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On the well-posedness of inhomogeneous hyperdissipative Navier-Stokes equations
1. | Department of Mathematics, Zhejiang University, Hangzhou 310027 |
2. | Department of Mathematics, Zhejiang University, Hangzhou, 310027, China |
References:
[1] |
H. Abidi, G. Gui and P. Zhang, On the decay and stability of global solutions to the 3D inhomogeneous Navier-Stokes equations, Comm. Pure Appl. Math., 64 (2011), 832-881.
doi: 10.1002/cpa.20351. |
[2] |
H. Abidi, G. Gui and P. Zhang, On the Wellposedness of three-dimensional inhomogeneous Navier-Stokes equations in the critical spaces, Arch. Rational Mech. Anal., 204 (2012), 189-230.
doi: 10.1007/s00205-011-0473-4. |
[3] |
S.-N. Antontsev, A.-V. Kazhikhov and V.-N. Monakhov, "Boundary Value Problems in Mechanics of Nonhomogeneous Fluids,'' Translated from the Russian, Studies in Mathematics and its Applications, 22, North-Holland Publishing Co., Amsterdam, 1990. |
[4] |
H. Bahouri, J.-Y. Chemin and R. Danchin, "Fourier Analysis and Nonlinear Partial Differential Equations,'' Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 343, Springer, Heidelberg, 2011.
doi: 10.1007/978-3-642-16830-7. |
[5] |
J.-M. Bony, Calcul symbolique et propagation des singularités pour équations aux dérivées partielles non linéaires, Annales Scinentifiques de l'École Normale Supérieure, 14 (1981), 209-246. |
[6] |
J.-Y. Chemin, Localization in Fourier space and Navier-Stokes system, in "Phase Space Analysis of Partial Differential Equations,'' Vol. I, Pubbl. Cent. Ric. Mat. Ennio Giorgi, Scuola Norm. Sup., Pisa, (2004), 53-135. |
[7] |
J.-Y. Chemin and N. Lerner, Flot de champs de vecteurs non Lipschitziens et équations de Navier-Stokes, Journal of Differential Equations, 121 (1995), 314-228.
doi: 10.1006/jdeq.1995.1131. |
[8] |
R. Danchin, Density-dependent incompressible viscous fluids in critical spaces, Proc. Roy. Soc. Edinburgh Sect. A, 133 (2003), 1311-1334.
doi: 10.1017/S030821050000295X. |
[9] |
R. Danchin, Local and global well-posedness results for flows of inhomogeneous viscous fluids, Adv. Differential Equations, 9 (2004), 353-386. |
[10] |
R. Danchin, The inviscid limit for density-dependent incompressible fluids, Ann. Fac. Sci. Toulouse Math., 15 (2006), 637-688. |
[11] |
R. Danchin, Uniform estimates for transport-diffusion equations, J. Hyperbolic Differ. Equ., 4 (2007), 1-17.
doi: 10.1142/S021989160700101X. |
[12] |
B. Desjardins, Global existence results for the incompressible density-dependent Navier-Stokes equations in the whole space, Differential Integral Equations, 10 (1997), 587-598. |
[13] |
B. Desjardins, Linear transport equations with initial values in Sobolev spaces and application to the Navier-Stokes equations, Differential Integral Equations, 10 (1997), 577-586. |
[14] |
B. Desjardins, Regularity results for two-dimensional flows of multiphase viscous fluids, Arch. Rational Mech. Anal., 137 (1997), 135-158. |
[15] |
R. J. DiPerna and P.-L. Lions, Ordinary differential equations,transport theory and Sobolev spaces, Invent. Math., 98 (1989), 511-547.
doi: 10.1007/BF01393835. |
[16] |
H. Fujita and T. Kato, On the Navier-Stokes initial value problems. I, Arch. Rational Mech. Anal., 16 (1964), 269-315. |
[17] |
G. Gui, J. Huang and P. Zhang, Large global solutions to 3-D inhomogeneous Navier-Stokes equations slowly varying in one variable, J. Funct. Anal., 261 (2011), 3181-3210.
doi: 10.1016/j.jfa.2011.07.026. |
[18] |
G. Gui and P. Zhang, Global smooth solutions to the 2-D inhomogeneous Navier-Stokes equations with variable viscosity, Chin. Ann. Math. Ser. B, 30 (2009), 607-630.
doi: 10.1007/s11401-009-0027-3. |
[19] |
S. Itoh and A. Tani, Solvability of nonstationary problems for nonhomogeneous incompressible fluids and the convergence with vanishing viscosity, Tokyo Joural of Mathematics, 22 (1999), 17-42.
doi: 10.3836/tjm/1270041610. |
[20] |
N. Katz and N. Pavlović, A cheap Caffarelli-Kohn-Nirenberg inequality for the Navier-Stokes equation with hyper-dissipation, Geom. Funct. Anal., 12 (2002), 355-379.
doi: 10.1007/s00039-002-8250-z. |
[21] |
O. Ladyzhenskaja and V. Solonnikov, The unique solvability of an initial-boundary value problem for viscous incompressible inhomogeneous fluids, Journal of Soviet Mathematics, 9 (1978), 697-749. |
[22] |
P.-L. Lions, "Mathematical Topics in Fluid Mechanics. Vol. 1. Incompressible Models,'' Oxford Lecture Series in Mathematics and its Applications, 3, Oxford University Press, The Clarendon Press, Oxford University Press, New York, 1996. |
[23] |
T. Tao, Global regularity for a logarithmically supercritical hyperdissipative Navier-Stokes equation, Anal. PDE, 2 (2009), 361-366.
