Optimal $$L^{2}$$ decay of the magneto-micropolar system in $${\mathbb {R}}^{3}$$

Decay estimates for a class of non‐Newtonian 3D magneto‐micropolar fluids

“…Moreover, for the micro‐rotational field $${\bm w}$$, it can improve and extend the decay result in Cruz and Novais [7] and Guterres et al. [11] due to the linear damping term $$2 {\bm w}$$.…”

“…On the other hand, Li et al [15] shown that the decay has algebraic rate, ‖(𝒖, 𝒘, 𝒃)(𝑡)‖ 2 𝐿 2 ≤ 𝐶(1 + 𝑡) − 3 2 for (𝒖 0 , 𝒘 0 , 𝒃 0 ) ∈ (𝐿 1 ∩ 𝐿 2 )(ℝ 3 ) based on the classical Fourier splitting method. And also, Cruz et al [7] proved that the decay rate for the micro-rotational field with the same initial data can be improved to…”

“…And also, Cruz et al. [7] proved that the decay rate for the micro‐rotational field with the same initial data can be improved to $$ \Vert {\bm w}(t) \Vert _{L^2} ^2 \le C (1 + t) ^{- \frac{5}{2}}$$ (see also e.g., Refs. [6, 24, 27, 30]).…”

Existence and time asymptotic profiles of weak solutions to the nonhomogeneous magneto‐micropolar equations

“…For classical magneto‐micropolar equations with initial data $$$ \left({u}_0,{w}_0,{b}_0\right)\in {L}^1\left({\mathrm{\mathbb{R}}}^3\right)\cap {L}^2\left({\mathrm{\mathbb{R}}}^3\right) $$$, Li and Shang [10] proved the decay properties of weak solutions. Furthermore, Cruz and Novais [11] proved that the micro‐rotational velocity $$$ w $$$ decays faster than $$$ \left(u,b\right) $$$, specifically, $$$$ {\left\Vert \left(u,b\right)(t)\right\Vert}_{L^2}\le C{\left(1+t\right)}^{-\frac{3}{4}}\kern15.12pt \mathrm{and}\kern15.12pt {\left\Vert w(t)\right\Vert}_{L^2}\le C{\left(1+t\right)}^{-\frac{5}{4}}. $$$$ For more decay properties of related models, it is referred to previous studies [12–18] and the references therein.…”

“…The nonhomogeneous magneto‐micropolar equations that we studied in this paper are more complicated than the classical ones in previous studies [6, 10, 11]. There are two main difficulties in studying the decay properties of weak solutions: •Firstly, for (), we should consider the momentum $$$ \rho u $$$ rather than the velocity $$$ u $$$, but the lower order terms (e.g., $$$ \operatorname{curl}\kern3.0235pt w,\operatorname{curl}\kern3.0235pt u $$$, and $$$ w $$$) in () are independent of $$$ \rho $$$. •Moreover, before revising the decay rates of $$$ \left(u,b\right) $$$, we need a faster decay rate of the micro‐rotational velocity $$$ w $$$ which should be obtained by other ways. …”