Weak solutions to the Bloch equations with distant dipolar field

Abstract

The distant dipolar field (DDF) is a long-range, nonlocal contribution to spin dynamics in liquids that arises from intermolecular dipolar couplings and can generate multiple-quantum coherences in liquids and produce novel MRI contrast. Its nonlocal, sign-changing kernel makes Bloch–DDF dynamics strongly geometry-dependent, and common FFT-based dipolar convolutions are naturally aligned with periodic boxes or padded Cartesian grids rather than bounded samples with reflective diffusion boundaries. We study the Bloch equations with the DDF on bounded domains under homogeneous Neumann diffusion conditions. We derive a conforming finite-element weak formulation that allows spatially varying diffusion and relaxation parameters and uses a short-distance regularization of the secular DDF kernel with length a > 0. For fixed a, we prove boundedness of the induced DDF operator, establish an L2 energy balance in which precession is neutral while diffusion and transverse relaxation are dissipative, and obtain local well-posedness with continuous dependence on the data (with global existence under energy-neutral transport conditions). For the Galerkin semi-discretization, we show a discrete energy identity that mirrors the continuum estimate. For computation, we evaluate the DDF in real space with a matrix-free near/far scheme and advance in time with a second-order implicit–explicit splitting method that treats diffusion and relaxation implicitly and treats precession explicitly. The explicit stage applies a Rodrigues rotation at DDF quadrature points, followed by an L2 projection, which supports stable multi-cycle lab-frame calculations. We validate against three closed-form benchmarks that isolate distinct model components, and we quantify curved-boundary effects by comparing mapped-geometry finite elements with a voxel-mask finite-difference baseline on a spherical Neumann eigenmode decay. These results provide an analyzable and reproducible route for Bloch–DDF dynamics on bounded domains with complex geometry.