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 and 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[14]

L. Zhang, Decay of solutions to 2-dimensional Navier-Stokes equations, Chinese Advances in Mathematics, 22 (1993), 469-472.

[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.

[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.

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[14]

L. Zhang, Decay of solutions to 2-dimensional Navier-Stokes equations, Chinese Advances in Mathematics, 22 (1993), 469-472.

[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.

[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.

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