Navier–Stokes equations with vorticity in Besov spaces of negative regular indices

Global existence of weak solution and regularity criteria for the 2D Bénard system with partial dissipation

“…In particular, if θ = Const., then system (1.1) reduces to the classical Navier-Stokes system which describes the motion of incompressible viscous fluid flows and has been extensively studied by many authors; see [18][19][20][21][22][23][24] and the references therein. In addition to this, the reader is referred to [25][26][27][28][29] to find more results about the related fluid flow equations.…”

Regularity criteria for 3D shear‐thinning fluids in terms of two components of vorticity

“…in [15], and provided two components of vorticity ω = (𝜔 1 , 𝜔 2 , 0) satisfy condition (1.2) in [16], and ω ∈ L 1 (0, T; BMO) in [17]. For more advances, we refer to [18][19][20][21][22][23][24]. Very recently, for system (1.1) with p ≠ 2, there have been studied the regularity criteria in terms of velocity or direction of vorticity in [8,[25][26][27][28].…”

“…Thereafter, the regularity problem for the 3D NSEs is solved provided $$$ \omega \in {L}^1\left(0,T; BMO\right) $$$ in [14] and $$$$ \omega \in {L}^p\left(0,T;{L}^q\right),\kern0.30em \frac{2}{p}+\frac{3}{q}\le 2,\kern0.30em 1\le p<\infty $$$$ in [15], and provided two components of vorticity $$$ \tilde{\omega}=\left({\omega}_1,{\omega}_2,0\right) $$$ satisfy condition () in [16], and $$$ \tilde{\omega}\in {L}^1\left(0,T; BMO\right) $$$ in [17]. For more advances, we refer to [18–24].…”

Global regularity criterion for the Navier‐Stokes equations based on the direction of vorticity

“…From the scaling point of view, is important in the sense that if u is a solution of , then u λ ( x , t )= λu ( λx , λ 2 t ) with λ >0 is a parameter of solutions of ; moreover, Regularity criterion was later generalized by Beir o da Veiga to be The Prodi‐Serrin conditions was improved from classical Lebesgue spaces larger Besov spaces by Kozono‐Shimada: Zhang and Yang extended and simultaneously as …”