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We investigate the global-in-time existence and uniqueness of weak solutionsfor a family of equations introduced by Moffatt to model magnetic relaxation.These equations are topology-preserving and admit all stationary solutions tothe classical incompressible Euler equations as steady states. In the work ofBeekie, Friedlander and Vicol, global regularity results have been establishedfor initial magnetic field B0 $\in$ Hs(Td)(s > d/2+1) when the regularizationparameter $\gamma$ in the equations satisfies $\gamma$ > $\gamma$c := d/2+1.Global regularity for $\gamma$ $\in$ [0, $\gamma$c] is left as an open problem,as well as the existence of weak solutions with rough initial data for any$\gamma$ $\ge$ 0. In this paper, we show that for any solenoidal magnetic fieldB0 $\in$ L2(Td) there exists a unique global weak solution when $\gamma$ >$\gamma$c. Moreover, the solution can propagate higher-order Sobolevregularity. These results hold true for the borderline case $\gamma$ =$\gamma$c only if B0 $\in$ L2+(Td). Finally, we discuss Moffatt's magneticrelaxation procedure.To prove the existence and uniqueness of weak solutions,we establish a key compactness result for an active vector equation and providesome estimates for fractional Laplacian after the Lagrangian change ofvariables, which may have independent interest.

Weak solutions of Moffatt's magnetic relaxation equations