December  2016, 9(4): 777-796. doi: 10.3934/krm.2016016

A blowup criterion for the 2D $k$-$\varepsilon$ model equations for turbulent flows

1. 

School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo 454000, Henan

2. 

School of Mathematics and Information Science, Henan Polytechnic University, Henan 454000, China

Received  September 2015 Revised  May 2016 Published  September 2016

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 $T$.
Citation: Baoquan Yuan, Guoquan Qin. A blowup criterion for the 2D $k$-$\varepsilon$ model equations for turbulent flows. Kinetic & Related Models, 2016, 9 (4) : 777-796. doi: 10.3934/krm.2016016
References:
[1]

S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I, Comm. Pure Appl. Math., 12 (1959), 623-727. doi: 10.1002/cpa.3160120405.  Google Scholar

[2]

D. F. Bian and B. L. Guo, Global existence of smooth solutions to the $k$-$\varepsilon$ model equations for turbulent flows, Comm. Math. Sci., 12 (2014), 707-721. doi: 10.4310/CMS.2014.v12.n4.a6.  Google Scholar

[3]

T. Cazenave, An introduction to Nonlinear Schrödinger Equations, UFRJ, Rio de Janeiro: RJ:Instituto de Matemática, 1996. Google Scholar

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J. S. Fan, S. Jiang and Y. B. Ou, A blow-up criterion for compressible viscous heat-conductive flows, Ann. I. H. Poincaré-AN, 27 (2010), 337-350. doi: 10.1016/j.anihpc.2009.09.012.  Google Scholar

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D. Y. Fang, R. Z. Zi and T. Zhang, A blow-up criterion for two dimensional compressible viscous heat-conductive flows, Nonlinear Anal., 75 (2012), 3130-3141. doi: 10.1016/j.na.2011.12.011.  Google Scholar

[6]

X. D. Huang, J. Li and Y. Wang, Serrin-type blowup criterion for the full compressible Navier-Stokes system, Arch. Ration. Mech. Anal., 207 (2013), 303-316. doi: 10.1007/s00205-012-0577-5.  Google Scholar

[7]

X. D. Huang, J. Li and Z. P. Xin, Blowup criterion for viscous baratropic flows with vacuum states, Comm. Math. Phys., 301 (2011), 23-35. doi: 10.1007/s00220-010-1148-y.  Google Scholar

[8]

X. D. Huang and Z. P. Xin, A blow-up criterion for classical solutions to the compressible Navier-Stokes equations, Sci. China Math., 53 (2010), 671-686. doi: 10.1007/s11425-010-0042-6.  Google Scholar

[9]

X. D. Huang, J. Li and Z. P. Xin, Serrin type criterion for the three-dimensional viscous compressible flows, SIAM J. Math. Anal., 43 (2011), 1872-1886. doi: 10.1137/100814639.  Google Scholar

[10]

X. D. Huang and Y. Wang, Global strong solution to the 2D nonhomogeneous incompressible MHD system, J. Differential Equations, 254 (2013), 511-527. doi: 10.1016/j.jde.2012.08.029.  Google Scholar

[11]

S. Jiang and Y. B. Ou, A blow-up criterion for compressible viscous heat-conductive flows, Acta Math. Sci. Ser. B Engl. Ed., 30 (2010), 1851-1864. doi: 10.1016/S0252-9602(10)60178-6.  Google Scholar

[12]

B. E. Launder and D. B. Spalding, Mathematical Models of Turbulence, Academic Press, London and New York, 1972. doi: 10.1002/zamm.19730530619.  Google Scholar

[13]

O. Rozanova, Blow-up of smooth highly decreasing at infinity solutions to the compressible Navier-Stokes equations, J. Differential Equations, 245 (2008), 1762-1774. doi: 10.1016/j.jde.2008.07.007.  Google Scholar

[14]

Y. Z. Sun, C. Wang and Z. F. Zhang, A Beale-Kato-Majda blow-up criterion for the 3-D compressible Navier-Stokes equations, J. Math. Pures Appl., 95 (2011), 36-47. doi: 10.1016/j.matpur.2010.08.001.  Google Scholar

[15]

Y. Z. Sun, C. Wang and Z. F. Zhang, A Beale-Kato-Majda criterion for three dimensional compressible viscous heat-conductive flows, Arch. Ration. Mech. Anal., 201 (2011), 727-742. doi: 10.1007/s00205-011-0407-1.  Google Scholar

[16]

Y. Wang, One new blowup criterion for the 2D full compressible Navier-Stokes system, Nonlinear Anal. Real World Appl., 16 (2014), 214-226. doi: 10.1016/j.nonrwa.2013.09.020.  Google Scholar

[17]

T. Wang, A regularity criterion of strong solutions to the 2D compressible magnetohydrodynamic equations, Nonlinear Anal. Real World Appl., 31 (2016), 100-118. doi: 10.1016/j.nonrwa.2016.01.011.  Google Scholar

[18]

H. Y. Wen and C. J. Zhu, Blow-up criterions of strong solutions to 3D compressible Navier-Stokes equations with vacuum, Adv. Math., 248 (2013), 534-572. doi: 10.1016/j.aim.2013.07.018.  Google Scholar

[19]

Z. P. Xin, Blowup of smooth solutions to the compressible Navier-Stokes equations with compact density, Comm. Pure Appl. Math., 51 (1998), 229-240. doi: 10.1002/(SICI)1097-0312(199803)51:3<229::AID-CPA1>3.0.CO;2-C.  Google Scholar

[20]

B. Q. Yuan and G. Q. Qin, Local existence of strong solutions to the $k$-$\varepsilon$ model equations for turbulent flows, Bound. Value Probl., 27 (2016), 1-26. doi: 10.1186/s13661-016-0532-8.  Google Scholar

