American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Random data Cauchy problem for the nonlinear Schrödinger equation with derivative nonlinearity
  • DCDS Home
  • This Issue
  • Next Article
    Global existence and general decay estimates for the viscoelastic equation with acoustic boundary conditions
December  2016, 36(12): 6921-6941. doi: 10.3934/dcds.2016101

Global well-posedness for inhomogeneous Navier-Stokes equations with logarithmical hyper-dissipation

Bin Han 1, and  Changhua Wei 2,

1. 

Department of Mathematics, Hangzhou Dianzi University, Hangzhou 310018, China
2. 

Department of Mathematics, Zhejiang Sci-Tech University, Hangzhou 310018, China

Received  October 2015 Revised  August 2016 Published  October 2016

This paper is devoted to studying the global well-posedness for 3D inhomogeneous logarithmical hyper-dissipative Navier-Stokes equations with dissipative terms $D^2u$. Here we consider the supercritical case, namely, the symbol of the Fourier multiplier $D$ takes the form $h(\xi)=|\xi|^{\frac{5}{4}}/g(\xi)$, where $g(\xi)=\log^{\frac{1}{4}}(2+|\xi|^2)$. This generalizes the work of Tao [17] to the inhomogeneous system, and can also be viewed as a generalization of Fang and Zi [12], in which they considered the critical case $h(\xi)=|\xi|^{\frac{5}{4}}$.
Keywords: Critical spaces, energy estimates., global solution.
Mathematics Subject Classification: 76D03, 76D0.
Citation: Bin Han, Changhua Wei. Global well-posedness for inhomogeneous Navier-Stokes equations with logarithmical hyper-dissipation. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 6921-6941. doi: 10.3934/dcds.2016101
References:
[1]

H. Bahouri, J. Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations,, Grundlehren der mathematischen Wissenschaften, (2011).  doi: 10.1007/978-3-642-16830-7.  Google Scholar

[2]

D. Barbato, F. Morandin and M. Romito, Global regularity for a slightly supercritical hyperdissipative Navier-Stokes system,, Analysis and PDE, 7 (2014), 2009.  doi: 10.2140/apde.2014.7.2009.  Google Scholar

[3]

J. M. Bony, Calcul symbolique et propagation des singularités pour équations aux dérivées partielles nonlinéaires,, Annales Scinentifiques de l'école Normale Supérieure, 14 (1981), 209.   Google Scholar

[4]

J. Y. Chemin, Localization in Fourier space and Navier-Stokes system,, Phase Space Analysis of partial Differential Equations, 1 (2004), 53.   Google Scholar

[5]

J. Y. Chemin and N. Lerner, Flot de champs de vecteurs non Lipschitziens et équations de Navier-Stokes,, J. Differential Equations, 121 (1995), 314.  doi: 10.1006/jdeq.1995.1131.  Google Scholar

[6]

R. Danchin, Density-dependent incompressible Viscous fluids in critical spaces,, Proc. Roy. Soc. Edinburgh Sect.A, 133 (2003), 1311.  doi: 10.1017/S030821050000295X.  Google Scholar

[7]

R. Danchin, The inviscid limit for density-dependent incompressible fluids,, Ann. Fac. Sci. Toulouse Math. Ser., 15 (2006), 637.  doi: 10.5802/afst.1133.  Google Scholar

[8]

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

[9]

R. Danchin, Well-posedness in critical spaces for barotropic viscous fluids with truly not constant density,, Communications in Partial Differential Equations, 32 (2007), 1373.  doi: 10.1080/03605300600910399.  Google Scholar

[10]

R. J. DiPerna and P. L. Lions, Ordinary differential equations transport theory and Sobolev spaces,, Invent. Math., 98 (1989), 511.  doi: 10.1007/BF01393835.  Google Scholar

[11]

H. Fujita and T. Kato, On the Navier-Stokes initial value problem I,, Arch. Roational Mech. Anal. 16 (1964), 16 (1964), 269.  doi: 10.1007/BF00276188.  Google Scholar

[12]

D. Fang and Rui. Z. Zi, On the well-posedness of inhomogeneous hyperdissipative Navier-Stokes equations,, Discrete and continuous Dynamical systems, 33 (2013), 3517.  doi: 10.3934/dcds.2013.33.3517.  Google Scholar

[13]

S. Itoh and A. Tani, Solvability of nonstationary problems for nonhomogeneous incompressible uids and the convergence with vanishing viscosity,, Tokyo Joural of Mathematics, 22 (1999), 17.  doi: 10.3836/tjm/1270041610.  Google Scholar

[14]

