On the global existence of classical solutions for compressible Navier-Stokes equations with vacuum

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February 2016, 36(2): 1085-1103. doi: 10.3934/dcds.2016.36.1085

On the global existence of classical solutions for compressible Navier-Stokes equations with vacuum

Peixin Zhang ^1, , Jianwen Zhang ^2, and Junning Zhao ^2,

1.	School of Mathematical Sciences, Huaqiao University, Quanzhou 362021, China
2.	School of Mathematical Sciences, Xiamen University, Xiamen 361005, China

Received March 2014 Revised November 2014 Published August 2015

Abstract
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In this paper we present a sufficient condition for the global well-posedness of classical solutions to an initial value problem of compressible isentropic Navier-Stokes equations in the whole space $\mathbb{R}^3$. As an immediate result, the main theorem obtained implies that the Cauchy problem of compressible Navier-Stokes equations with vacuum has a global unique classical solution, provided the initial energy is sufficiently small, or the shear viscosity coefficient is sufficiently large, or the upper bound of the initial density is suitably small and the adiabatic exponent $\gamma\in (1,3/2)$. These results particularly extend the recent ones due to Huang-Li-Xin [7], where the global well-posedness of classical solutions with small initial energy was established.

Keywords: global classical solution, vacuum., large initial data, Compressible Navier-Stokes equations.

Mathematics Subject Classification: Primary: 76N10; Secondary: 35M1.

Citation: Peixin Zhang, Jianwen Zhang, Junning Zhao. On the global existence of classical solutions for compressible Navier-Stokes equations with vacuum. Discrete and Continuous Dynamical Systems, 2016, 36 (2) : 1085-1103. doi: 10.3934/dcds.2016.36.1085

References:

[1]	R. A. Adams, Sobolev Space, Academic Press, New York, 1975.
[2]	Y. Cho, H. J. Choe and H. Kim, Unique solvability of the initial boundary value problems for compressible viscous fluid, J. Math. Pures Appl., 83 (2004), 243-275. doi: 10.1016/j.matpur.2003.11.004.
[3]	Y. Cho and H. Kim, Strong solutions of the Navier-Stokes equations for isentropic compressible fluids, J. Differ. Eqns., 190 (2003), 504-523. doi: 10.1016/S0022-0396(03)00015-9.
[4]	Y. Cho and H. Kim, On classical solutions of the compressible Navier-Stokes equations with nonnegative intial densities, Manuscript Math., 120 (2006), 91-129. doi: 10.1007/s00229-006-0637-y.
[5]	E. Feiresl, A. Novotny and H. Petzeltová, On the existence of globally defined weak solutions to the Navier-Stokes equations, J. Math. Fluid Mech., 3 (2001), 358-392. doi: 10.1007/PL00000976.
[6]	X. D. Huang, J. Li and Z. P. Xin, Serrin type criterion for the three-dimensional compressible flows, SIAM J. Math. Anal., 43 (2011), 1872-1886. doi: 10.1137/100814639.
[7]	X. D. Huang, J. Li and Z. P. Xin, Global well-posedness of classical solutions with large oscillations and vacuum to the three-dimensional isentropic compressible Navier-Stokes equations, Comm. Pure Appl. Math., 65 (2012), 549-585. doi: 10.1002/cpa.21382.
[8]	D. Hoff, Global solutions of the Navier-Stokes equations for multidimendional compressible flow with disconstinuous initial data, J. Differ. Eqs., 120 (1995), 215-254. doi: 10.1006/jdeq.1995.1111.
[9]	D. Hoff, Strong convergence to global solutions for multidimensional flows of compressible, viscous fluids with polytropic equations of state and discontinuous initial data, Arch. Rational Mech. Anal., 132 (1995), 1-14. doi: 10.1007/BF00390346.
[10]	J. Leray, Sur le mouvement d'un kiquide visqueux emplissant l'espace, Acta Math., 63 (1934), 193-248. doi: 10.1007/BF02547354.
[11]	O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Fluids, $2^{nd}$ edition, Gordon and Breach, New York, 1969.
[12]	J. L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, (French), Gauthier-Villars, Paris, 1969.
[13]	P. L. Lions, Mathematical Topics in Fluid Mechanics, Vol. 2 Compressible Models, Oxford University Press, New York, 1998.
[14]	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 (1980), 67-104.
[15]	J. Nash, Le problème de Cauchy pour les équations différentielles d'un fluide général, Bull. Soc. Math. France., 90 (1962), 487-497.
[16]	R. Salvi and I. Straskraba, Global existence for viscous compressible fluids and their behavior as $t\rightarrow \infty$, J. Fac. Sci. Univ. Tokyo Sect. IA. Math., 40 (1993), 17-51.
[17]	J. Serrin, On the uniqueness of compressible fluid motion, Arch. Rational. Mech. Anal., 3 (1959), 271-288.
[18]	Y. Z. Sun, C. Wang and Z. F. Zhang, A Beale-Kato-Majda blow-up criterion for the 3-D compressible Navier-Stokes equations, J. Math. Pures Appl., 95 (2011), 36-47. doi: 10.1016/j.matpur.2010.08.001.
[19]	Z. P. Xin, Blowup of smooth solutions to the compressible Navier-Stokes equation with compact density, Comm. Pure Appl. Math., 51 (1998), 229-240. doi: 10.1002/(SICI)1097-0312(199803)51:3<229::AID-CPA1>3.0.CO;2-C.
[20]	A. A. Zlotnik, Uniform estimates and stabilization of symmetric solutions of a system of quasilinear equations, Diff. Equations, 36 (2000), 701-716. doi: 10.1007/BF02754229.

