On Compressible Smooth Viscous Fluids in Slowly Expanding Balls
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摘要
In [17] and [19, 20], the global existence and large time behaviors of smooth compressible fluids(including inviscid gases of Euler equations, viscous gases of Navier-Stokes equations, and rarified gases of Boltzmann equation, respectively) have been established in an infinitely expanding ball with a constant expansion speed. This paper concerns with the viscous fluids in a slowly expanding ball. By involved analysis on the density function and the weighted energy estimates, we show that the fluid in the slowly expanding ball smoothly tends to a vacuum state and there is no appearance of vacuum in any part of the expansive ball. Our present result is a meaningful supplement to the one in [19].
In [17] and [19, 20], the global existence and large time behaviors of smooth compressible fluids(including inviscid gases of Euler equations, viscous gases of Navier-Stokes equations, and rarified gases of Boltzmann equation, respectively) have been established in an infinitely expanding ball with a constant expansion speed. This paper concerns with the viscous fluids in a slowly expanding ball. By involved analysis on the density function and the weighted energy estimates, we show that the fluid in the slowly expanding ball smoothly tends to a vacuum state and there is no appearance of vacuum in any part of the expansive ball. Our present result is a meaningful supplement to the one in [19].
In [17] and [19, 20], the global existence and large time behaviors of smooth compressible fluids(including inviscid gases of Euler equations, viscous gases of Navier-Stokes equations, and rarified gases of Boltzmann equation, respectively) have been established in an infinitely expanding ball with a constant expansion speed. This paper concerns with the viscous fluids in a slowly expanding ball. By involved analysis on the density function and the weighted energy estimates, we show that the fluid in the slowly expanding ball smoothly tends to a vacuum state and there is no appearance of vacuum in any part of the expansive ball. Our present result is a meaningful supplement to the one in [19].
引文
[1] Cho Yonggeun, Choe Hi Jun and Kim Hyunseok, Unique solvability of the initial boundary value problems for compressible viscous fluids, J. Math. Pures Appl., 83(2004), 243–275.
[2] Cho Yonggeun and Kim Hyunseok, On classical solutions of the compressible Navier-Stokes equations with nonnegative initial densities, Manuscript Math., 120(2006), 91–129.
[3] R. Danchin, Global existence in critical spaces for compressible Navier-Stokes equations,Invent. Math., 141(2000), 579–614.
[4] E. Feireisl, Dynamics of Viscous Compressible Fluids, Oxford University Press, New York,2004.
[5] E. Feireisl, O. Kreml, S. Necasova, J. Neustupa and J. Stebel, Weak solutions to the barotropic Navier-Stokes system with slip boundary conditions in time dependent domains, J. Differential Equations, 254(1)(2013), 125–140.
[6] D. Hoff, Discontinuous solution of the Navier-Stokes equations for multi-dimensional heatconducting fluids, Arch. Ration. Mech. Anal., 193(1997), 303–354.
[7] Huang Xiangdi, Li Jing and Xin Zhouping, Global well-posedness of classical solutions with large oscillations and vacuum to the three-dimensional isentropic compressible NavierStokes equations, Commun. Pure Appl. Math., 65(2012), 549–585.
[8] Jiang Song, Global spherically symmetric solutions to the equations of a viscous polytropic ideal gas in an exterior domain, Commun. Math. Phys., 178(1996), 339–374.
[9] Jiang Song and Zhang Ping, Global spherically symmetric solutions of the compressible isentropic Navier-Stokes equations, Commun. Math. Phys., 215(2001), 559–581.
[10] Y. Kagei and S. Kawashima, Stability of planar stationary solutions to the compressible Navier-Stokes equation on the half space, Commun. Math. Phys., 266(2006), 401–430.
[11] P. L. Lions, Mathematical Topics in Fluid Mechanics, Vol. 2. Compressible Models, Oxford University Press, New York, 1998.
[12] A. Matsumura and T. Nishida, The initial value problem for the equations of motion of viscous and heat-conductive gases, J. Math. Kyoto Univ., 20(1)(1980), 67–104.
[13] A. Matsumura and T. Nishida, Initial boundary value problems for the equations of motion of compressible viscous and heat conductive fluids, Commun. Math. Phys., 89(1983), 445–464.
[14] J. Nash, Le probleme de Cauchy pour les equations differentielles d’un fluide general, Bull.Soc. Math. France, 90(1962), 487–497.
[15] D. Serre, Solutions faibles globales des équations de Navier-Stokes pour un fluide compressible, C. R. Acad. Sci. Paris Sér. I Math., 303(1986), 639–642.
[16] V. A. Vaigant and A. V. Kazhikhov, On the existence of global solutions of two-dimensional Navier-Stokes equations of a compressible viscous fluid,(Russian)Sibirsk. Mat. Zh., 36(1995), 1283–1316, ii; translation in Siberian Math. J., 36(1995), 1108–1141.
