On Woltjer’s force free minimizers and Moffatt’s
magnetic relaxation

In [18], Moffatt considered the following evolution equations of a viscous and perfectly conductive magneto-fluid

where $\rho$ is the fluid density (assumed uniform), $\mu$ viscosity (in [18] it is assumed sufficiently large when compared with Reynolds number associated with the flow) $p(\mathbf{x},t)$ is the pressure field. The second equation in (4) assures that $\mathbf{B}_{t}=\mathbf{B}(\mathbf{x},t)$ is transported with the flow $\phi_{\mathbf{v}}$ of $\mathbf{v}$, i.e.

where $\phi_{\mathbf{v}}(\mathbf{x},t)_{\ast}$ denotes a pushforward of the field under the diffeomorhism $\phi_{\mathbf{v}}$. Moffatt further shows that as long as $\mathbf{v}\neq 0$, the $L^{2}$ energy of $\mathbf{v}$ and $\mathbf{B}$ decreases as $t\to\infty$, by means of the following formula

Further, he asserts that a minimizing sequence $\mathbf{B}_{t}$, $t\to\infty$ (c.f. [6, 22]) should yield a stationary state $\mathbf{B}=\mathbf{B}_{\infty}$ satisfying the Euler’s equations. In [22], Nishiyama observes that a rigorous justification of convergence of $\mathbf{B}_{t}$ to the stationary state is problematic due to the perfect conductivity of the magnetofluid, and introduces, guided by Vallis et.al, [27], an alternative to (3)–(6) system which admits a measure-valued solution in the sense of DiPerna and Majda [7].

Since the relaxation of the field $\mathbf{B}_{0}$ according to $\eqref{eq:MHD-equation-1}$–$\eqref{eq:bdry-conditions}$ decreases its energy, a general question studied by Moffatt in [18] is as follows

As illustrated in [18], one expects the minimizers to be different. In particular, in the case of fields with zero helicity, the force free minimizer of (2) is the zero field, however a nontrivial topology of the initial field $\mathbf{B}_{0}$ can be still prevent a complete energy relaxation. The easiest examples where this situation occurs are the vector fields modeled on essential links and knots in ${\mathbb{R}}^{3}$ (see Appendix), Figure 1 shows an example of a field modeled on the pair of Hopf links. In [18], among other examples, Moffatt considers the field modeled on Borromean rings $\mathbf{B}_{Borr}$ and observes that the energy $E(\mathbf{B}_{Borr})$ cannot be decreased to zero under (4), (7) thanks to the lower bound of Freedman and He in [12] (see further remarks after the proof of Theorem A). These considerations however require a strong $L^{2}$ convergence of a minimizing sequence $\mathbf{B}_{t}$ obtained from (8), for $t\to\infty$, which is in general problematic as discussed above.

In order to circumvent these problems, we observe that any minimizing sequence $\{\mathbf{B}_{t}\}$ obtained from (8) always has a weakly convergent subsequence. Thus we consider a variational problem associated with (3)–(6) which, asks to minimize the $L^{2}$–energy; $E$ over the subset

of divergence free fields obtained from $\mathbf{B}_{0}\in\mathsf{L}^{2}_{\operatorname{curl}}(\Omega)$ via pushforwards by volume preserving diffeomorphisms of $\Omega$ which are identity when restricted to $\partial\Omega$ (the $C^{1}$ diffeomorphisms are denoted by $\operatorname{Diff}_{0}(\Omega,d\mathbf{x})$). This is consistent with (4), since every vector field in $\mathsf{M}(\Omega,\mathbf{B}_{0})$ has the same topology as the initial field $\mathbf{B}_{0}$, and (4) simply defines a path¹¹1for the perfectly conducting magneto–fluid the long time existence is not known [22]. in $\mathsf{M}(\Omega,\mathbf{B}_{0})$. A clear difference with Woltjer’s problem is that the helicity constraint provides only a “mild” restriction on a topology of a field, whereas vector fields in $\mathsf{M}(\Omega,\mathbf{B}_{0})$ have equivalent topology to the initial $\mathbf{B}_{0}$.

In order to put it in the general setting, recall that a usual variational problem asks to minimize a weakly lower semicontinuous functional $E$ over a weakly compact class of functions $W$, [8, 16]. One then considers a minimizing sequence $f_{n}\in W$ weakly convergent to $f\in W$, then $E(f)\leq\lim\inf E(f_{n})$. By the extreme value theorem for the weakly lower semicontinuous functions, $f$ is a minimizer of $E$ over $W$. Note that $\mathsf{W}_{c_{\ast}}(\Omega)$ in (2), is weakly closed in $\mathsf{L}^{2}(\Omega)$ [15, p. 1237]. Since $\mathsf{M}(\Omega,\mathbf{B}_{0})$ is not weakly closed, we consider its weak $\mathsf{L}^{2}$ closure $\overline{\mathsf{M}}^{w}(\Omega,\mathbf{B}_{0})$ and ask to

