# American Institute of Mathematical Sciences

December  2016, 9(6): 2181-2200. doi: 10.3934/dcdss.2016091

## Decay estimates with sharp rates of global solutions of nonlinear systems of fluid dynamics equations

 1 Department of Mathematics, Lehigh University, 14 East Packer Avenue, Bethlehem, Pennsylvania 18015

Received  June 2015 Revised  September 2016 Published  November 2016

Consider the Cauchy problems for the $n$-dimensional incompressible Navier-Stokes equations \begin{eqnarray*} \frac{\partial{\bf u}}{\partial t}-\alpha\triangle{\bf u}+({\bf u}\cdot\nabla){\bf u}+\nabla p={\bf f}({\bf x},t),\qquad {\bf u}({\bf x},0)={\bf u}_0({\bf x}). \end{eqnarray*} In this system, the dimension $n\geq 3$, ${\bf u}({\bf x},t)=(u_1({\bf x},t),u_2({\bf x},t),\cdots,u_n({\bf x},t))$ and ${\bf f}({\bf x},t)=(f_1({\bf x},t),f_2({\bf x},t),\cdots,f_n({\bf x},t))$ are real vector valued functions of ${\bf x}=(x_1,x_2,\cdots,x_n)$ and $t$. Additionally, $\alpha>0$ is a positive constant. Suppose that the initial function and the external force satisfy appropriate conditions.
The main purpose of this paper is to make complete use of the uniform energy estimates of the global smooth solutions and couple together a well known Gronwall's inequality to improve the Fourier splitting method to accomplish the decay estimates with sharp rates. The decay estimates with sharp rates of the global smooth solutions of the Cauchy problems for the $n$-dimensional magnetohydrodynamics equations may be established very similarly.
Citation: Linghai Zhang. Decay estimates with sharp rates of global solutions of nonlinear systems of fluid dynamics equations. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 2181-2200. doi: 10.3934/dcdss.2016091
##### References:
 [1] J.-Y. Chemin, About weak-strong uniqueness for the 3D incompressible Navier-Stokes system, Communications in Pure and Applied Mathematics, 64 (2011), 1587-1598. doi: 10.1002/cpa.20386.  Google Scholar [2] J.-Y. Chemin, I. Gallagher and M. Paicu, Global regularity for some classes of large solutions to the Navier-Stokes equations, Annals of Mathematics, 173 (2011), 983-1012. doi: 10.4007/annals.2011.173.2.9.  Google Scholar [3] B. Guo and L. Zhang, Decay of solutions to magnetohydrodynamics equations in two space dimensions, Proceedings of the Royal Society of London, Series A: Mathematical and Physical Sciences, 449 (1995), 79-91. doi: 10.1098/rspa.1995.0033.  Google Scholar [4] T. Hou, Z. Lei and C. Li, Global regularity of the three-dimensional axi-symmetric Navier-Stokes equations with anisotropic data, Communications in Partial Differential Equations, 33 (2008), 1622-1637. doi: 10.1080/03605300802108057.  Google Scholar [5] Z. Lei and F. Lin, Global mild solutions of Navier-Stokes equations, Communications in Pure and Applied Mathematics, 64 (2011), 1297-1304. doi: 10.1002/cpa.20361.  Google Scholar [6] Z. Lei, F. Lin and Y. Zhou, Structure of helicity and global solution of incompressible Navier-Stokes equations, Archive for Rational Mechanics and Analysis, 218 (2015), 1417-1430. doi: 10.1007/s00205-015-0884-8.  Google Scholar [7] F. Lin, A new proof of the Caffarelli-Kohn-Nirenberg theorem, Communications in Pure and Applied Mathematics, 51 (1998), 241-257. doi: 10.1002/(SICI)1097-0312(199803)51:3<241::AID-CPA2>3.0.CO;2-A.  Google Scholar [8] W. Peng and Y. Zhou, Global large solutions to incompressible Navier-Stokes equations with gravity, Mathematical Methods in Applied Sciences, 38 (2015), 590-597. doi: 10.1002/mma.3088.  Google Scholar [9] M. E. Schonbek, $L^2$ decay for weak solutions of the nonlinear Navier-Stokes equations, Archive for Rational Mechanics and Analysis, 88 (1985), 209-222. doi: 10.1007/BF00752111.  Google Scholar [10] M. E. Schonbek, Large time behaviour to the Navier-Stokes equations, Communications in Partial Differential Equations, 11 (1986), 733-763. doi: 10.1080/03605308608820443.  Google Scholar [11] R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Second Edition, Applied Mathematical Sciences, Volume 68, Spring-Verlag, New York 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar [12] R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, Reprint of the 1984 edition. AMS Chelsea Publishing, Providence, Rhode Island, 2001. doi: 10.1090/chel/343.  Google Scholar [13] G. Tian and Z. Xin, Gradient estimation on Navier-Stokes equations, Communications in Analysis and Geometry, 7 (1999), 221-257. doi: 10.4310/CAG.1999.v7.n2.a1.  Google Scholar [14] L. Zhang, Decay of solutions to 2-dimensional Navier-Stokes equations, Chinese Advances in Mathematics, 22 (1993), 469-472.  Google Scholar [15] L. Zhang, Decay estimates for the solutions of some nonlinear evolution equations, Journal of Differential Equations, 116 (1995), 31-58. doi: 10.1006/jdeq.1995.1028.  Google Scholar [16] L. Zhang, Sharp rate of decay of solutions to 2-dimensional Navier-Stokes equations, Communications in Partial Differential Equations, 20 (1995), 119-127. doi: 10.1080/03605309508821089.  Google Scholar

