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

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

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.]
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]

Comm. Pure Appl. Math., 64 (2011), 832-881. doi: 10.1002/cpa.20351.  Google Scholar

[2]

Arch. Rational Mech. Anal., 204 (2012), 189-230. doi: 10.1007/s00205-011-0473-4.  Google Scholar

[3]

Studies in Mathematics and its Applications, 22, North-Holland Publishing Co., Amsterdam, 1990.  Google Scholar

[4]

Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 343, Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-16830-7.  Google Scholar

[5]

Annales Scinentifiques de l'École Normale Supérieure, 14 (1981), 209-246.  Google Scholar

[6]

Vol. I, Pubbl. Cent. Ric. Mat. Ennio Giorgi, Scuola Norm. Sup., Pisa, (2004), 53-135.  Google Scholar

[7]

Journal of Differential Equations, 121 (1995), 314-228. doi: 10.1006/jdeq.1995.1131.  Google Scholar

[8]

Proc. Roy. Soc. Edinburgh Sect. A, 133 (2003), 1311-1334. doi: 10.1017/S030821050000295X.  Google Scholar

[9]

Adv. Differential Equations, 9 (2004), 353-386.  Google Scholar

[10]

Ann. Fac. Sci. Toulouse Math., 15 (2006), 637-688.  Google Scholar

[11]

J. Hyperbolic Differ. Equ., 4 (2007), 1-17. doi: 10.1142/S021989160700101X.  Google Scholar

[12]

Differential Integral Equations, 10 (1997), 587-598.  Google Scholar

[13]

Differential Integral Equations, 10 (1997), 577-586.  Google Scholar

[14]

Arch. Rational Mech. Anal., 137 (1997), 135-158. Google Scholar

[15]

Invent. Math., 98 (1989), 511-547. doi: 10.1007/BF01393835.  Google Scholar

[16]

Arch. Rational Mech. Anal., 16 (1964), 269-315.  Google Scholar

[17]

J. Funct. Anal., 261 (2011), 3181-3210. doi: 10.1016/j.jfa.2011.07.026.  Google Scholar

[18]

Chin. Ann. Math. Ser. B, 30 (2009), 607-630. doi: 10.1007/s11401-009-0027-3.  Google Scholar

[19]

Tokyo Joural of Mathematics, 22 (1999), 17-42. doi: 10.3836/tjm/1270041610.  Google Scholar

[20]

Geom. Funct. Anal., 12 (2002), 355-379. doi: 10.1007/s00039-002-8250-z.  Google Scholar

[21]

Journal of Soviet Mathematics, 9 (1978), 697-749.  Google Scholar

[22]

Oxford Lecture Series in Mathematics and its Applications, 3, Oxford University Press, The Clarendon Press, Oxford University Press, New York, 1996.  Google Scholar

[23]

Anal. PDE, 2 (2009), 361-366. doi: 10.2140/apde.2009.2.361.  Google Scholar

show all references

References:
[1]

Comm. Pure Appl. Math., 64 (2011), 832-881. doi: 10.1002/cpa.20351.  Google Scholar

[2]

Arch. Rational Mech. Anal., 204 (2012), 189-230. doi: 10.1007/s00205-011-0473-4.  Google Scholar

[3]

Studies in Mathematics and its Applications, 22, North-Holland Publishing Co., Amsterdam, 1990.  Google Scholar

[4]

Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 343, Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-16830-7.  Google Scholar

[5]

Annales Scinentifiques de l'École Normale Supérieure, 14 (1981), 209-246.  Google Scholar

[6]

Vol. I, Pubbl. Cent. Ric. Mat. Ennio Giorgi, Scuola Norm. Sup., Pisa, (2004), 53-135.  Google Scholar

[7]

Journal of Differential Equations, 121 (1995), 314-228. doi: 10.1006/jdeq.1995.1131.  Google Scholar

[8]

Proc. Roy. Soc. Edinburgh Sect. A, 133 (2003), 1311-1334. doi: 10.1017/S030821050000295X.  Google Scholar

[9]

Adv. Differential Equations, 9 (2004), 353-386.  Google Scholar

[10]

Ann. Fac. Sci. Toulouse Math., 15 (2006), 637-688.  Google Scholar

[11]

J. Hyperbolic Differ. Equ., 4 (2007), 1-17. doi: 10.1142/S021989160700101X.  Google Scholar

[12]

Differential Integral Equations, 10 (1997), 587-598.  Google Scholar

[13]

Differential Integral Equations, 10 (1997), 577-586.  Google Scholar

[14]

Arch. Rational Mech. Anal., 137 (1997), 135-158. Google Scholar

[15]

Invent. Math., 98 (1989), 511-547. doi: 10.1007/BF01393835.  Google Scholar

[16]

Arch. Rational Mech. Anal., 16 (1964), 269-315.  Google Scholar

[17]

J. Funct. Anal., 261 (2011), 3181-3210. doi: 10.1016/j.jfa.2011.07.026.  Google Scholar

[18]

Chin. Ann. Math. Ser. B, 30 (2009), 607-630. doi: 10.1007/s11401-009-0027-3.  Google Scholar

[19]

Tokyo Joural of Mathematics, 22 (1999), 17-42. doi: 10.3836/tjm/1270041610.  Google Scholar