doi: 10.2140/apde.2009.2.361. |
show all references
References:
[1] |
H. Abidi, G. Gui and P. Zhang, On the decay and stability of global solutions to the 3D inhomogeneous Navier-Stokes equations, Comm. Pure Appl. Math., 64 (2011), 832-881.
doi: 10.1002/cpa.20351. |
[2] |
H. Abidi, G. Gui and P. Zhang, On the Wellposedness of three-dimensional inhomogeneous Navier-Stokes equations in the critical spaces, Arch. Rational Mech. Anal., 204 (2012), 189-230.
doi: 10.1007/s00205-011-0473-4. |
[3] |
S.-N. Antontsev, A.-V. Kazhikhov and V.-N. Monakhov, "Boundary Value Problems in Mechanics of Nonhomogeneous Fluids,'' Translated from the Russian, Studies in Mathematics and its Applications, 22, North-Holland Publishing Co., Amsterdam, 1990. |
[4] |
H. Bahouri, J.-Y. Chemin and R. Danchin, "Fourier Analysis and Nonlinear Partial Differential Equations,'' Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 343, Springer, Heidelberg, 2011.
doi: 10.1007/978-3-642-16830-7. |
[5] |
J.-M. Bony, Calcul symbolique et propagation des singularités pour équations aux dérivées partielles non linéaires, Annales Scinentifiques de l'École Normale Supérieure, 14 (1981), 209-246. |
[6] |
J.-Y. Chemin, Localization in Fourier space and Navier-Stokes system, in "Phase Space Analysis of Partial Differential Equations,'' Vol. I, Pubbl. Cent. Ric. Mat. Ennio Giorgi, Scuola Norm. Sup., Pisa, (2004), 53-135. |
[7] |
J.-Y. Chemin and N. Lerner, Flot de champs de vecteurs non Lipschitziens et équations de Navier-Stokes, Journal of Differential Equations, 121 (1995), 314-228.
doi: 10.1006/jdeq.1995.1131. |
[8] |
R. Danchin, Density-dependent incompressible viscous fluids in critical spaces, Proc. Roy. Soc. Edinburgh Sect. A, 133 (2003), 1311-1334.
doi: 10.1017/S030821050000295X. |
[9] |
R. Danchin, Local and global well-posedness results for flows of inhomogeneous viscous fluids, Adv. Differential Equations, 9 (2004), 353-386. |
[10] |
R. Danchin, The inviscid limit for density-dependent incompressible fluids, Ann. Fac. Sci. Toulouse Math., 15 (2006), 637-688. |
[11] |
R. Danchin, Uniform estimates for transport-diffusion equations, J. Hyperbolic Differ. Equ., 4 (2007), 1-17.
doi: 10.1142/S021989160700101X. |
[12] |
B. Desjardins, Global existence results for the incompressible density-dependent Navier-Stokes equations in the whole space, Differential Integral Equations, 10 (1997), 587-598. |
[13] |
B. Desjardins, Linear transport equations with initial values in Sobolev spaces and application to the Navier-Stokes equations, Differential Integral Equations, 10 (1997), 577-586. |
[14] |
B. Desjardins, Regularity results for two-dimensional flows of multiphase viscous fluids, Arch. Rational Mech. Anal., 137 (1997), 135-158. |
[15] |
R. J. DiPerna and P.-L. Lions, Ordinary differential equations,transport theory and Sobolev spaces, Invent. Math., 98 (1989), 511-547.
doi: 10.1007/BF01393835. |
[16] |
H. Fujita and T. Kato, On the Navier-Stokes initial value problems. I, Arch. Rational Mech. Anal., 16 (1964), 269-315. |
[17] |
G. Gui, J. Huang and P. Zhang, Large global solutions to 3-D inhomogeneous Navier-Stokes equations slowly varying in one variable, J. Funct. Anal., 261 (2011), 3181-3210.
doi: 10.1016/j.jfa.2011.07.026. |
[18] |
G. Gui and P. Zhang, Global smooth solutions to the 2-D inhomogeneous Navier-Stokes equations with variable viscosity, Chin. Ann. Math. Ser. B, 30 (2009), 607-630.
doi: 10.1007/s11401-009-0027-3. |
[19] |
S. Itoh and A. Tani, Solvability of nonstationary problems for nonhomogeneous incompressible fluids and the convergence with vanishing viscosity, Tokyo Joural of Mathematics, 22 (1999), 17-42.
doi: 10.3836/tjm/1270041610. |
[20] |
N. Katz and N. Pavlović, A cheap Caffarelli-Kohn-Nirenberg inequality for the Navier-Stokes equation with hyper-dissipation, Geom. Funct. Anal., 12 (2002), 355-379.
doi: 10.1007/s00039-002-8250-z. |
[21] |
O. Ladyzhenskaja and V. Solonnikov, The unique solvability of an initial-boundary value problem for viscous incompressible inhomogeneous fluids, Journal of Soviet Mathematics, 9 (1978), 697-749. |
[22] |
P.-L. Lions, "Mathematical Topics in Fluid Mechanics. Vol. 1. Incompressible Models,'' Oxford Lecture Series in Mathematics and its Applications, 3, Oxford University Press, The Clarendon Press, Oxford University Press, New York, 1996. |
[23] |
T. Tao, Global regularity for a logarithmically supercritical hyperdissipative Navier-Stokes equation, Anal. PDE, 2 (2009), 361-366.
doi: 10.2140/apde.2009.2.361. |
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