show all references

##### References:
 [1] J.-Y. Chemin, About weak-strong uniqueness for the 3D incompressible Navier-Stokes system, Communications in Pure and Applied Mathematics, 64 (2011), 1587-1598. doi: 10.1002/cpa.20386.  Google Scholar [2] J.-Y. Chemin, I. Gallagher and M. Paicu, Global regularity for some classes of large solutions to the Navier-Stokes equations, Annals of Mathematics, 173 (2011), 983-1012. doi: 10.4007/annals.2011.173.2.9.  Google Scholar [3] B. Guo and L. Zhang, Decay of solutions to magnetohydrodynamics equations in two space dimensions, Proceedings of the Royal Society of London, Series A: Mathematical and Physical Sciences, 449 (1995), 79-91. doi: 10.1098/rspa.1995.0033.  Google Scholar [4] T. Hou, Z. Lei and C. Li, Global regularity of the three-dimensional axi-symmetric Navier-Stokes equations with anisotropic data, Communications in Partial Differential Equations, 33 (2008), 1622-1637. doi: 10.1080/03605300802108057.  Google Scholar [5] Z. Lei and F. Lin, Global mild solutions of Navier-Stokes equations, Communications in Pure and Applied Mathematics, 64 (2011), 1297-1304. doi: 10.1002/cpa.20361.  Google Scholar [6] Z. Lei, F. Lin and Y. Zhou, Structure of helicity and global solution of incompressible Navier-Stokes equations, Archive for Rational Mechanics and Analysis, 218 (2015), 1417-1430. doi: 10.1007/s00205-015-0884-8.  Google Scholar [7] F. Lin, A new proof of the Caffarelli-Kohn-Nirenberg theorem, Communications in Pure and Applied Mathematics, 51 (1998), 241-257. doi: 10.1002/(SICI)1097-0312(199803)51:3<241::AID-CPA2>3.0.CO;2-A.  Google Scholar [8] W. Peng and Y. Zhou, Global large solutions to incompressible Navier-Stokes equations with gravity, Mathematical Methods in Applied Sciences, 38 (2015), 590-597. doi: 10.1002/mma.3088.  Google Scholar [9] M. E. Schonbek, $L^2$ decay for weak solutions of the nonlinear Navier-Stokes equations, Archive for Rational Mechanics and Analysis, 88 (1985), 209-222. doi: 10.1007/BF00752111.  Google Scholar [10] M. E. Schonbek, Large time behaviour to the Navier-Stokes equations, Communications in Partial Differential Equations, 11 (1986), 733-763. doi: 10.1080/03605308608820443.  Google Scholar [11] R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Second Edition, Applied Mathematical Sciences, Volume 68, Spring-Verlag, New York 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar [12] R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, Reprint of the 1984 edition. AMS Chelsea Publishing, Providence, Rhode Island, 2001. doi: 10.1090/chel/343.  Google Scholar [13] G. Tian and Z. Xin, Gradient estimation on Navier-Stokes equations, Communications in Analysis and Geometry, 7 (1999), 221-257. doi: 10.4310/CAG.1999.v7.n2.a1.  Google Scholar [14] L. Zhang, Decay of solutions to 2-dimensional Navier-Stokes equations, Chinese Advances in Mathematics, 22 (1993), 469-472.  Google Scholar [15] L. Zhang, Decay estimates for the solutions of some nonlinear evolution equations, Journal of Differential Equations, 116 (1995), 31-58. doi: 10.1006/jdeq.1995.1028.  Google Scholar [16] L. Zhang, Sharp rate of decay of solutions to 2-dimensional Navier-Stokes equations, Communications in Partial Differential Equations, 20 (1995), 119-127. doi: 10.1080/03605309508821089.  Google Scholar