[17] Xu Gang and Yin Huicheng, The global existence and large time behavior of smooth compressible fluid in an infinitely expanding ball, I:3D Euler equations, Phys. Scr., 93(10)(2018),105001.
[18] Yang Tong, Yao Zhengan and Zhu Changjiang, Compressible Navier-Stokes equations with density dependent viscosity and vacuum, Commun. Partial Differential Equations, 26(5-6)(2001), 965–981.
[19] Yin Huicheng and Zhang Lin, The global existence and large time behavior of smooth compressible fluid in an infinitely expanding ball, II:3D Navier-Stokes equations, Discrete and Continuous Dynamical System-A, 38(3)(2018), 1063–1102.
[20] Yin Huicheng and Zhao Wenbin, The global existence and large time behavior of smooth compressible fluid in an infinitely expanding ball, III:3D Boltzmann equations, J. Differential Equations, 264(2018), 30–81.
[2] Cho Yonggeun and Kim Hyunseok, On classical solutions of the compressible Navier-Stokes equations with nonnegative initial densities, Manuscript Math., 120(2006), 91–129.
[3] R. Danchin, Global existence in critical spaces for compressible Navier-Stokes equations,Invent. Math., 141(2000), 579–614.
[4] E. Feireisl, Dynamics of Viscous Compressible Fluids, Oxford University Press, New York,2004.
[5] E. Feireisl, O. Kreml, S. Necasova, J. Neustupa and J. Stebel, Weak solutions to the barotropic Navier-Stokes system with slip boundary conditions in time dependent domains, J. Differential Equations, 254(1)(2013), 125–140.
[6] D. Hoff, Discontinuous solution of the Navier-Stokes equations for multi-dimensional heatconducting fluids, Arch. Ration. Mech. Anal., 193(1997), 303–354.
[7] Huang Xiangdi, Li Jing and Xin Zhouping, Global well-posedness of classical solutions with large oscillations and vacuum to the three-dimensional isentropic compressible NavierStokes equations, Commun. Pure Appl. Math., 65(2012), 549–585.
[8] Jiang Song, Global spherically symmetric solutions to the equations of a viscous polytropic ideal gas in an exterior domain, Commun. Math. Phys., 178(1996), 339–374.
[9] Jiang Song and Zhang Ping, Global spherically symmetric solutions of the compressible isentropic Navier-Stokes equations, Commun. Math. Phys., 215(2001), 559–581.
[10] Y. Kagei and S. Kawashima, Stability of planar stationary solutions to the compressible Navier-Stokes equation on the half space, Commun. Math. Phys., 266(2006), 401–430.
[11] P. L. Lions, Mathematical Topics in Fluid Mechanics, Vol. 2. Compressible Models, Oxford University Press, New York, 1998.
[12] A. Matsumura and T. Nishida, The initial value problem for the equations of motion of viscous and heat-conductive gases, J. Math. Kyoto Univ., 20(1)(1980), 67–104.
[13] A. Matsumura and T. Nishida, Initial boundary value problems for the equations of motion of compressible viscous and heat conductive fluids, Commun. Math. Phys., 89(1983), 445–464.
[14] J. Nash, Le probleme de Cauchy pour les equations differentielles d’un fluide general, Bull.Soc. Math. France, 90(1962), 487–497.
[15] D. Serre, Solutions faibles globales des équations de Navier-Stokes pour un fluide compressible, C. R. Acad. Sci. Paris Sér. I Math., 303(1986), 639–642.
[16] V. A. Vaigant and A. V. Kazhikhov, On the existence of global solutions of two-dimensional Navier-Stokes equations of a compressible viscous fluid,(Russian)Sibirsk. Mat. Zh., 36(1995), 1283–1316, ii; translation in Siberian Math. J., 36(1995), 1108–1141.
[17] Xu Gang and Yin Huicheng, The global existence and large time behavior of smooth compressible fluid in an infinitely expanding ball, I:3D Euler equations, Phys. Scr., 93(10)(2018),105001.
[18] Yang Tong, Yao Zhengan and Zhu Changjiang, Compressible Navier-Stokes equations with density dependent viscosity and vacuum, Commun. Partial Differential Equations, 26(5-6)(2001), 965–981.
[19] Yin Huicheng and Zhang Lin, The global existence and large time behavior of smooth compressible fluid in an infinitely expanding ball, II:3D Navier-Stokes equations, Discrete and Continuous Dynamical System-A, 38(3)(2018), 1063–1102.
[20] Yin Huicheng and Zhao Wenbin, The global existence and large time behavior of smooth compressible fluid in an infinitely expanding ball, III:3D Boltzmann equations, J. Differential Equations, 264(2018), 30–81.