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Global well-posedness and large-time decay for the 2D micropolar equations. (English) Zbl 1361.35143
The authors are concerned with the \(2D\) micropolar equations with only angular velocity dissipation,
\(\partial_tu + \kappa u-2\kappa \nabla \times w + \nabla \pi + u\cdot \nabla u =0,\)
\(\nabla \cdot u =0,\)
\(\partial_t w-\gamma \Delta w+4\kappa w-2\kappa \nabla \times u+u\cdot \nabla w=0,\)
where \(u=(u_1,u_2)\) and \(w\) are the usual unknown functions. The first result (Theorem 1.1) establishes the global existence and uniqueness of the solutions to the above PDE system. The next result (Theorem 1.2) establishes explicit time decay rates for the solutions of the same PDE system, without the term \(4\kappa w\), under the condition \(\gamma > 4\kappa\). Theorem 1.3 is an additional result on the decay rate of \(\| w(\cdot , t)\|_{L^2}\) as \(t\rightarrow \infty\).

35Q35 PDEs in connection with fluid mechanics
35B40 Asymptotic behavior of solutions to PDEs
35B65 Smoothness and regularity of solutions to PDEs
Full Text: DOI
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