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References:

[1]	R. A. Adams, Sobolev Space, Academic Press, New York, 1975.
[2]	Y. Cho, H. J. Choe and H. Kim, Unique solvability of the initial boundary value problems for compressible viscous fluid, J. Math. Pures Appl., 83 (2004), 243-275. doi: 10.1016/j.matpur.2003.11.004.
[3]	Y. Cho and H. Kim, Strong solutions of the Navier-Stokes equations for isentropic compressible fluids, J. Differ. Eqns., 190 (2003), 504-523. doi: 10.1016/S0022-0396(03)00015-9.
[4]	Y. Cho and H. Kim, On classical solutions of the compressible Navier-Stokes equations with nonnegative intial densities, Manuscript Math., 120 (2006), 91-129. doi: 10.1007/s00229-006-0637-y.
[5]	E. Feiresl, A. Novotny and H. Petzeltová, On the existence of globally defined weak solutions to the Navier-Stokes equations, J. Math. Fluid Mech., 3 (2001), 358-392. doi: 10.1007/PL00000976.
[6]	X. D. Huang, J. Li and Z. P. Xin, Serrin type criterion for the three-dimensional compressible flows, SIAM J. Math. Anal., 43 (2011), 1872-1886. doi: 10.1137/100814639.
[7]	X. D. Huang, J. Li and Z. P. Xin, Global well-posedness of classical solutions with large oscillations and vacuum to the three-dimensional isentropic compressible Navier-Stokes equations, Comm. Pure Appl. Math., 65 (2012), 549-585. doi: 10.1002/cpa.21382.
[8]	D. Hoff, Global solutions of the Navier-Stokes equations for multidimendional compressible flow with disconstinuous initial data, J. Differ. Eqs., 120 (1995), 215-254. doi: 10.1006/jdeq.1995.1111.
[9]	D. Hoff, Strong convergence to global solutions for multidimensional flows of compressible, viscous fluids with polytropic equations of state and discontinuous initial data, Arch. Rational Mech. Anal., 132 (1995), 1-14. doi: 10.1007/BF00390346.
[10]	J. Leray, Sur le mouvement d'un kiquide visqueux emplissant l'espace, Acta Math., 63 (1934), 193-248. doi: 10.1007/BF02547354.
[11]	O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Fluids, $2^{nd}$ edition, Gordon and Breach, New York, 1969.
[12]	J. L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, (French), Gauthier-Villars, Paris, 1969.
[13]	P. L. Lions, Mathematical Topics in Fluid Mechanics, Vol. 2 Compressible Models, Oxford University Press, New York, 1998.
[14]	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 (1980), 67-104.
[15]	J. Nash, Le problème de Cauchy pour les équations différentielles d'un fluide général, Bull. Soc. Math. France., 90 (1962), 487-497.
[16]	R. Salvi and I. Straskraba, Global existence for viscous compressible fluids and their behavior as $t\rightarrow \infty$, J. Fac. Sci. Univ. Tokyo Sect. IA. Math., 40 (1993), 17-51.
[17]	J. Serrin, On the uniqueness of compressible fluid motion, Arch. Rational. Mech. Anal., 3 (1959), 271-288.
[18]	Y. Z. Sun, C. Wang and Z. F. Zhang, A Beale-Kato-Majda blow-up criterion for the 3-D compressible Navier-Stokes equations, J. Math. Pures Appl., 95 (2011), 36-47. doi: 10.1016/j.matpur.2010.08.001.
[19]	Z. P. Xin, Blowup of smooth solutions to the compressible Navier-Stokes equation with compact density, Comm. Pure Appl. Math., 51 (1998), 229-240. doi: 10.1002/(SICI)1097-0312(199803)51:3<229::AID-CPA1>3.0.CO;2-C.
[20]	A. A. Zlotnik, Uniform estimates and stabilization of symmetric solutions of a system of quasilinear equations, Diff. Equations, 36 (2000), 701-716. doi: 10.1007/BF02754229.

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