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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

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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

R. Danchin, Local and global well-posedness results for flows of inhomogeneous viscous fluids, Adv. Differential Equations, 9 (2004), 353-386.  Google Scholar

[10]

R. Danchin, The inviscid limit for density-dependent incompressible fluids, Ann. Fac. Sci. Toulouse Math., 15 (2006), 637-688.  Google Scholar

[11]

R. Danchin, Uniform estimates for transport-diffusion equations, J. Hyperbolic Differ. Equ., 4 (2007), 1-17. doi: 10.1142/S021989160700101X.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

B. Desjardins, Regularity results for two-dimensional flows of multiphase viscous fluids, Arch. Rational Mech. Anal., 137 (1997), 135-158. Google Scholar

[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.  Google Scholar

[16]

H. Fujita and T. Kato, On the Navier-Stokes initial value problems. I, Arch. Rational Mech. Anal., 16 (1964), 269-315.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

R. Danchin, Local and global well-posedness results for flows of inhomogeneous viscous fluids, Adv. Differential Equations, 9 (2004), 353-386.  Google Scholar

[10]

R. Danchin, The inviscid limit for density-dependent incompressible fluids, Ann. Fac. Sci. Toulouse Math., 15 (2006), 637-688.  Google Scholar

[11]

R. Danchin, Uniform estimates for transport-diffusion equations, J. Hyperbolic Differ. Equ., 4 (2007), 1-17. doi: 10.1142/S021989160700101X.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

B. Desjardins, Regularity results for two-dimensional flows of multiphase viscous fluids, Arch. Rational Mech. Anal., 137 (1997), 135-158. Google Scholar

[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.  Google Scholar

[16]

H. Fujita and T. Kato, On the Navier-Stokes initial value problems. I, Arch. Rational Mech. Anal., 16 (1964), 269-315.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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