Energy Decay in the Dynamics of Complex Bodies with Spreading Microstructures Represented by 3D Vectors

We analyze complex bodies with active microstructure described by a vector field with val- ues ν ∈ R3, which complements the displacement u ∈ R3. We consider linear elastic consti- tutive structures for macroscopic and microstructural stresses, diffusion of microstructures, and a viscoelastic-substrate-type effect, which has a regularizing influence that we eliminate at a later stage of our analysis. For weak solutions of the balance equations, we prove decay inequalities in W1,2 and also for the p-power of the norms of u and ν in Lp spaces under slip-without-friction-type boundary conditions.