Partial suitable solutions for the micropolar equations and regularity properties

The incompressible Micropolar system is given by two coupled equations: the first equation gives the evolution of the velocity field <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover accent="true"> <mml:mi>u</mml:mi> <mml:mo>→</mml:mo> </mml:mover> </mml:math> while the second equation gives the evolution of the microrotation field <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover accent="true"> <mml:mi>ω</mml:mi> <mml:mo>→</mml:mo> </mml:mover> </mml:math> . In this article we will consider regularity problems for weak solutions of this system. For this we will introduce the new notion of partial suitable solutions, which imposes a specific behavior for the velocity field <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover accent="true"> <mml:mi>u</mml:mi> <mml:mo>→</mml:mo> </mml:mover> </mml:math> only, and under some classical hypotheses over the pressure, we will obtain a hölderian gain for both variables <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover accent="true"> <mml:mi>u</mml:mi> <mml:mo>→</mml:mo> </mml:mover> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover accent="true"> <mml:mi>ω</mml:mi> <mml:mo>→</mml:mo> </mml:mover> </mml:math> .