This formulation meets the requirements of the usual variational problem. Clearly, the caveat of replacing $\mathsf{M}(\Omega,\mathbf{B}_{0})$ by $\overline{\mathsf{M}}^{w}(\Omega,\mathbf{B}_{0})$ is that the field line topology is no longer preserved and it is possible that

examples of paths in $\mathsf{M}(\Omega,\mathbf{B}_{0})$ where this possibility is realized were constructed in [16]. However, the relaxation (3)–(6), under the assumption of the long time existence of classical solutions, always takes place in $\mathsf{M}(\Omega,\mathbf{B}_{0})$, therefore the field line topology is preserved for “all time” except in the limit where it can be lost. Moffatt [18] defines fields topologically accessible from $\mathbf{B}_{0}$ (i.e. limits of minimizing sequences $\mathbf{B}_{t}$ obtaind from (4), (8)), here we simply allow such fields to be all possible weak $L^{2}$ limits of fields from $\mathsf{M}(\Omega,\mathbf{B}_{0})$. Equivalently, one could say that under Moffatt’s assumption, topologically accessible fields are in the strong $L^{2}$ closure of $\mathsf{M}(\Omega,\mathbf{B}_{0})$, whereas in general we may only show that they are in the weak $L^{2}$ closure $\overline{\mathsf{M}}^{w}(\Omega,\mathbf{B}_{0})$ of $\mathsf{M}(\Omega,\mathbf{B}_{0})$. Theorem A, presented in this work, shows that the global minimizer (Woltjer’s minimizer) is not in $\overline{\mathsf{M}}^{w}(\Omega,\mathbf{B}_{0})$ for a field $\mathbf{B}_{0}$ with a nontrivial linkage of flowlines.

Our notation for the function spaces in the next section is as follows: $\mathsf{C}^{\infty}_{0,\operatorname{div}}(\Omega)$ smooth (test) divergence free compactly supported vector fields on $\Omega$, $\mathsf{L}^{2}(\Omega)=\{X\ |\ X\in L^{2}(\Omega)\times L^{2}(\Omega)\times L^{2}(\Omega)\}$, the square integrable vector fields, $\mathsf{H}^{1}(\Omega)=\{X\ |\ X\in H^{1}(\Omega)\times H^{1}(\Omega)\times H^{1}(\Omega)\}$ the Sobolev space of $L^{2}$ vector fields with $L^{2}$ weak derivatives.

In a simply connected domain $\Omega\subset{\mathbb{R}}^{3}$ with smooth connected boundary, let us consider as an initial vector field $\mathbf{B}_{h}$ (i.e. $\mathbf{B}_{0}=\mathbf{B}_{h}$) modeled on a pair of Hopf links $L=L^{+}_{h}\cup L^{-}_{h}$ (see Appendix A) as shown in Figure 1 and supported on the tubes $\{\mathcal{T}^{\pm}_{1},\mathcal{T}^{\pm}_{2}\}$. It follows from the well known flux helicity formula [18], that the total helicity of $\mathbf{B}_{h}$ is zero (we review this in the proof below) and therefore the force free minimizer of (2) is zero, [15].

Also, without loss of generality, we may scale the fields to obtain the unit fluxes i.e. $\Phi(\mathbf{B}^{\pm}_{h,\ast})=1$ and $\mathcal{H}(\mathbf{B}^{\pm}_{h,1},\mathbf{B}^{\pm}_{h,2})=\pm 1$. Further, $\mathcal{H}$ is a symmetric bilinear, thus the above identities yield,

If one could assume a strong $L^{2}$ convergence $\mathbf{B}^{\pm}_{h,n}\longrightarrow\overline{\mathbf{B}}^{\pm}_{h}$ in the proof of Theorem A then the classical energy–helicity estimate [1]: $c_{1}(\Omega)|\mathcal{H}(\mathbf{B})|\leq E(\mathbf{B})$, can be used to argue that $\overline{\mathbf{B}}_{h}\neq 0$. Alternatively, one can use the asymptotic crossing number estimate of [10]. Therefore, these classical energy estimates suffice to show that there is a gap between the global minimum (the zero field, in this case) and the minimum over the strong $L^{2}$ closure of $\mathsf{M}(\Omega,\mathbf{B}_{0})$. Theorem A shows a stronger fact i.e. that there is a gap between the global minimum and the minimum over the weak $L^{2}$ closure $\overline{\mathsf{M}}^{w}(\Omega,\mathbf{B}_{0})$. Clearly, this result applies beyond the Moffatt’s relaxation to any other relaxation process which evolves a divergence free field by means of energy-decreasing diffeomorphisms, such as Vallis [27] and Nishiyama [22]. In particular, Theorem A answers positively the question posed in [22, p. 417]. In the context of Taylor’s conjecture [26] it imples that the Woltjer’s minimizer cannot be reached during the perfectly conductive relaxation phase, if the inital field has a nontrivial topology. In the follow–up work we address this problem in the case of vector fields modeled on links exhibiting a higher order linkage (such as Borromean rings).