On the Incompressible Navier-Stokes Equations with Damping

Wenyan Zhao, Zhibo Zheng

Department of Mathematics, Baoshan University, Baoshan, China.

DOI: 10.4236/am.2013.44089                

Abstract

We consider dynamics system with damping, which are obtained by some transformations from the system of incompressible Navier-Stokes equations. These have similar properties to original Navier-Stokes equations the scaling invariance. Due to the presence of the damping term, conclusions are different with proving the origin of the incompressible Navier-Stokes equations and get some new conclusions. For one form of dynamics system with damping we prove the existence of solution, and get the existence of the attractors. Moreover, we discuss with limit-behavior the deformations of the Navier-Stokes equation.

Keywords

Incompressible Navier-Stokes Equation; Solution; Maximal Attractor; Limit-Behavior

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	C. Foias and R. Teman, “Attractor Representing Tulent Flows,” Memoirs of Applied Mathematical Sciences, Vol. 53, No. 314, 1985.
[2]	C. Foias and R. Teman, “On the Dimension of the Attractors in Two-Demensional Turbulence,” Physica D, Vol. 30, No. 3, 1988, pp. 284-296. doi:10.1016/0167-2789(88)90022-X
[3]	C. Foias and R. Teman, “On the Large-Time Galerkin Approximation of the Navier-Stokes Equations,” SIAM Journal on Numerical Analysis, Vol. 21, No. 4, 1984, pp. 615-634. doi:10.1137/0721043
[4]	J. E. Marsden, L. Sirovich and F. John, “Infinite-Dimensional Dynamical Systems in Mechanics and Physics,” Applied Mathematical Sciences, Vol. 68, 1997, Springer Verlag, New York, pp. 15-25.
[5]	F. Abergel, “Attractor for a Navier-Stokes Flow in Unbounded Domain,” Mathematical Modelling and Numerical Analysis, Vol. 23, No. 3, 1989, pp. 359-370.
[6]	C. S. Zhao and K. T. Li, “The Global Attractor of N-S Equation with Linear Dampness on the Whole Two Dimensional Space and Estimates of Its Demensions,” ACTA Mathematical Application Sinica, Vol. 23, No. 1, 2000, pp. 90-96.
[7]	A. V. Babin and M. I. Vishik, “Maximal Attractors of Semigroups Corresponding to Evolution Differential Equations,” Mathematics of the USSR-Sbornik, Vol. 54, No. 2, 1986, pp. 387-408. doi:10.1070/SM1986v054n02ABEH002976
[8]	F. Abergel, “Existence and Finite Dimensionality of Global Attractor for Evolution Equations on Unbounded Domains,” Journal of Differential Equations, Vol. 83, No. 1, 1990, pp. 85-108. doi:10.1016/0022-0396(90)90070-6
[9]	R. S. Adms, “Sobolve Space,” Academic Press, New York, 1975.
[10]	A. Pazy, “Semigroups of Linear Operator and Application to Partial Differential Equation,” Applied Mathematical Sciences, Springer-Verlag, New York, 2006, pp. 1-38.