arXiv subjects

Susan Friedlander

Publications and source records attributed to Susan Friedlander.

At least 19 recordsLinked to original sources

arXiv2026

Long time dynamics and anomalous dissipation of energy in viscous forced active scalar equations

We study an abstract family of advection-diffusion equations within the framework of the fractional Laplacian. The system involves two independent diffusion parameters: one introduced via a damping operator acting on the scalar unknown and the other as the coefficient of the fractional Laplacian. We establish existence and convergence results in specific parameter regimes and limits. In particular, we demonstrate the absence of anomalous energy dissipation for long-time averaged solutions. Moreover, we investigate the long time dynamics and prove the existence of a unique global attractor. These results are then applied to two specific classes of active scalar equations in geophysical fluid dynamics, namely the surface quasigeostrophic equation and the magnetogeostrophic equation.

math.AP↗

arXiv2023

Non-uniqueness of forced active scalar equations with even drift operators

We consider forced active scalar equations with even and homogeneous degree 0 drift operator on $\mathbb T^d$. Inspired by the non-uniqueness construction for dyadic fluid models, by implementing a sum-difference convex integration scheme we obtain non-unique weak solutions for the active scalar equation in space $C_t^0C_x^\alpha$ with $\alpha<\frac{1}{2d+1}$. We note that in 1D, the regularity $\alpha<\frac13$ is sharp as the energy identity is satisfied for solutions in $C^\alpha$ with $\alpha>\frac13$. Without external forcing, Isett and Vicol constructed non-unique weak solutions for such active scalar equations with spatial regularity $C_x^\alpha$ for $\alpha<\frac{1}{4d+1}$.

math.AP↗

arXiv2022

Ill/well-posedness of non-diffusive active scalar equations with physical applications

We consider a general class of non-diffusive active scalar equations with constitutive laws obtained via an operator $\mathbf{T}$ that is singular of order $r_0\in[0,2]$. For $r_0\in(0,1]$ we prove well-posedness in Gevrey spaces $G^s$ with $s\in[1,\frac{1}{r_0})$, while for $r_0\in[1,2]$ and further conditions on $\mathbf{T}$ we prove ill-posedness in $G^s$ for suitable $s$. We then apply the ill/well-posedness results to several specific non-diffusive active scalar equations including the magnetogeostrophic equation, the incompressible porous media equation and the singular incompressible porous media equation.

math.AP↗

arXiv2022

Dyadic models for fluid equations: a survey

Over the centuries mathematicians have been challenged by the partial differential equations (PDEs) that describe the motion of fluids in many physical contexts. Important and beautiful results were obtained in the past one hundred years, including the groundbreaking work of Ladyzhenskaya on the Navier-Stokes equations. However crucial questions such as the existence, uniqueness and regularity of the three dimensional Navier-Stokes equations remain open. Partly because of this mathematical challenge and partly motivated by the phenomena of turbulence, insights into the full PDEs have been sought via the study of simpler approximating systems that retain some of the original nonlinear features. One such simpler system is an infinite dimensional coupled set of nonlinear ordinary differential equations referred to a dyadic model. In this survey we provide a brief overview of dyadic models and describe recent results. In particular, we discuss results for certain dyadic models in the context of existence, uniqueness and regularity of solutions.

math.AP↗

arXiv2021

Uniqueness and non-uniqueness results for dyadic MHD models

We construct non-unique Leray-Hopf solutions for some dyadic models for magnetohydrodynamics when the intermittency dimension $\delta$ is less than 1. In contrast, uniqueness of Leray-Hopf solution is established in the case of $\delta\geq 1$. Analogous results on uniqueness and non-uniqueness of Leray-Hopf solution are also obtained for dyadic models of MHD with fractional diffusion.

math.AP↗

arXiv2021

On Moffatt's magnetic relaxation equations

We investigate the stability properties for a family of equations introduced by Moffatt to model magnetic relaxation. These models preserve the topology of magnetic streamlines, contain a cubic nonlinearity, and yet have a favorable $L^2$ energy structure. We consider the local and global in time well-posedness of these models and establish a difference between the behavior as $t\to \infty$ with respect to weak and strong norms.

math.AP↗

arXiv2021

Dyadic models for ideal MHD

We study two dyadic models for incompressible ideal magnetohydrodynamics, one with a uni-directional energy cascade and the other one with both forward and backward energy cascades. Global existence of weak solutions and local well-posedness are established for both models. In addition, solutions to the model with uni-directional energy cascade associated with positive initial data are shown to develop blow-up at a finite time. Moreover, a set of fixed points is found for each model. Linear instability about some particular fixed points is proved.

math.AP↗

arXiv2020

On a class of forced active scalar equations with small diffusive parameters

Many equations that model fluid behaviour are derived from systems that encompass multiple physical forces. When the equations are written in non dimensional form appropriate to the physics of the situation, the resulting partial differential equations often contain several small parameters. We study a general class of such PDEs called active scalar equations which in specific parameter regimes produce certain well known models for fluid motion. We address various mathematical questions relating to well-posedness, regularity and long time behaviour of the solutions to this general class including vanishing limits of several diffusive parameters.

math.AP↗

arXiv2020

Vanishing diffusion limits and long time behaviour of a class of forced active scalar equations

