Abstract
This paper proves a blow-up criterion for the strong solutions with vacuum to the density-dependent Navier–Stokes–Korteweg equations over a bounded smooth domain in \(\mathbb{R}^{2}\), which only in terms of the density.
Similar content being viewed by others
References
Burtea, C., Charve, F.: Lagrangian methods for a general inhomogeneous incompressible Navier–Stokes–Korteweg system with variable capillarity and viscosity coefficients. SIAM J. Math. Anal. 49(5), 3476–3495 (2017)
Cho, Y., Kim, H.: Unique solvability for the density-dependent Navier–Stokes equations. Nonlinear Anal. 59(4), 465–489 (2004)
Desjardins, B.: Regularity results for two-dimensional flows of multiphase viscous fluids. Arch. Ration. Mech. Anal. 137, 135–158 (1997)
Danchin, R., Desjardins, B.: Existence of solutions for compressible fluid models of Korteweg type. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 18, 97–133 (2001)
Hattori, H., Li, D.: Solutions for two dimensional system for materials of Korteweg type. SIAM J. Math. Anal. 25, 85–98 (1994)
Hattori, H., Li, D.: Global solutions of a high-dimensional system for Korteweg materials. J. Math. Anal. Appl. 198, 84–97 (1996)
Huang, X.D., Wang, Y.: Global strong solution to the 2D nonhomogeneous incompressible MHD system. J. Differ. Equ. 254(2), 511–527 (2013)
Huang, X.D., Wang, Y.: Global strong solution with vacuum to the two dimensional density-dependent Navier–Stokes system. SIAM J. Math. Anal. 46(3), 1771–1788 (2014)
Huang, X.D., Wang, Y.: Global strong solution of 3D inhomogeneous Navier–Stokes equations with density-dependent viscosity. J. Differ. Equ. 259(4), 1606–1627 (2015)
Kostin, I., Marion, M., Texier-Picard, R., Volpert Vitaly, A.: Modelling of miscible liquids with the Korteweg stress. Math. Model. Numer. Anal. 37(5), 741–753 (2003)
Sy, M., Bresch, D., Guillén-González, F., Lemoine, J., Rodríguez-Bellido, M.A.: Local strong solution for the incompressible Korteweg model. C. R. Math. Acad. Sci. Paris 342(3), 169–174 (2006)
Tan, Z., Wang, Y.J.: Strong solutions for the incompressible fluid models of Korteweg type. Acta Math. Sci. Ser. B Engl. Ed. 30(3), 799–809 (2010)
Wang, T.: Unique solvability for the density-dependent incompressible Navier–Stokes–Korteweg system. J. Math. Anal. Appl. 455(1), 606–618 (2017)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Li, H. A Blow-up Criterion for the Density-Dependent Navier–Stokes–Korteweg Equations in Dimension Two. Acta Appl Math 166, 73–83 (2020). https://doi.org/10.1007/s10440-019-00255-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10440-019-00255-3