On the well-posedness of inhomogeneous hyperdissipative Navier-Stokes equations

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August 2013, 33(8): 3517-3541. doi: 10.3934/dcds.2013.33.3517

On the well-posedness of inhomogeneous hyperdissipative Navier-Stokes equations

Daoyuan Fang ^1, and Ruizhao Zi ^2,

1.	Department of Mathematics, Zhejiang University, Hangzhou 310027
2.	Department of Mathematics, Zhejiang University, Hangzhou, 310027, China

Received September 2012 Revised November 2012 Published January 2013

Abstract
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This paper is devoted to the study of the inhomogeneous hyperdissipative Navier-Stokes equations on the whole space $\mathbb{R}^N,N\geq3$. Compared with the classical inhomogeneous Navier-Stokes, the dissipative term $- Δ u$ here is replaced by $D^2u$, where $D$ is a Fourier multiplier whose symbol is $m(\xi)=|\xi|^{\frac{N+2}{4}}$. For arbitrary small positive constants $ε$ and $δ$, global well-posedness is showed for the data $(\rho_0, u_0)$ such that $(ρ_{0} - 1, u_0)∈ H^{\frac{N}{2}+ε} × H^{δ}$ with $\inf_{x\in \mathbb{R}^N}\rho_0>0$. To our best knowledge, this is the first result on the inhomogeneous hyperdissipative Navier-Stokes equations, and it can also be viewed as the high-dimensional generalization of the 2D result for classical inhomogeneous Navier-Stokes equations given by Danchin [Local and global well-posedness results for flows of inhomogeneous viscous fluids. Adv. Differential Equations 9 (2004), 353--386.]

Keywords: inhomogeneous, existence, uniqueness., Navier-Stokes equations, hyperdissipative.

Mathematics Subject Classification: Primary: 76D03, 76D0.

Citation: Daoyuan Fang, Ruizhao Zi. On the well-posedness of inhomogeneous hyperdissipative Navier-Stokes equations. Discrete and Continuous Dynamical Systems, 2013, 33 (8) : 3517-3541. doi: 10.3934/dcds.2013.33.3517

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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