N. Katz and N. Pavlović, A cheap Caffarelli-Kohn-Nirenberg inequality for the Navier-Stokes equaiton with hyper-dissipation,, Geom. Funct. Anal., 12 (2002), 355.  doi: 10.1007/s00039-002-8250-z.  Google Scholar

[15]

P. L. Lions, Mathematical Topics in Fluid Mechanics. Vol. 1. Incompressible Models,, Oxford Lecture Series in Mathematics and its Applications, (1996).   Google Scholar

[16]

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.   Google Scholar

[17]

T. Tao, Global regularity for a logarithmically supercritical hyperdissipative Navier-Stokes equation,, Anal. PDE, 2 (2009), 361.  doi: 10.2140/apde.2009.2.361.  Google Scholar

show all references

References:
[1]

H. Bahouri, J. Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations,, Grundlehren der mathematischen Wissenschaften, (2011).  doi: 10.1007/978-3-642-16830-7.  Google Scholar

[2]

D. Barbato, F. Morandin and M. Romito, Global regularity for a slightly supercritical hyperdissipative Navier-Stokes system,, Analysis and PDE, 7 (2014), 2009.  doi: 10.2140/apde.2014.7.2009.  Google Scholar

[3]

J. M. Bony, Calcul symbolique et propagation des singularités pour équations aux dérivées partielles nonlinéaires,, Annales Scinentifiques de l'école Normale Supérieure, 14 (1981), 209.   Google Scholar

[4]

J. Y. Chemin, Localization in Fourier space and Navier-Stokes system,, Phase Space Analysis of partial Differential Equations, 1 (2004), 53.   Google Scholar

[5]

J. Y. Chemin and N. Lerner, Flot de champs de vecteurs non Lipschitziens et équations de Navier-Stokes,, J. Differential Equations, 121 (1995), 314.  doi: 10.1006/jdeq.1995.1131.  Google Scholar

[6]

R. Danchin, Density-dependent incompressible Viscous fluids in critical spaces,, Proc. Roy. Soc. Edinburgh Sect.A, 133 (2003), 1311.  doi: 10.1017/S030821050000295X.  Google Scholar

[7]

R. Danchin, The inviscid limit for density-dependent incompressible fluids,, Ann. Fac. Sci. Toulouse Math. Ser., 15 (2006), 637.  doi: 10.5802/afst.1133.  Google Scholar

[8]

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

[9]

R. Danchin, Well-posedness in critical spaces for barotropic viscous fluids with truly not constant density,, Communications in Partial Differential Equations, 32 (2007), 1373.  doi: 10.1080/03605300600910399.  Google Scholar

[10]

R. J. DiPerna and P. L. Lions, Ordinary differential equations transport theory and Sobolev spaces,, Invent. Math., 98 (1989), 511.  doi: 10.1007/BF01393835.  Google Scholar

[11]

H. Fujita and T. Kato, On the Navier-Stokes initial value problem I,, Arch. Roational Mech. Anal. 16 (1964), 16 (1964), 269.  doi: 10.1007/BF00276188.  Google Scholar

[12]

D. Fang and Rui. Z. Zi, On the well-posedness of inhomogeneous hyperdissipative Navier-Stokes equations,, Discrete and continuous Dynamical systems, 33 (2013), 3517.  doi: 10.3934/dcds.2013.33.3517.  Google Scholar

[13]

S. Itoh and A. Tani, Solvability of nonstationary problems for nonhomogeneous incompressible uids and the convergence with vanishing viscosity,, Tokyo Joural of Mathematics, 22 (1999), 17.  doi: 10.3836/tjm/1270041610.  Google Scholar

[14]

N. Katz and N. Pavlović, A cheap Caffarelli-Kohn-Nirenberg inequality for the Navier-Stokes equaiton with hyper-dissipation,, Geom. Funct. Anal., 12 (2002), 355.  doi: 10.1007/s00039-002-8250-z.  Google Scholar

[15]

P. L. Lions, Mathematical Topics in Fluid Mechanics. Vol. 1. Incompressible Models,, Oxford Lecture Series in Mathematics and its Applications, (1996).   Google Scholar

[16]

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.   Google Scholar

[17]

T. Tao, Global regularity for a logarithmically supercritical hyperdissipative Navier-Stokes equation,, Anal. PDE, 2 (2009), 361.  doi: 10.2140/apde.2009.2.361.  Google Scholar

[1]

Xinghong Pan, Jiang Xu. Global existence and optimal decay estimates of the compressible viscoelastic flows in $ L^p $ critical spaces. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 2021-2057. doi: 10.3934/dcds.2019085
[2]