 [1] J. Huang, Marius Paicu. Decay estimates of global solution to 2D incompressible Navier-Stokes equations with variable viscosity. Discrete & Continuous Dynamical Systems, 2014, 34 (11) : 4647-4669. doi: 10.3934/dcds.2014.34.4647 [2] Cheng Wang. Convergence analysis of Fourier pseudo-spectral schemes for three-dimensional incompressible Navier-Stokes equations. Electronic Research Archive, 2021, 29 (5) : 2915-2944. doi: 10.3934/era.2021019 [3] Kaitai Li, Yanren Hou. Fourier nonlinear Galerkin method for Navier-Stokes equations. Discrete & Continuous Dynamical Systems, 1996, 2 (4) : 497-524. doi: 10.3934/dcds.1996.2.497 [4] 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 [5] Ben-Yu Guo, Yu-Jian Jiao. Mixed generalized Laguerre-Fourier spectral method for exterior problem of Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2009, 11 (2) : 315-345. doi: 10.3934/dcdsb.2009.11.315 [6] Yingshan Chen, Shijin Ding, Wenjun Wang. Global existence and time-decay estimates of solutions to the compressible Navier-Stokes-Smoluchowski equations. Discrete & Continuous Dynamical Systems, 2016, 36 (10) : 5287-5307. doi: 10.3934/dcds.2016032 [7] Marco Cappiello, Fabio Nicola. Sharp decay estimates and smoothness for solutions to nonlocal semilinear equations. Discrete & Continuous Dynamical Systems, 2016, 36 (4) : 1869-1880. doi: 10.3934/dcds.2016.36.1869 [8] Hyukjin Kwean. Kwak transformation and Navier-Stokes equations. Communications on Pure & Applied Analysis, 2004, 3 (3) : 433-446. doi: 10.3934/cpaa.2004.3.433 [9] Qi S. Zhang. An example of large global smooth solution of 3 dimensional Navier-Stokes equations without pressure. Discrete & Continuous Dynamical Systems, 2013, 33 (11&12) : 5521-5523. doi: 10.3934/dcds.2013.33.5521 [10] Mohamad Rachid. Incompressible Navier-Stokes-Fourier limit from the Landau equation. Kinetic & Related Models, 2021, 14 (4) : 599-638. doi: 10.3934/krm.2021017 [11] Xue-Li Song, Yan-Ren Hou. Attractors for the three-dimensional incompressible Navier-Stokes equations with damping. Discrete & Continuous Dynamical Systems, 2011, 31 (1) : 239-252. doi: 10.3934/dcds.2011.31.239 [12] Mehdi Badra, Fabien Caubet, Jérémi Dardé. Stability estimates for Navier-Stokes equations and application to inverse problems. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2379-2407. doi: 10.3934/dcdsb.2016052 [13] Hi Jun Choe, Hyea Hyun Kim, Do Wan Kim, Yongsik Kim. Meshless method for the stationary incompressible Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2001, 1 (4) : 495-526. doi: 10.3934/dcdsb.2001.1.495 [14] Hi Jun Choe, Do Wan Kim, Yongsik Kim. Meshfree method for the non-stationary incompressible Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2006, 6 (1) : 17-39. doi: 10.3934/dcdsb.2006.6.17 [15] Keyan Wang. On global regularity of incompressible Navier-Stokes equations in $\mathbf R^3$. Communications on Pure & Applied Analysis, 2009, 8 (3) : 1067-1072. doi: 10.3934/cpaa.2009.8.1067 [16] Lukáš Poul. Existence of weak solutions to the Navier-Stokes-Fourier system on Lipschitz domains. Conference Publications, 2007, 2007 (Special) : 834-843. doi: 10.3934/proc.2007.2007.834 [17] Peter Anthony, Sergey Zelik. Infinite-energy solutions for the Navier-Stokes equations in a strip revisited. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1361-1393. doi: 10.3934/cpaa.2014.13.1361 [18] Bum Ja Jin, Kyungkeun Kang. Caccioppoli type inequality for non-Newtonian Stokes system and a local energy inequality of non-Newtonian Navier-Stokes equations without pressure. Discrete & Continuous Dynamical Systems, 2017, 37 (9) : 4815-4834. doi: 10.3934/dcds.2017207 [19] Daniel Pardo, José Valero, Ángel Giménez. Global attractors for weak solutions of the three-dimensional Navier-Stokes equations with damping. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3569-3590. doi: 10.3934/dcdsb.2018279 [20] 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

2020 Impact Factor: 2.425