On Moffatt’s Magnetic Relaxation Equations

Optimal Transport, Convection, Magnetic Relaxation and Generalized Boussinesq Equations

We establish a connection between optimal transport theory (see Villani in Topics in optimal transportation. Graduate studies in mathematics, vol. 58, AMS, Providence, 2003, for instance) and classical convection theory for geophysical flows (Pedlosky, in Geophysical fluid dynamics, Springer, New York, 1979). Our starting point is the model designed few years ago by Angenent, Haker, and Tannenb...

Fractional relaxation equations on Banach spaces

We study existence and qualitative properties of solutions for the abstract fractional relaxation equation (0.1) u′(t)−ADα t u(t) + u(t) = f(t), 0 < α < 1, t ≥ 0, u(0) = 0, on a complex Banach space X, where A is a closed linear operator, Dα t is the Caputo derivative of fractional order α ∈ (0, 1), and f is an X-valued function. We also study conditions under which the solution operator has th...

Topological constraints on magnetic field relaxation

Magnetic field relaxation is determined by both the field’s geometry and its topology. For relaxation processes, however, it turns out that its topology is a much more stringent constraint. As quantifier for the topology we use magnetic helicity and test whether it is a stronger condition than the linking of field lines. Further, we search for evidence of other topological invariants, which giv...

Waveform Relaxation for Functional-diierential Equations Waveform Relaxation for Functional-diierential Equations Waveform Relaxation for Functional-differential Equations

The convergence of waveform relaxation techniques for solving functional-diierential equations is studied. New error estimates are derived that hold under linear and nonlinear conditions for the right-hand side of the equation. Sharp error bounds are obtained under generalized time-dependent Lipschitz conditions. The convergence of the waveform method and the quality of the a priori error bound...

Relaxation in magnetic nanostructures