Dynamics of a fractional hydrodynamical equation for the Heisenberg paramagnet

In the present work, we study the global solvability and large time dynamics for a fractional generalization of the hydrodynamical equation modeling the soft micromagnetic materials. Introducing a cancellation property, we prove the existence of weak solutions and establish a uniqueness criterion. A maximal principle is obtained and the global existence and uniqueness of smooth solutions are proved by some a priori estimates. Finally, we analyze the asymptotic behavior of the solutions within the theory of infinite dimensional dissipative dynamical systems. We prove that the problem generates a strongly continuous semigroup on a suitable phase space and show the existence of a maximal global attractor A in this phase space. Moreover, in absence of external force, global attractor A converges exponentially to a single equilibrium.

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