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Yuan, Baoquan

On regularity criteria for weak solutions to the micropolar fluid equations in Lorentz space. (English) Zbl 1191.35217

Proc. Am. Math. Soc. 138, No. 6, 2025-2036 (2010).
Summary: The regularity of weak solutions and blow-up criteria for smooth solutions to the micropolar fluid equations in three dimensional space are studied in the Lorentz space \(L^{p,\infty}(\mathbb R^3)\). We obtain that if \(u\in L^q(0,T;L^{p,\infty}(\mathbb R^3))\) for \(\frac 2q+\frac 3p\leq 1\) with \(3<p\leq\infty\), or if \(\nabla u\in L^q(0,T;L^{p,\infty}(\mathbb R^3))\) for \(\frac 2q+ \frac 3p\leq 2\) with \(\frac 32<p\leq\infty\), or if the pressure \(P\in L^q(0,T;L^{p,\infty}(\mathbb R^3))\) for \(\frac 2q+ \frac 3p\leq 2\) with \(\frac 32<p\leq\infty\), or if \(\nabla P\in L^q(0,T;L^{p,\infty}(\mathbb R^3))\) for \(\frac 2q+\frac 3p\leq 3\) with \(1<p\leq\infty\), then the weak solution \((u,\omega)\) satisfying the energy inequality is a smooth solution on \([0,T)\).

Cited in 42 Documents

MSC:

35Q35 PDEs in connection with fluid mechanics
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
35B44 Blow-up in context of PDEs
76W05 Magnetohydrodynamics and electrohydrodynamics

Keywords:

micropolar fluid equations; regularity of weak solutions; Lorentz spaces
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References:

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