KRM
A blowup criterion for the 2D $k$-$\varepsilon$ model equations for turbulent flows
Baoquan Yuan Guoquan Qin
Kinetic & Related Models 2016, 9(4): 777-796 doi: 10.3934/krm.2016016
We establish a blow up criterion for the two-dimensional $k$-$\varepsilon$ model equations for turbulent flows in a bounded smooth domain $\Omega$. It is shown that for the initial-boundary value problem of the 2D $k$-$\varepsilon$ model equations in a bounded smooth domain, if $\|\nabla u\|_{L^{1}(0, T; L^{\infty})}+\|\nabla\rho\|_{L^{2}(0, T; L^{\infty})} +\|\varepsilon\|_{L^{2}(0, T; L^{\infty})}$ $<\infty$, then the strong solution $(\rho, u, h,k, \varepsilon)$ can be extended beyond $T$.
keywords: $k$-$\varepsilon$ model equations compressible flows turbulence. blow up criterion strong solution
KRM
Blowup of smooth solutions to the full compressible MHD system with compact density
Baoquan Yuan Xiaokui Zhao
Kinetic & Related Models 2014, 7(1): 195-203 doi: 10.3934/krm.2014.7.195
This paper studies the blowup of smooth solutions to the full compressible MHD system with zero resistivity on $\mathbb{R}^{d}$, $d\geq 1$. We obtain that the smooth solutions to the MHD system will blow up in finite time, if the initial density is compactly supported.
keywords: Full compressible MHD system smooth solution blowup.
DCDS-S
Blow-up criteria of smooth solutions to the three-dimensional micropolar fluid equations in Besov space
Baoquan Yuan Xiao Li
Discrete & Continuous Dynamical Systems - S 2016, 9(6): 2167-2179 doi: 10.3934/dcdss.2016090
In this paper, we investigate the blow-up criteria of smooth solutions and the regularity of weak solutions to the micropolar fluid equations in three dimensions. We obtain that if $ \nabla_{h}u,\nabla_{h}\omega\in L^{1}(0,T;\dot{B}^{0}_{\infty,\infty})$ or $ \nabla_{h}u,\nabla_{h}\omega\in L^{\frac{8}{3}}(0,T;\dot{B}^{-1}_{\infty,\infty})$ then the solution $(u,\omega)$ can be extended smoothly beyond $t=T$.
keywords: Micropolar fluid equations blow-up criteria regularity criteria smooth solutions Besov space.
DCDS
Note on the blowup criterion of smooth solution to the incompressible viscoelastic flow
Baoquan Yuan
Discrete & Continuous Dynamical Systems - A 2013, 33(5): 2211-2219 doi: 10.3934/dcds.2013.33.2211
We study the blowup criterion of smooth solution to the Oldroyd model. Let $(u(t,x), F(t,x)$ be a smooth solution in $[0,T)$, it is shown that the solution $(u(t,x), F(t,x)$ does not appear breakdown until $t=T$ provided $∇ u(t,x)∈ L^1([0,T]; L^∞(\mathbb{R}^n))$ for $n=2,3$.
keywords: Oldroyd model Incompressible viscoelastic fluids blowup criterion of smooth solution.

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