We investigate the properties of an abstract family of advection diffusion equations in the context of the fractional Laplacian. Two independent diffusion parameters enter the system, one via the constitutive law for the drift velocity and one as the prefactor of the fractional Laplacian. We obtain existence and convergence results in certain parameter regimes and limits. We study the long time behaviour of solutions to the general problem and prove the existence of a unique global attractor. We apply results to two particular active scalar equations arising in geophysical fluid dynamics, namely the surface quasigeostrophic equation and the magnetogeostrophic equation.

math.AP↗

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↗

arXiv2017

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 $\{u^\nu\}_{\nu_\ge0}\subset L^\infty_t BMO^{-1}_x$, and we obtain strong convergence of solutions as the viscosity $\nu$ vanishes. We then apply our results to a family of active scalar equations which includes the three dimensional magneto-geostrophic $\{$MG$^\nu\}_{\nu\ge0}$ 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 $\{\mathcal{A}^\nu\}_{\nu\ge0}$ in $L^2(\mathbb{T}^3)$ for the MG$^\nu$ equations including the critical equation where $\nu=0$. Furthermore, we obtain the upper semicontinuity of the global attractor as $\nu$ vanishes.

math.AP↗

arXiv2016

Asymptotic Analysis for Randomly Forced MHD

We consider the three-dimensional magnetohydrodynamics (MHD) equations in the presence of a spatially degenerate stochastic forcing as a model for magnetostrophic turbulence in the Earth's fluid core. We examine the multi-parameter singular limit of vanishing Rossby number $\epsilon$ and magnetic Reynold's number $\delta$, and establish that: (i) the limiting stochastically driven active scalar equation (with $\epsilon =\delta=0$) possesses a unique ergodic invariant measure, and (ii) any suitable sequence of statistically invariant states of the full MHD system converge weakly, as $\epsilon,\delta \rightarrow 0$, to the unique invariant measure of the limit equation. This latter convergence result does not require any conditions on the relative rates at which $\varepsilon, \delta$ decay. Our analysis of the limit equation relies on a recently developed theory of hypo-ellipticity for infinite-dimensional stochastic dynamical systems. We carry out a detailed study of the interactions between the nonlinear and stochastic terms to demonstrate that a H\"{o}rmander bracket condition is satisfied, which yields a contraction property for the limit equation in a suitable Wasserstein metric. This contraction property reduces the convergence of invariant states in the multi-parameter limit to the convergence of solutions at finite times. However, in view of the phase space mismatch between the small parameter system and the limit equation, and due to the multi-parameter nature of the problem, further analysis is required to establish the singular limit. In particular, we develop methods to lift the contraction for the limit equation to the extended phase space, including the velocity and magnetic fields. Moreover, for the convergence of solutions at finite times we make use of a probabilistic modification of the Gr\"onwall inequality, relying on a delicate stopping time argument.

math.AP↗

arXiv2014

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 $\theta(x,t)$ produces two orders of smoothing. We study the implications of this property. For example, we prove that in the case of the non-diffusive ($\varepsilon_\kappa=0$) active scalar equation, initial data $\theta_0\in L^3$ implies the existence of unique, global weak solutions. If $\theta_0\in W^{s,3}$ with $s>0$, then the solution $\theta(x,t)\in W^{s,3}$ for all time. In the case of positive diffusivity ($\varepsilon_\kappa>0$), even for singular initial data $\theta_0\in L^3$, 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

Inviscid Limits for a Stochastically Forced Shell Model of Turbulent Flow

We establish the anomalous mean dissipation rate of energy in the inviscid limit for a stochastic shell model of turbulent fluid flow. The proof relies on viscosity independent bounds for stationary solutions and on establishing ergodic and mixing properties for the viscous model. The shell model is subject to a degenerate stochastic forcing in the sense that noise acts directly only through one wavenumber. We show that it is hypo-elliptic (in the sense of Hormander) and use this property to prove a gradient bound on the Markov semigroup.

math-ph↗

arXiv2014

H\"{o}lder 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↗

arXiv2013

The magneto-geostrophic equations: a survey

We discuss recent results obtained by the authors, regarding the analysis of the magneto-geostrophic equation: a model proposed by Moffat and Loper to study the geodynamo and turbulence in the Earth's fluid core. We conclude this review by indicating some open problems around the MG equation, that remain to be addressed in the future.

math.AP↗

arXiv2012

On the second iterate for active scalar equations

We consider an iterative resolution scheme for a broad class of active scalar equations with a fractional power \gamma of the Laplacian and focus our attention on the second iterate. The main objective of our work is to analyze boundedness properties of the resulting bilinear operator, especially in the super-critical regime. Our results are two-fold: we prove continuity of the bilinear operator in BMO^{1-2\gamma} - a fractional analogue of the Koch-Tataru space; for equations with an even symbol we show that the B^{-\gamma}_{\infty,q} -regularity, where q > 2, is in a sense a minimal necessary requirement on the solution.

math.AP↗

arXiv2012

On a singular incompressible porous media equation

In this paper we study a singularly modified version of the incompressible porous media equation. We investigate the implications for the local well-posedness of the equations by modifying, with a fractional derivative, the constitutive relation between the scalar density and the convecting divergence free velocity vector. Our analysis is motivated by recent work \cite{CCCGW} where it is shown that for the surface quasi-geostrophic equation such a singular modification of the constitutive law for the velocity, quite surprisingly still yields a locally well-posed problem. In contrast, for the singular active scalar equation discussed in this paper, local well-posedness does not hold for smooth solutions, but it does hold for certain weak solutions.

math.AP↗