On hydrodynamics of incompressible magneto-viscoelastic flows

Abstract In this paper, we investigate the well-posedness of the incompressible magneto-viscoelastic model without any artificial damping effects on the deformation tensor. First, a blow-up criterion in terms of the maximum norms of the gradient of the velocity and the magnetisation is established. With the blow-up criterion, we then prove the global solutions for the Cauchy problems in both two- and three dimensions under H s ( s ⩾ 2 ) small perturbations near equilibrium, in particular, we only require the initial small assumptions on the velocity and deformation tensor in H 3 -norms and the gradient of the magnetisation in H 2 -norm when s ⩾ 3 . Moreover, based on the rotation-strain decomposition of the deformation tensor, we also show that the smallness of the strain tensor (with possibly large perturbations on the deformation tensor itself) is enough to ensure the global regularity of the system in two dimensions. Finally, the weak-strong uniqueness is obtained by combining a relative energy approach with an elaborate dissipation estimate for the magnetisation.