Global Well-Posedness to the Cauchy Problem of Two-Dimensional Nonhomogeneous Heat Conducting Navier-Stokes Equations
Tóm tắt
In the present paper, we consider the Cauchy problem of nonhomogeneous heat conducting Navier–Stokes equations in the whole space
$${\mathbb {R}}^2$$
. Combining delicate energy estimates and a logarithmic interpolation inequality, we derive the global existence and uniqueness of strong solutions. In particular, the initial data can be arbitrarily large.
Tài liệu tham khảo
Bahouri, H., Chemin, J.-Y., Danchin, R.: Fourier analysis and nonlinear partial differential equations. Springer, Heidelberg (2011)
Cho, Y., Kim, H.: Existence result for heat-conducting viscous incompressible fluids with vacuum. J. Korean Math. Soc. 45, 645–681 (2008)
Craig, W., Huang, X., Wang, Y.: Global wellposedness for the 3D inhomogeneous incompressible Navier-Stokes equations. J. Math. Fluid Mech. 15(4), 747–758 (2013)
Danchin, R., Mucha, P.B.: The incompressible Navier-Stokes equations in vacuum. Commun. Pure Appl. Math. 72(7), 1351–1385 (2019)
Desjardins, B.: Regularity of weak solutions of the compressible isentropic Navier-Stokes equations. Commun. Partial Differ. Equ. 22(5–6), 977–1008 (1997)
Desjardins, B.: Regularity results for two-dimensional flows of multiphase viscous fluids. Arch. Ration. Mech. Anal. 137(2), 135–158 (1997)
Evans, L.C.: Partial differential equations, 2nd edn. American Mathematical Society, Providence (2010)
Galdi, G.P.: An introduction to the mathematical theory of the Navier-Stokes equations. Steady-state problems, 2nd edn. Springer, New York (2011)
Guo, Z., Li, Q.: Global existence and large time behaviors of the solutions to the full incompressible Navier-Stokes equations with temperature-dependent coefficients. J. Differ. Equ. 274, 876–923 (2021)
Huang, X., Wang, Y.: Global strong solution to the 2D nonhomogeneous incompressible MHD system. J. Differ. Equ. 254(2), 511–527 (2013)
Itoh, S., Tani, A.: The initial value problem for the non-homogeneous Navier-Stokes equations with general slip boundary condition. Proc. R. Soc. Edinb. A 130(4), 827–835 (2000)
Kim, H.: A blow-up criterion for the nonhomogeneous incompressible Navier-Stokes equations. SIAM J. Math. Anal. 37(5), 1417–1434 (2006)
Lions, P.L.: Mathematical topics in fluid mechanics, vol. I: incompressible models. Oxford University Press, Oxford (1996)
Lucas, C.F., Ferreira, G. Planas., Villamizar-Roa, E.J.: On the nonhomogeneous Navier-Stokes system with Navier friction boundary conditions. SIAM J. Math. Anal. 45(4), 2576–2595 (2013)
Łukaszewicz, G., Kalita, P.: Navier-Stokes equations. An introduction with applications. Springer, Cham (2016)
Luo, Z.: Local existence of classical solutions to the two-dimensional viscous compressible flows with vacuum. Commun. Math. Sci. 10(2), 527–554 (2012)
Lü, B., Shi, X., Zhong, X.: Global existence and large time asymptotic behavior of strong solutions to the Cauchy problem of 2D density-dependent Navier-Stokes equations with vacuum. Nonlinearity 31(6), 2617–2632 (2018)
Malý, J., Ziemer, W.P.: Fine regularity of solutions of elliptic partial differential equations. American Mathematical Society, Providence (1997)
Nirenberg, L.: On elliptic partial differential equations. Ann. Scuola Norm. Sup. Pisa 13, 115–162 (1959)
Simon, J.: Nonhomogeneous viscous incompressible fluids: existence of velocity, density, and pressure. SIAM J. Math. Anal. 21(5), 1093–1117 (1990)
Wang, W., Yu, H., Zhang, P.: Global strong solutions for 3D viscous incompressible heat conducting Navier-Stokes flows with the general external force. Math. Methods Appl. Sci. 41(12), 4589–4601 (2018)
Zhang, X., Tan, Z.: The global wellposedness of the 3D heat-conducting viscous incompressible fluids with bounded density. Nonlinear Anal. Real World Appl. 22, 129–147 (2015)
Zhong, X.: Global strong solution for 3D viscous incompressible heat conducting Navier-Stokes flows with non-negative density. J. Differ. Equ. 263(8), 4978–4996 (2017)
Zhong, X.: Global well-posedness to the 2D Cauchy problem of nonhomogeneous heat conducting magnetohydrodynamic equations with large initial data and vacuum. Calc. Var. Part. Differ. Equ. 60(2), 1–24 (2021)
Zhong, X.: Global well-posedness to the 3D Cauchy problem of nonhomogeneous heat conducting Navier-Stokes equations with vacuum and large oscillations. J. Math. Fluid Mech. 24(1), 1–17 (2022)
Zhong, X.: Global existence and large time behavior of strong solutions for nonhomogeneous heat conducting Navier-Stokes equations with large initial data and vacuum. Commun. Math. Sci. 20(5) (2022)