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Parabolic elliptic type Keller-Segel system on the whole space case
On the global existence of classical solutions for compressible Navier-Stokes equations with vacuum
1. | School of Mathematical Sciences, Huaqiao University, Quanzhou 362021, China |
2. | School of Mathematical Sciences, Xiamen University, Xiamen 361005, China |
References:
[1] | |
[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. |
show all references
References:
[1] | |
[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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