Global unique solvability of nonhomogeneous asymmetric fluids: A Lagrangian approach

Global existence and large time behavior of classical solutions to the two‐dimensional micropolar equations with large initial data and vacuum

“…Then, the Cauchy problem (1.2)-(1.3) admits at least one global weak solution [ , u, w] satisfying the following regularity and integrability: for any T ∈ (0, ∞), 4 3 . Furthermore, it seems that > 1 is the extremal case for the well-posedness of solutions to the micropolar equations.…”

“…1 Then, many mathematicians paid much attention to this model. For the incompressible case, Dong et al 2 studied the global regularity and large time behavior of solutions to the two-dimensional micropolar equations and generalized the global regularity for the two-dimensional micropolar equations with fractional dissipation in Dong et al 3 Recently, Braz e Silva et al 4 considered the variable density micropolar equations and proved the uniqueness of the solution under L ∞ assumption to the initial density. For the compressible case, the pioneering works were made by Mujaković. For the one-dimensional case, she proved the local existence and global existence to an initial-boundary value problem.…”

“…studied the global regularity and large time behavior of solutions to the two‐dimensional micropolar equations and generalized the global regularity for the two‐dimensional micropolar equations with fractional dissipation in Dong et al 3 . Recently, Braz e Silva et al 4 . considered the variable density micropolar equations and proved the uniqueness of the solution under L ∞ assumption to the initial density.…”

Global well-posedness and exponential decay for 3D nonhomogeneous magneto-micropolar fluid equations with vacuum

“…When the initial density is strictly away from vacuum (i.e., ρ 0 is strictly positive), the authors [4] proved some existence and uniqueness results for strong solutions. Meanwhile, Braz e Silva et al [6] investigated global existence and uniqueness of solutions for 3D Cauchy problem through a Lagrangian approach. On the other hand, for the initial density allowing vacuum states, Lukaszewicz [18] (see also [19,Chapter 3]) obtained short-time existence of weak solutions provided that the initial functions u 0 and w 0 are in H 1 0 and that the initial density ρ 0 is uniformly bounded and satisfies ρ −1 0 L 3 < ∞, while Braz e Silva and Santos [12] established the global existence of weak solutions.…”

Global strong solution to the nonhomogeneous micropolar fluid equations with large initial data and vacuum

“…When the initial density is strictly away from vacuum (i.e., ρ 0 is strictly positive), the authors [1] proved some existence and uniqueness results for strong solutions. Meanwhile, Braz e Silva et al [2] investigated global existence and uniqueness of solutions for 3D Cauchy problem through a Lagrangian approach. On the other hand, for the initial density allowing vacuum states, Lukaszewicz [18] (see also [19,Chapter 3]) obtained short-time existence of weak solutions provided that the initial functions u 0 and w 0 are in H 1 0 and that the initial density ρ 0 is uniformly bounded and satisfies ∥ρ −1 0 ∥ L 3 < ∞, while Braz e Silva and Santos [9] established the global existence of weak solutions.…”