# American Institute of Mathematical Sciences

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 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.
 [1] Giovanni P. Galdi. Existence and uniqueness of time-periodic solutions to the Navier-Stokes equations in the whole plane. Discrete and 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 and 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 and 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 and 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 and Control Theory, 2012, 1 (1) : 217-234. doi: 10.3934/eect.2012.1.217 [6] Zaihong Jiang, Li Li, Wenbo Lu. Existence of axisymmetric and homogeneous solutions of Navier-Stokes equations in cone regions. Discrete and Continuous Dynamical Systems - S, 2021, 14 (12) : 4231-4258. doi: 10.3934/dcdss.2021126 [7] Bin Han, Changhua Wei. Global well-posedness for inhomogeneous Navier-Stokes equations with logarithmical hyper-dissipation. Discrete and Continuous Dynamical Systems, 2016, 36 (12) : 6921-6941. doi: 10.3934/dcds.2016101 [8] 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 and Continuous Dynamical Systems - S, 2010, 3 (2) : 199-219. doi: 10.3934/dcdss.2010.3.199 [9] Ariane Piovezan Entringer, José Luiz Boldrini. A phase field $\alpha$-Navier-Stokes vesicle-fluid interaction model: Existence and uniqueness of solutions. Discrete and Continuous Dynamical Systems - B, 2015, 20 (2) : 397-422. doi: 10.3934/dcdsb.2015.20.397 [10] Tomás Caraballo, Xiaoying Han. A survey on Navier-Stokes models with delays: Existence, uniqueness and asymptotic behavior of solutions. Discrete and Continuous Dynamical Systems - S, 2015, 8 (6) : 1079-1101. doi: 10.3934/dcdss.2015.8.1079 [11] Pavel I. Plotnikov, Jan Sokolowski. Compressible Navier-Stokes equations. Conference Publications, 2009, 2009 (Special) : 602-611. doi: 10.3934/proc.2009.2009.602 [12] Jan W. Cholewa, Tomasz Dlotko. Fractional Navier-Stokes equations. Discrete and Continuous Dynamical Systems - B, 2018, 23 (8) : 2967-2988. doi: 10.3934/dcdsb.2017149 [13] Reinhard Farwig, Yasushi Taniuchi. Uniqueness of backward asymptotically almost periodic-in-time solutions to Navier-Stokes equations in unbounded domains. Discrete and Continuous Dynamical Systems - S, 2013, 6 (5) : 1215-1224. doi: 10.3934/dcdss.2013.6.1215 [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 and 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 and Applied Analysis, 2004, 3 (3) : 353-366. doi: 10.3934/cpaa.2004.3.353 [16] Stefano Scrobogna. Global existence and convergence of nondimensionalized incompressible Navier-Stokes equations in low Froude number regime. Discrete and Continuous Dynamical Systems, 2020, 40 (9) : 5471-5511. doi: 10.3934/dcds.2020235 [17] 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 and Applied Analysis, 2014, 13 (4) : 1613-1627. doi: 10.3934/cpaa.2014.13.1613 [18] Wojciech M. Zajączkowski. Long time existence of regular solutions to non-homogeneous Navier-Stokes equations. Discrete and Continuous Dynamical Systems - S, 2013, 6 (5) : 1427-1455. doi: 10.3934/dcdss.2013.6.1427 [19] 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 and Continuous Dynamical Systems, 2014, 34 (10) : 4323-4341. doi: 10.3934/dcds.2014.34.4323 [20] Zhilei Liang, Jiangyu Shuai. Existence of strong solution for the Cauchy problem of fully compressible Navier-Stokes equations in two dimensions. Discrete and Continuous Dynamical Systems - B, 2021, 26 (10) : 5383-5405. doi: 10.3934/dcdsb.2020348

2020 Impact Factor: 1.392