On the long-time behavior for a damped Navier-Stokes-Bardina model

In this paper, we consider a damped Navier-Stokes-Bardinamodel posed on the whole three-dimensional. These equations have an important physical motivation and they arise from some oceanic model. From the mathematical point of view, they write down as the well-know Navier-Stokes equations with an additional nonlocal operator in their nonlinear transport term, and moreover, with an additional damping term depending of a parameter β > 0. We study first the existence and uniqueness of global in time weak solutions in the energy space. Thereafter, our main objective is to describe the long time behavior of these solutions. For this, we use some tools in the theory of dynamical systems to prove the existence of a global attractor, which is compact subset in the energy space attracting all the weak solutions when the time goes to infinity. Moreover, we derive an upper bound for the fractal dimension of the global attractor associated to these equations. Finally, we find a range of values for the damping parameter β > 0, where we are able to give an acutely description of the internal structure of the global attractor. More precisely, we prove that the global attractor only contains the stationary (time-independing) solution of the damped Navier-StokesBardina equations. manuel.cortez@epn.edu.ec corresponding author: oscar.jarrin@udla.edu.ec

[1]  Nikolaus A. Adams,et al.  Deconvolution Methods for Subgrid-Scale Approximation in Large-Eddy Simulation , 2001 .

[2]  Alexei Ilyin,et al.  A modified-Leray-α subgrid scale model of turbulence , 2006 .

[3]  F. Ramos,et al.  Inviscid limit for damped and driven incompressible , 2007 .

[4]  J. Holton.  Geophysical fluid dynamics. , 1983, Science.

[5]  Loukas Grafakos,et al.  Modern Fourier Analysis , 2008 .

[6]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[7]  William Layton,et al.  On a well-posed turbulence model , 2005 .

[8]  Hui Liu,et al.  Decay of solutions for the 3D Navier-Stokes equations with damping , 2017, Appl. Math. Lett..

[9]  P. Lemarié–Rieusset,et al.  On the Kolmogorov Dissipation Law in a Damped Navier–Stokes Equation , 2019, Journal of Dynamics and Differential Equations.

[10]  William J. Layton,et al.  A simple and stable scale-similarity model for large Eddy simulation: Energy balance and existence of weak solutions , 2003, Appl. Math. Lett..

[11]  R. Temam,et al.  Navier-Stokes equations: theory and numerical analysis: R. Teman North-Holland, Amsterdam and New York. 1977. 454 pp. US $45.00 , 1978 .

[12]  S. Zelik,et al.  Upper bounds for the attractor dimension of damped Navier-Stokes equations in $\mathbb R^2$ , 2015, 1503.03415.

[13]  R. Temam.  Infinite Dimensional Dynamical Systems in Mechanics and Physics Springer Verlag , 1993 .

[14]  V. Chepyzhov,et al.  On the fractal dimension of invariant sets; applications to Navier-Stokes equations. , 2003 .

[15]  Peter Constantin,et al.  Global Lyapunov Exponents, Kaplan-Yorke Formulas and the Dimension of the Attractors for 2D Navier-Stokes Equations , 1985 .

[16]  F. Chow,et al.  Mountain Weather Research and Forecasting: Recent Progress and Current Challenges , 2013 .

[17]  J. Ferziger,et al.  Improved subgrid-scale models for large-eddy simulation , 1980 .

[18]  Hantaek Bae.  Navier-Stokes equations , 1992 .

[19]  S. Nodari,et al.  Orbital Stability: Analysis Meets Geometry , 2014, 1407.5951.

[20]  Lieb-Thirring Inequalities , 2000, math-ph/0003039.

[21]  P. Lemarié–Rieusset.  The Navier-Stokes Problem in the 21st Century , 2016 .

[22]  E. S. Titi,et al.  Global well-posedness of the three-dimensional viscous and inviscid simplified Bardina turbulence models , 2006 .

[23]  L. Berselli,et al.  On the Bardina’s Model in the Whole Space , 2017, 1709.06425.

[24]  T. Coakley,et al.  Turbulence Modeling Validation, Testing, and Development , 1997 .