Applications of Mathematics, first online, pp. 1-13


A blow-up criterion for the strong solutions to the nonhomogeneous Navier-Stokes-Korteweg equations in dimension three

Huanyuan Li

Received September 6, 2019.   Published online April 6, 2020.

Abstract:  This paper proves a Serrin's type blow-up criterion for the 3D density-dependent Navier-Stokes-Korteweg equations with vacuum. It is shown that if the density $\rho$ and velocity field $u$ satisfy $\|\nabla\rho\|_{L^{\infty}(0,T; W^{1,q})} + \| u\|_{L^s(0,T; L^r_{\omega})}< \infty$ for some $q>3$ and any $(r,s)$ satisfying $2/s+3/r \le1$, $3 <r \le\infty,$ then the strong solutions to the density-dependent Navier-Stokes-Korteweg equations can exist globally over $[0,T]$. Here $L^r_{\omega}$ denotes the weak $L^r$ space.
Keywords:  Navier-Stokes-Korteweg equations; capillary fluid; blow-up criterion; vacuum; strong solutions
Classification MSC:  35Q35, 76D45, 35D35
DOI:  10.21136/AM.2020.0228-19

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References:
[1] S. Bosia, V. Pata, J. C. Robinson: A weak-$L^p$ Prodi-Serrin type regularity criterion for the Navier-Stokes equations. J. Math. Fluid Mech. 16 (2014), 721-725. DOI 10.1007/s00021-014-0182-5 | MR 3267544 | Zbl 1307.35186
[2] Y. Cho, H. Kim: Unique solvability for the density-dependent Navier-Stokes equations. Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 59 (2004), 465-489. DOI 10.1016/j.na.2004.07.020 | MR 2094425 | Zbl 1066.35070
[3] L. Grafakos: Classical Fourier Analysis. Graduate Texts in Mathematics 249, Springer, New York (2008). DOI 10.1007/978-0-387-09432-8 | MR 2445437 | Zbl 1220.42001
[4] X. Huang, Y. Wang: Global strong solution to the 2D nonhomogeneous incompressible MHD system. J. Differ. Equations 254 (2013), 511-527. DOI 10.1016/j.jde.2012.08.029 | MR 2990041 | Zbl 1253.35121
[5] X. Huang, Y. Wang: Global strong solution with vacuum to the two dimensional density-dependent Navier-Stokes system. SIAM J. Math. Anal. 46 (2014), 1771-1788. DOI 10.1137/120894865 | MR 3200422 | Zbl 1302.35294
[6] X. Huang, Y. Wang: Global strong solution of 3D inhomogeneous Navier-Stokes equations with density-dependent viscosity. J. Differ. Equations 259 (2015), 1606-1627. DOI 10.1016/j.jde.2015.03.008 | MR 3345862 | Zbl 1318.35064
[7] H. Kim: A blow-up criterion for the nonhomogeneous incompressible Navier-Stokes equations. SIAM J. Math. Anal. 37 (2006), 1417-1434. DOI 10.1137/S0036141004442197 | MR 2215270 | Zbl 1141.35432
[8] O. A. Ladyzhenskaya, V. A. Solonnikov, N. N. Ural'tseva: Linear and Quasilinear Equations of Parabolic Type. Translations of Mathematical Monographs 23, American Mathematical Society, Providence (1968). DOI 10.1090/mmono/023 | MR 0241822 | Zbl 0174.15403
[9] H. Li: A blow-up criterion for the density-dependent Navier-Stokes-Korteweg equations in dimension two. Acta Appl. Math. 166 (2020), 73-83. DOI 10.1007/s10440-019-00255-3 | MR 4077229 | Zbl 07181473
[10] H. Sohr: The Navier-Stokes Equations. An Elementary Functional Analytic Approach. Birkhäuser Advanced Texts, Birkhäuser, Basel (2001). DOI 10.1007/978-3-0348-0551-3 | MR 3013225 | Zbl 0983.35004
[11] Z. Tan, Y. Wang: Strong solutions for the incompressible fluid models of Korteweg type. Acta Math. Sci., Ser. B, Engl. Ed. 30 (2010), 799-809. DOI 10.1016/S0252-9602(10)60079-3 | MR 2675787 | Zbl 1228.76038
[12] T. Wang: Unique solvability for the density-dependent incompressible Navier-Stokes-Korteweg system. J. Math. Anal. Appl. 455 (2017), 606-618. DOI 10.1016/j.jmaa.2017.05.074 | MR 3665121 | Zbl 1373.35257
[13] X. Xu, J. Zhang: A blow-up criterion for 3D compressible magnetohydrodynamic equations with vacuum. Math. Models Methods Appl. Sci. 22 (2012), Article ID 1150010, 23 pages. DOI 10.1142/S0218202511500102 | MR 2887666 | Zbl 1388.76452

Affiliations:   Huanyuan Li, School of Mathematics and Statistics, Zhengzhou University, Zhengzhou, 450001, People's Republic of China, e-mail: lihuanyuan1111@163.com


 
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