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 ϱ and velocity field u satisfy \(\Vert\nabla\varrho\Vert_{L^{\infty}(0,T;W^{1,q})}+\Vert u\Vert_{L^{s}(0,T;L_{\omega}^{r})}<\infty\) for some q > 3 and any (r, s) satisfying 2/s + 3/r ⩽ 1, 3 < r ⩽ ∞, then the strong solutions to the density-dependent Navier-Stokes-Korteweg equations can exist globally over [0, T]. Here L rω denotes the weak Lr space.
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References
S. Bosia, V. Pata, J. C. Robinson: A weak-Lp Prodi-Serrin type regularity criterion for the Navier-Stokes equations. J. Math. Fluid Mech. 16 (2014), 721–725.
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.
L. Grafakos: Classical Fourier Analysis. Graduate Texts in Mathematics 249, Springer, New York, 2008.
X. Huang, Y. Wang: Global strong solution to the 2D nonhomogeneous incompressible MHD system. J. Differ. Equations 254 (2013), 511–527.
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.
X. Huang, Y. Wang: Global strong solution of 3D inhomogeneous Navier-Stokes equations with density-dependent viscosity. J. Differ. Equations 259 (2015), 1606–1627.
H. Kim: A blow-up criterion for the nonhomogeneous incompressible Navier-Stokes equations. SIAM J. Math. Anal. 37 (2006), 1417–1434.
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.
H. Li: A blow-up criterion for the density-dependent Navier-Stokes-Korteweg equations in dimension two. Acta Appl. Math. 166 (2020), 73–83.
H. Sohr: The Navier-Stokes Equations. An Elementary Functional Analytic Approach. Birkhäuser Advanced Texts, Birkhäuser, Basel, 2001.
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.
T. Wang: Unique solvability for the density-dependent incompressible Navier-Stokes-Korteweg system. J. Math. Anal. Appl. 455 (2017), 606–618.
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.
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This work was supported by the start-up grant from Zhengzhou University.
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Li, H. A Blow-Up Criterion for the Strong Solutions to the Nonhomogeneous Navier-Stokes-Korteweg Equations in Dimension Three. Appl Math 66, 43–55 (2021). https://doi.org/10.21136/AM.2020.0228-19
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DOI: https://doi.org/10.21136/AM.2020.0228-19