Attractors of the velocity–vorticity–Voigt model of the 3D Navier–Stokes equations with damping

Existence and long‐time behavior of solutions to the velocity–vorticity–Voigt model of the 3D Navier–Stokes equations with damping and memory

“…In this section, we recall some notations about function spaces and preliminary results. We can find it, for example, in Ue and Wang, Grasselli and Pata, and Robinson 4,14,15 …”

“…We can find it, for example, in Ue and Wang, Grasselli and Pata, and Robinson. 4,14,15 For 1 3 , and H k 0 (Ω) = (H k 0 (Ω)) 3 will denote the Lebesgue and Sobolev spaces of vector-valued functions on Ω as usual, where H k = W k,2 is a Hilbert space. We also denote by ||.|| and ⟨•, •⟩ the normal and scalar product in L 2 (Ω), respectively.…”

“…In 2020, Yue and Wang 4 also considered VVV model as in Larios et al, 3 but added the damping term $$$ \lambda w $$$ to the second equation, which parameterizes the extra dissipation occurring in the planetary boundary layer (see Zelik 5 ), such as $$$$ \left\{\begin{array}{cc}\kern0.30em & {u}_t-{\alpha}^2\Delta {u}_t-\vartheta \Delta u+w\times u+\nabla p=f,\\ {}\kern0.30em & {w}_t-\vartheta \Delta w+\left(u\cdotp \nabla \right)w-\left(w\cdotp \nabla \right)u+\lambda w=\nabla \times f,\\ {}\kern0.30em & \nabla \cdotp u=0,\nabla \cdotp w=0.\end{array}\right. $$$$ …”

Global and exponential attractors for a class of non‐Newtonian micropolar fluids

“…Now we recall some definitions and results of general dynamical systems (see, e.g., previous works 24–26,28,37–41 ). We briefly describe some results for an abstract autonomous evolutionary problem in a Banach space X , where F : X → X is a nonlinear operator, and u 0 ∈ X .…”

“…The authors 24 constructed the exponential attractor for some autonomous evolution equations by the squeezing property, and the authors 27 established the existence of an exponential attractor for a nonlinear reaction‐diffusion system by using the smoothing property of the difference of two solutions. Moreover, the authors 28 proved the existence of an exponential attractor for the velocity‐vorticity‐Voigt model of the 3D Navier–Stokes equations with damping by the method of semigroup decomposition.…”