$ vanishes.

math.AP↗

arXiv2015

Existence, uniqueness and regularity results for the viscous magneto-geostrophic equation

We study the three dimensional active scalar equation called the magneto-geostropic equation which was proposed by Moffatt and Loper as a model for the geodynamo processes in the Earth's fluid core. When the viscosity of the fluid is positive, the constitutive law that relates the drift velocity $u(x,t)$ and the scalar temperature $θ(x,t)$ produces two orders of smoothing. We study the implications of this property. For example, we prove that in the case of the non-diffusive ($\varepsilon_κ=0$) active scalar equation, initial data $θ_0\in L^3$ implies the existence of unique, global weak solutions. If $θ_0\in W^{s,3}$ with $s>0$, then the solution $θ(x,t)\in W^{s,3}$ for all time. In the case of positive diffusivity ($\varepsilon_κ>0$), even for singular initial data $θ_0\in L^3$, the global solution is instantaneously $C^\infty$-smoothed and satisfies the drift-diffusion equation classically for all $t>0$. We demonstrate, via a particular example, that the viscous magneto-geostrophic equation permits exponentially growing "dynamo type" instabilities.

math.AP↗

arXiv2014

Hölder continuity of solutions to the kinematic dynamo equations

We study the propagation of regularity of solutions to a three dimensional system of linear parabolic PDE known as the kinematic dynamo equations. The divergence free drift velocity is assumed to be at the critical regularity level with respect to the natural scaling of the equations.

math.AP↗

arXiv2012

A blow-up criterion for the 3D compressible magnetohydrodynamics in terms of density

We study an initial boundary value problem for the 3D magnetohydrodynamics (MHD) equations of compressible fluids in $\R^3$. We establish a blow-up criterion for the local strong solutions in terms of the density and magnetic field. Namely, if the density is away from vacuum ($ρ= 0$) and the concentration of mass ($ρ=\infty$) and if the magnetic field is bounded above in terms of $L^\infty$-norm, then a local strong solution can be continued globally in time.

math.AP↗

arXiv2012

Global Solutions of the Navier-Stokes Equations for Isentropic Flow with Large External Potential Force

We prove the global-in-time existence of weak solutions to the Navier-Stokes equations of compressible isentropic flow in three space dimensions with adiabatic exponent $γ\ge1$. Initial data and solutions are small in $L^2$ around a non-constant steady state with densities being positive and essentially bounded. No smallness assumption is imposed on the external forces when $γ=1$. A great deal of information about partial regularity and large-time behavior is obtained.

math.AP↗

arXiv2012

Nonexistence of Self-Similar Singularities in the Ideal Viscoelastic Flow

We prove the nonexistence of finite time self-similar singularities in an ideal viscoelastic flow in $R^3$. We exclude the occurrence of Leray-type self-similar singularities under suitable integrability conditions on velocity and deformation tensor. We also prove the nonexistence of asymptotically self-similar singularities in our system. The present work extends the results obtained by Chae in the case of magnetohydrodynamics (MHD).

math.AP↗