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.

We consider forced active scalar equations with even and homogeneous degree 0 drift operator on Td. Inspired by the non-uniqueness construction for dyadic fluid models (Dai and Friedlander, 2022; Filonov and Khodunov, 2021), by implementing a sum-difference convex integration scheme we obtain non-unique weak solutions for the active scalar equation in space Ct0Cxα with α<12d+1. Without external forcing, Isett and Vicol (2015) constructed non-unique weak solutions for such active scalar equations with spatial regularity Cxα for α<14d+1.

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.

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.

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 $$t\rightarrow \infty $$ with respect to weak and strong norms.

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.

Abstract. We construct non-unique Leray-Hopf solutions for some dyadic models for magnetohydrodynamics when the intermittency dimension δ is less than 1. In contrast, uniqueness of Leray-Hopf solution is established in the case of δ ≥ 1. In addition, for a dyadic model of the generalized Navier-Stokes equation with diffusion term (−∆)u, non-unique Leray-Hopf solutions are constructed provided α < 1

Abstract. We study a dyadic model for incompressible ideal magnetohydrodynamics. Besides establishing global existence of weak solutions and local well-posedness, a set of fixed points are found. Solutions starting from positive data near a positive fixed point are shown to remain positive for all the time. Solutions with positive data are also shown to develop blow-up at finite time. In addition, we prove that the pure fluid steady state (with zero magnetic field) attracts solutions with stronger perturbation in the fluid component than that of the magnetic field; vice versa, the non-fluid steady state attracts solutions with stronger perturbation in the magnetic field. The solutions attracted by a fixed point dissipate energy.

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 the results to two particular active scalar equations arising in geophysical fluid dynamics, namely the surface quasigeostrophic equation and the magnetogeostrophic equation.

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".

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.

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 _≥ 0⊂ L^∞ _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≥ 0 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 {𝒜^ν}_ν≥ 0 in L^2(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.

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 multiparameter singular limit of vanishing Rossby number $\varepsilon$ and magnetic Reynold's number $\delta$, and establish that (i) the formal limit (with $\varepsilon=\delta=0$), a stochastically driven active scalar equation, possesses a unique ergodic invariant measure, and (ii) any suitable sequence of statistically invariant states of the full MHD system converges weakly, as $\varepsilon,\delta \rightarrow 0$, 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 hypoellipticity 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 Hörmander bracket condition is satisfied for the limit equation, which yields a contraction property in a suitable Wasserstein metric. This contraction property reduces the convergence of invariant states in the multiparameter 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 multiparameter 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 Grönwall inequality, relying on a delicate stopping time argument.