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Please how to show the local existence for the solution of the 3D micropolar equations in a Besov space setting ?

$\left\{ \begin{array}{l} \partial_tu-(\nu+k)\Delta u-2k\nabla\times w+u\nabla u+\nabla\Pi=0\\ \nabla\cdot u = 0\\ \partial_tw+4kw- \mu\nabla\nabla\cdot w-2k\nabla\times u+u\nabla u = 0 \end{array} \right.$

where:

$u=u(x,t)$ is a fluid velosity

$w=w(x,t)$ is the field of microrotation

$\Pi$ is the scalar pressure

$\nu$ is the Newtonian kinetic velocity

$k$ is the microrotation velocity

$\mu$ is the angular velocity

Thanks in advance.

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closed as off-topic by Glitch, mrtaurho, Shaun, Riccardo.Alestra, Taroccoesbrocco Mar 15 at 16:58

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  • $\begingroup$ This is a very hard problem and I'm pretty sure no complete solution is known. 😀 $\endgroup$ – Robert Lewis Mar 12 at 0:08
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    $\begingroup$ Hi @RobertLewis, is there a method that we always follow for this type of questions ? or please is there a PDF papers that I can read and follow ? Thanks $\endgroup$ – Tom Mar 12 at 0:10
  • $\begingroup$ I don't know I'm googling around and will let you know if I find anything. My previous comment is based on general. knowledge. But a nonlinear coupled system of PDEs can't be too easy. $\endgroup$ – Robert Lewis Mar 12 at 0:33
  • $\begingroup$ By the way, are $\Pi$, $\nu$, $k$, $\mu$ constants? Or pre-defined, known functions? $\endgroup$ – Robert Lewis Mar 12 at 0:40
  • $\begingroup$ Hi @RobertLewis ,the unknowns for the system of equations are : $\Pi$ which is a scalar function, $u=u(x,t)$ and $w=w(x,t)$ but the other variables are known constants. $\endgroup$ – Tom Mar 12 at 0:48