[20]

Geom. Funct. Anal., 12 (2002), 355-379. doi: 10.1007/s00039-002-8250-z.  Google Scholar

[21]

Journal of Soviet Mathematics, 9 (1978), 697-749.  Google Scholar

[22]

Oxford Lecture Series in Mathematics and its Applications, 3, Oxford University Press, The Clarendon Press, Oxford University Press, New York, 1996.  Google Scholar

[23]

Anal. PDE, 2 (2009), 361-366. doi: 10.2140/apde.2009.2.361.  Google Scholar

[1]

Giovanni P. Galdi. Existence and uniqueness of time-periodic solutions to the Navier-Stokes equations in the whole plane. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1237-1257. doi: 10.3934/dcdss.2013.6.1237

[2]

G. Deugoué, T. Tachim Medjo. The Stochastic 3D globally modified Navier-Stokes equations: Existence, uniqueness and asymptotic behavior. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2593-2621. doi: 10.3934/cpaa.2018123

[3]

Daoyuan Fang, Bin Han, Matthias Hieber. Local and global existence results for the Navier-Stokes equations in the rotational framework. Communications on Pure & Applied Analysis, 2015, 14 (2) : 609-622. doi: 10.3934/cpaa.2015.14.609

[4]

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

[5]

Reinhard Racke, Jürgen Saal. Hyperbolic Navier-Stokes equations II: Global existence of small solutions. Evolution Equations & Control Theory, 2012, 1 (1) : 217-234. doi: 10.3934/eect.2012.1.217

[6]

Bin Han, Changhua Wei. Global well-posedness for inhomogeneous Navier-Stokes equations with logarithmical hyper-dissipation. Discrete & Continuous Dynamical Systems, 2016, 36 (12) : 6921-6941. doi: 10.3934/dcds.2016101

[7]

Luigi C. Berselli. An elementary approach to the 3D Navier-Stokes equations with Navier boundary conditions: Existence and uniqueness of various classes of solutions in the flat boundary case.. Discrete & Continuous Dynamical Systems - S, 2010, 3 (2) : 199-219. doi: 10.3934/dcdss.2010.3.199

[8]

Ariane Piovezan Entringer, José Luiz Boldrini. A phase field $\alpha$-Navier-Stokes vesicle-fluid interaction model: Existence and uniqueness of solutions. Discrete & Continuous Dynamical Systems - B, 2015, 20 (2) : 397-422. doi: 10.3934/dcdsb.2015.20.397

[9]

Tomás Caraballo, Xiaoying Han. A survey on Navier-Stokes models with delays: Existence, uniqueness and asymptotic behavior of solutions. Discrete & Continuous Dynamical Systems - S, 2015, 8 (6) : 1079-1101. doi: 10.3934/dcdss.2015.8.1079

[10]

Pavel I. Plotnikov, Jan Sokolowski. Compressible Navier-Stokes equations. Conference Publications, 2009, 2009 (Special) : 602-611. doi: 10.3934/proc.2009.2009.602

[11]

Jan W. Cholewa, Tomasz Dlotko. Fractional Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 2967-2988. doi: 10.3934/dcdsb.2017149

[12]

Reinhard Farwig, Yasushi Taniuchi. Uniqueness of backward asymptotically almost periodic-in-time solutions to Navier-Stokes equations in unbounded domains. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1215-1224. doi: 10.3934/dcdss.2013.6.1215

[13]

Zhilei Liang, Jiangyu Shuai. Existence of strong solution for the Cauchy problem of fully compressible Navier-Stokes equations in two dimensions. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020348

[14]

Yuming Qin, Lan Huang, Zhiyong Ma. Global existence and exponential stability in $H^4$ for the nonlinear compressible Navier-Stokes equations. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1991-2012. doi: 10.3934/cpaa.2009.8.1991

[15]

Joel Avrin. Global existence and regularity for the Lagrangian averaged Navier-Stokes equations with initial data in $H^{1//2}$. Communications on Pure & Applied Analysis, 2004, 3 (3) : 353-366. doi: 10.3934/cpaa.2004.3.353

[16]

Jian-Guo Liu, Zhaoyun Zhang. Existence of global weak solutions of $ p $-Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021051

[17]

Stefano Scrobogna. Global existence and convergence of nondimensionalized incompressible Navier-Stokes equations in low Froude number regime. Discrete & Continuous Dynamical Systems, 2020, 40 (9) : 5471-5511. doi: 10.3934/dcds.2020235

[18]

Michele Campiti, Giovanni P. Galdi, Matthias Hieber. Global existence of strong solutions for $2$-dimensional Navier-Stokes equations on exterior domains with growing data at infinity. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1613-1627. doi: 10.3934/cpaa.2014.13.1613

[19]

Wojciech M. Zajączkowski. Long time existence of regular solutions to non-homogeneous Navier-Stokes equations. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1427-1455. doi: 10.3934/dcdss.2013.6.1427

[20]

Takeshi Taniguchi. The existence and decay estimates of the solutions to $3$D stochastic Navier-Stokes equations with additive noise in an exterior domain. Discrete & Continuous Dynamical Systems, 2014, 34 (10) : 4323-4341. doi: 10.3934/dcds.2014.34.4323

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (61)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]