[21]

B. Q. Yuan and X. K. Zhao, Blow-up criteria for the 2D full compressible MHD system, Appl. Anal.: An International Journal, 93 (2014), 1339-1357. doi: 10.1080/00036811.2013.831076.  Google Scholar

show all references

References:
[1]

S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I, Comm. Pure Appl. Math., 12 (1959), 623-727. doi: 10.1002/cpa.3160120405.  Google Scholar

[2]

D. F. Bian and B. L. Guo, Global existence of smooth solutions to the $k$-$\varepsilon$ model equations for turbulent flows, Comm. Math. Sci., 12 (2014), 707-721. doi: 10.4310/CMS.2014.v12.n4.a6.  Google Scholar

[3]

T. Cazenave, An introduction to Nonlinear Schrödinger Equations, UFRJ, Rio de Janeiro: RJ:Instituto de Matemática, 1996. Google Scholar

[4]

J. S. Fan, S. Jiang and Y. B. Ou, A blow-up criterion for compressible viscous heat-conductive flows, Ann. I. H. Poincaré-AN, 27 (2010), 337-350. doi: 10.1016/j.anihpc.2009.09.012.  Google Scholar

[5]

D. Y. Fang, R. Z. Zi and T. Zhang, A blow-up criterion for two dimensional compressible viscous heat-conductive flows, Nonlinear Anal., 75 (2012), 3130-3141. doi: 10.1016/j.na.2011.12.011.  Google Scholar

[6]

X. D. Huang, J. Li and Y. Wang, Serrin-type blowup criterion for the full compressible Navier-Stokes system, Arch. Ration. Mech. Anal., 207 (2013), 303-316. doi: 10.1007/s00205-012-0577-5.  Google Scholar

[7]

X. D. Huang, J. Li and Z. P. Xin, Blowup criterion for viscous baratropic flows with vacuum states, Comm. Math. Phys., 301 (2011), 23-35. doi: 10.1007/s00220-010-1148-y.  Google Scholar

[8]

X. D. Huang and Z. P. Xin, A blow-up criterion for classical solutions to the compressible Navier-Stokes equations, Sci. China Math., 53 (2010), 671-686. doi: 10.1007/s11425-010-0042-6.  Google Scholar

[9]

X. D. Huang, J. Li and Z. P. Xin, Serrin type criterion for the three-dimensional viscous compressible flows, SIAM J. Math. Anal., 43 (2011), 1872-1886. doi: 10.1137/100814639.  Google Scholar

[10]

X. D. Huang and Y. Wang, Global strong solution to the 2D nonhomogeneous incompressible MHD system, J. Differential Equations, 254 (2013), 511-527. doi: 10.1016/j.jde.2012.08.029.  Google Scholar

[11]

S. Jiang and Y. B. Ou, A blow-up criterion for compressible viscous heat-conductive flows, Acta Math. Sci. Ser. B Engl. Ed., 30 (2010), 1851-1864. doi: 10.1016/S0252-9602(10)60178-6.  Google Scholar

[12]

B. E. Launder and D. B. Spalding, Mathematical Models of Turbulence, Academic Press, London and New York, 1972. doi: 10.1002/zamm.19730530619.  Google Scholar

[13]

O. Rozanova, Blow-up of smooth highly decreasing at infinity solutions to the compressible Navier-Stokes equations, J. Differential Equations, 245 (2008), 1762-1774. doi: 10.1016/j.jde.2008.07.007.  Google Scholar

[14]

Y. Z. Sun, C. Wang and Z. F. Zhang, A Beale-Kato-Majda blow-up criterion for the 3-D compressible Navier-Stokes equations, J. Math. Pures Appl., 95 (2011), 36-47. doi: 10.1016/j.matpur.2010.08.001.  Google Scholar

[15]

Y. Z. Sun, C. Wang and Z. F. Zhang, A Beale-Kato-Majda criterion for three dimensional compressible viscous heat-conductive flows, Arch. Ration. Mech. Anal., 201 (2011), 727-742. doi: 10.1007/s00205-011-0407-1.  Google Scholar

[16]

Y. Wang, One new blowup criterion for the 2D full compressible Navier-Stokes system, Nonlinear Anal. Real World Appl., 16 (2014), 214-226. doi: 10.1016/j.nonrwa.2013.09.020.  Google Scholar

[17]

T. Wang, A regularity criterion of strong solutions to the 2D compressible magnetohydrodynamic equations, Nonlinear Anal. Real World Appl., 31 (2016), 100-118. doi: 10.1016/j.nonrwa.2016.01.011.  Google Scholar

[18]

H. Y. Wen and C. J. Zhu, Blow-up criterions of strong solutions to 3D compressible Navier-Stokes equations with vacuum, Adv. Math., 248 (2013), 534-572. doi: 10.1016/j.aim.2013.07.018.  Google Scholar

[19]

Z. P. Xin, Blowup of smooth solutions to the compressible Navier-Stokes equations with compact density, Comm. Pure Appl. Math., 51 (1998), 229-240. doi: 10.1002/(SICI)1097-0312(199803)51:3<229::AID-CPA1>3.0.CO;2-C.  Google Scholar

[20]

B. Q. Yuan and G. Q. Qin, Local existence of strong solutions to the $k$-$\varepsilon$ model equations for turbulent flows, Bound. Value Probl., 27 (2016), 1-26. doi: 10.1186/s13661-016-0532-8.  Google Scholar

[21]

B. Q. Yuan and X. K. Zhao, Blow-up criteria for the 2D full compressible MHD system, Appl. Anal.: An International Journal, 93 (2014), 1339-1357. doi: 10.1080/00036811.2013.831076.  Google Scholar

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