33 problems

Taylor's conjecture. Magnetic helicity is approximately conserved for small resistivities. This conjecture concerns the robustness of magnetic helicity in turbulent regimes, and th…

Consider the incompressible MHD system … in $\mathbb R^2$, with $\alpha=1$ and a nontrivial background magnetic field. Resistivity-independent dissipation conjecture. The energy of…

Grad's conjecture. If there exists an MHD equilibrium $(X,p)$ on $M$ whose pressure has toroidally nested level sets, then the equilibrium is axisymmetric.

Let $B$ be a magnetic field satisfying the force-balance equation referenced in the source, on a general toroidal domain. Grad's conjecture. Smooth solutions to the force-balance e…

Let $\psi:\mathbb{T}^3\to\mathbb{R}^3$ be smooth, and let $B(t,x)$ solve the ideal kinematic dynamo equation with divergence-free initial datum. Define the perfect dynamo growth ra…

The convergence conjecture. As $\delta\to0$, the sequence ${\bf x}_n$ will converge to the approximating solutions obtained by the continuous-in-time nudging method given as Algori…

Let $(u,b)$ be a weak solution of the ideal MHD system, and define the total energy and cross helicity by … … Here $C_{x,t}^\alpha$ denotes joint Hölder regularity and…

Let a magnetically ideal plasma evolve from an initial state through ideal magnetic relaxation toward a force-free equilibrium. A tangential discontinuity is a discontinuity in the…

Buckmaster–Vicol magnetic helicity conservation conjecture. The space $L^3_{t,x}$ is the threshold regularity space for solutions of the ideal MHD system to conserve magnetic helic…

Let $\Omega\subset\mathbb{R}^3$ be a toroidal domain, meaning that $\partial\Omega$ is diffeomorphic to $\mathbb{T}^2$. A smooth steady Euler flow on $\Omega$ is a smooth solution…

Let $d$ be the spatial dimension, let $\frac{d}{2}+1<\sigma<\infty$, and let $(v,B,\Gamma)\in C([0,T);\mathbf{H}^{\sigma})$ be a solution to the free boundary MHD equations. Write…

The stochastic induction equation involves vector fields whose stretching term produces excess fluctuations in the scaling limit. A suitable geometry means a geometric arrangement…

Iterative existence conjecture. The iterative scheme of section 7 may provide the basis for a mathematical proof of the existence of MHD equilibria with a non-vanishing pressure gr…

Rotational-transform conjecture. If such limiting solutions exist, then solutions $\boldsymbol{B}=\nabla\Psi_{\infty}\times\nabla\Theta_{\infty}$ with different rotational transfor…

Existence conjecture. The derived results may provide a starting point for a mathematical proof of the existence of nontrivial MHD equilibria in toroidal domains without Euclidean…

The paper considers a compressible magnetohydrodynamic system with magnetic field ${\bf B}$ in a domain coupled to an exterior region, where the interface conditions include contin…

Let $B$ be the magnetic field of an ideal magnetohydrodynamic solution, and let $L^3_{t,x}$ denote the space of space-time functions integrable to the third power. Faraco–Lindberg–…

Helicity isoperimetric nonexistence conjecture. The corresponding helicity isoperimetric problem does not admit a solution.

Grad's symmetry conjecture. In three dimensions, the only stationary equilibria fibered by flux surfaces and without flow velocity must inherit Euclidean symmetries; for example, a…

A quasisymmetric vector field is a vector field satisfying the quasisymmetry conditions in a toroidal domain; the relevant construction reduces to solving a first-order nonlinear p…

Horizontal-component scattering conjecture. There exist final states $\mathcal Z_{1,\infty}^{\pm}$ and some $\kappa>0$ such that

Let $T\subset \mathbb{R}^3$ be a topologically toroidal or cylindrical domain. Consider a smooth, unforced magnetohydrostatic equilibrium on $T$ that is non-isolated and non-vanish…

Consider domains in which dissipative processes, including the decay of kinetic energy and Joule heating, are minimized. Symmetry conjecture. The domain shape that creates optimal…

Negentropy-cascade merger conjecture. The negentropy cascade of compressible MHD turbulence should merge with the kinetic cascade at scales below the ion mean free path.

Consider the two-dimensional magnetic hydrodynamical (MHD) system, with energy understood in the usual physical sense and with ohmic resistivity as the magnetic diffusivity paramet…