Ruizhao Zi. Global solution in critical spaces to the compressible Oldroyd-B model with non-small coupling parameter. Discrete & Continuous Dynamical Systems - A, 2017, 37 (12) : 6437-6470. doi: 10.3934/dcds.2017279
[3]

Qing Chen, Zhong Tan. Global existence in critical spaces for the compressible magnetohydrodynamic equations. Kinetic & Related Models, 2012, 5 (4) : 743-767. doi: 10.3934/krm.2012.5.743
[4]

Minghua Yang, Zunwei Fu, Jinyi Sun. Global solutions to Chemotaxis-Navier-Stokes equations in critical Besov spaces. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3427-3460. doi: 10.3934/dcdsb.2018284
[5]

Jie Jiang. Global stability of Keller–Segel systems in critical Lebesgue spaces. Discrete & Continuous Dynamical Systems - A, 2020, 40 (1) : 609-634. doi: 10.3934/dcds.2020025
[6]

Joachim Krieger, Kenji Nakanishi, Wilhelm Schlag. Global dynamics of the nonradial energy-critical wave equation above the ground state energy. Discrete & Continuous Dynamical Systems - A, 2013, 33 (6) : 2423-2450. doi: 10.3934/dcds.2013.33.2423
[7]

Kenji Nakanishi, Tristan Roy. Global dynamics above the ground state for the energy-critical Schrödinger equation with radial data. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2023-2058. doi: 10.3934/cpaa.2016026
[8]

Laiqing Meng, Jia Yuan, Xiaoxin Zheng. Global existence of almost energy solution to the two-dimensional chemotaxis-Navier-Stokes equations with partial diffusion. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 3413-3441. doi: 10.3934/dcds.2019141
[9]

J. Huang, Marius Paicu. Decay estimates of global solution to 2D incompressible Navier-Stokes equations with variable viscosity. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4647-4669. doi: 10.3934/dcds.2014.34.4647
[10]

Hongjie Dong. Dissipative quasi-geostrophic equations in critical Sobolev spaces: Smoothing effect and global well-posedness. Discrete & Continuous Dynamical Systems - A, 2010, 26 (4) : 1197-1211. doi: 10.3934/dcds.2010.26.1197
[11]

Xiaoping Zhai, Yongsheng Li, Wei Yan. Global well-posedness for the 3-D incompressible MHD equations in the critical Besov spaces. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1865-1884. doi: 10.3934/cpaa.2015.14.1865
[12]

Guangyu Xu, Jun Zhou. Global existence and blow-up of solutions to a singular Non-Newton polytropic filtration equation with critical and supercritical initial energy. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1805-1820. doi: 10.3934/cpaa.2018086
[13]

Yuri Latushkin, Valerian Yurov. Stability estimates for semigroups on Banach spaces. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5203-5216. doi: 10.3934/dcds.2013.33.5203
[14]

Casey Jao. Energy-critical NLS with potentials of quadratic growth. Discrete & Continuous Dynamical Systems - A, 2018, 38 (2) : 563-587. doi: 10.3934/dcds.2018025
[15]

Jiecheng Chen, Dashan Fan, Lijing Sun. Asymptotic estimates for unimodular Fourier multipliers on modulation spaces. Discrete & Continuous Dynamical Systems - A, 2012, 32 (2) : 467-485. doi: 10.3934/dcds.2012.32.467
[16]

Annegret Glitzky. Energy estimates for electro-reaction-diffusion systems with partly fast kinetics. Discrete & Continuous Dynamical Systems - A, 2009, 25 (1) : 159-174. doi: 10.3934/dcds.2009.25.159
[17]

Dorina Mitrea, Irina Mitrea, Marius Mitrea, Lixin Yan. Coercive energy estimates for differential forms in semi-convex domains. Communications on Pure & Applied Analysis, 2010, 9 (4) : 987-1010. doi: 10.3934/cpaa.2010.9.987
[18]

Rowan Killip, Changxing Miao, Monica Visan, Junyong Zhang, Jiqiang Zheng. The energy-critical NLS with inverse-square potential. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3831-3866. doi: 10.3934/dcds.2017162
[19]

Yinbin Deng, Wentao Huang. Least energy solutions for fractional Kirchhoff type equations involving critical growth. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 1929-1954. doi: 10.3934/dcdss.2019126
[20]

Jaeyoung Byeon, Sungwon Cho, Junsang Park. On the location of a peak point of a least energy solution for Hénon equation. Discrete & Continuous Dynamical Systems - A, 2011, 30 (4) : 1055-1081. doi: 10.3934/dcds.2011.30.1055

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (52)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences