20 problems

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…

High-capacitance oscillation conjecture. The surviving upper Hopf bifurcation moves toward

The gyrokinetic phase-space one-form is treated using canonical coordinates adapted to the magnetic-field geometry. In contrast with constructions under regularity or near-axisymme…

Approximate and non-smooth invariants conjecture. A satisfactory theory of approximate and/or non-smooth invariants may be possible.

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…

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…

Nontrivial-helicity conjecture. If $\Sigma$ has non-trivial homology, then there exists $v\in C^\infty\mathcal{V}_0(\Sigma)$ with $\mathcal{H}(v)\neq 0$.

Conjecture. Then $(\alpha_n,\psi_n)$ satisfies the hypothesis in. The conjecture concerns the asymptotic structure of variational solutions for superlinear free boundary problems a…

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…

Angular-momentum asymptotic-information conjecture. The distribution of the angular momentum is the only microscopic information retained in the time-asymptotic limit as…

Magnetic confinement approximation conjecture. The specular reflection effect of the boundary can be approximated by a sufficiently large external magnetic confinement field in the…

All-order field-representation conjecture. Such field representations can be derived at every order.

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…

Applicability conjecture. The result on linearization approximation for the Boltzmann equation with an external force should be helpful for investigating physical processes in low-…

Let $\gamma$ be the growth rate of the plasmoid instability and let $\tau_A$ be the Alfvénic timescale based on the length of the current sheet. Pucci–Velli conjecture. Current she…

An isotropic pair plasma model is a plasma model in which the particle pressure is isotropic, and anomalous resistivity is an effective resistivity arising from processes such as p…

The matrices under consideration may be non-real, and let $\langle u,v\rangle=u^{T}\cdot v$ denote the pseudo scalar product. A basis of eigenvectors is called orthogonal when its…