January  2019, 24(1): 363-386. doi: 10.3934/dcdsb.2018084

Long-time dynamics for a non-autonomous Navier-Stokes-Voigt equation in Lipschitz domains

1. 

College of Mathematics and Information Science, Henan Normal University, Xinxiang 453007, China

2. 

College of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu 611130, China

3. 

Instituto de Ciências Matemáticas e da Computação, Universidade de São Paulo-Campus de São Carlos, Caixa Postal 668, 13560-970 São Carlos SP, Brazil

4. 

Department of Mathematical Sciences, University of Cincinnati, Cincinnati, OH 45221, USA

5. 

Department of Applied and Computational Mathematics and Statistics, University of Notre Dame, Notre Dame, IN 46556, USA

* Corresponding author: Taige Wang

Received  October 2016 Revised  August 2017 Published  January 2019 Early access  March 2018

Fund Project: Xinguang Yang is supported by FAPESP, Grant 2014/17080-0, the Mainstay Fund from Henan Normal University, the Program for Science and Technology Innovation Talents in University of Henan Province (No. 142102210448); Baowei Feng is supported by National Natural Foundation of China (No. 11701465).

This article focuses on the optimal regularity and long-time dynamics of solutions of a Navier-Stoke-Voigt equation with non-autonomous body forces in non-smooth domains. Optimal regularity is considered, since the regularity $H_0^1\cap H^2$ cannot be achieved. Given the initial data in certain spaces, it can be shown that the problem generates a well-defined evolutionary process. Then we prove the existence of a uniform attractor consisting of complete trajectories.

Citation: Xinguang Yang, Baowei Feng, Thales Maier de Souza, Taige Wang. Long-time dynamics for a non-autonomous Navier-Stokes-Voigt equation in Lipschitz domains. Discrete and Continuous Dynamical Systems - B, 2019, 24 (1) : 363-386. doi: 10.3934/dcdsb.2018084
References:
[1]

R. M. BrownP. A. Perry and Z. Shen, On the dimension of the attractor of the non-homogeneous Navier-Stokes equations in non-smooth domains, Indiana Univ. Math. J., 49 (2000), 1-34. 

[2]

Y. P. CaoE. M. Lunasin and E. S. Titi, Global well-posedness of the three dimensional viscous and inviscid simplified Bardina turbulence models, Comm. Math. Sci., 4 (2006), 823-848.  doi: 10.4310/CMS.2006.v4.n4.a8.

[3]

A. O. ÇelebiV. K. Kalantarov and M. Polat, Global attractors for 2D Navier-Stokes-Voight equations in an unbounded domain, Appl. Anal., 88 (2009), 381-392.  doi: 10.1080/00036810902766682.

[4]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, Providence, RI: American Mathematical Society, 2002.

[5]

V. K. Kalantarov, Attractors for some nonlinear problems of mathematical physics, Zap. Nauchn. Sem. LOMI, 152 (1986), 50-54. 

[6]

V. K. Kalantarov, Global Behavior of Solutions of Nonlinear Equations of Mathematical Physics of Classical and Non-Classical Type, Postdoctoral Thesis, St. Petersburg, 1988.

[7]

V. K. KalantarovB. Levant and E. S. Titi, Gevrey regularity for the attractor of the 3D Navier-Stokes-Voight equations, J. Nonlinear Sci., 19 (2009), 133-152.  doi: 10.1007/s00332-008-9029-7.

[8]

V. K. Kalantarov and E. S. Titi, Global attractors and determining models for the 3D Navier-Stokes-Voight equations, Chinese Ann. Math. B, 30 (2009), 697-714.  doi: 10.1007/s11401-009-0205-3.

[9]

O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, New York: Gordon and Breach Science Publishers, 1969.

[10]

W. J. Layton and L. G. Rebholz, On relaxation times in the Navier-Stokes-Voigt model, Inter. J. Comp. Fluid Dyn., 27 (2013), 184-187.  doi: 10.1080/10618562.2013.766328.

[11]

S. Lu, Attractors for nonautonomous 2D Navier-Stokes equations with less regular normal forces, J. Diff. Equ., 230 (2006), 196-212.  doi: 10.1016/j.jde.2006.07.009.

[12]

S. LuH. Wu and C. Zhong, Attractors for nonautonomous $2D$ Navier-Stokes equations with normal external forces, Disc. Cont. Dyn. Syst., 13 (2005), 701-719.  doi: 10.3934/dcds.2005.13.701.

[13]

J. G. LuengoP. Marín-Rubio and J. Real, Pullback attractors for three-dimensional non-autonomous Navier-Stokes-Voigt equations, Nonlinearity, 25 (2012), 905-930.  doi: 10.1088/0951-7715/25/4/905.

[14]

Q. MaS. Wang and C. K. Zhong, Necessary and sufficient conditions for the existence of global attractors for semigroups and applications, Indiana Univ. Math. J., 51 (2002), 1542-1559.  doi: 10.1512/iumj.2002.51.2255.

[15]

A. Miranville and X. Wang, Upper bounded on the dimension of the attractor for nonhomogeneous Navier-Stokes equations, Disc. Cont. Dyn. Syst., 2 (1996), 95-110. 

[16]

A. Miranville and X. Wang, Attractors for non-autonomous nonhomogenerous Navier-Stokes equations, Nonlinearity, 10 (1997), 1047-1061.  doi: 10.1088/0951-7715/10/5/003.

[17]

A. P. Oskolkov, The uniqueness and solvability in the large of boundary value problems for the equations of motion of aqueous solutions of polymers, Zap. Nauchn. Sem., LOMI, 38 (1973), 98-136. 

[18]

Y. QinX. Yang and X. Liu, Averaging of 3D Navier-Stokes-Voigt equations with singularly oscillating force, Nonlinear Anal., RWA, 13 (2012), 893-904.  doi: 10.1016/j.nonrwa.2011.08.025.

[19]

J. C. Robinson, Infinite-dimensional Dynamical Systems-An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors, Cambridge Univ. Press, Cambridge, 2001.

[20]

J. C. Robinson, Attractors and finite-dimensional behaviour in the 2D Navier-Stokes equations, ISRN Math Anal., 203 (2013), Article ID 291823, 29pp.

[21]

R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Berlin: Springer, the 2nd editon, 1997.

[22]

R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, Studies in Mathematics and its Applications, Vol. 2. North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977.

[23]

W. Voigt, Ueber innere Reibung fester Krper, insbesondere der Metalle, Annalen der Physik, 283 (1892), 671-693. 

[24]

D. Wu and C. Zhong, The attractors for the nonhomogeneous nonautonomous Navier-Stokes equations, J. Math. Anal. Appl., 321 (2006), 426-444.  doi: 10.1016/j.jmaa.2005.08.044.

[25]

X. Yang, Y. Qin and T. F. Ma, Pullback attractors for the 2D non-autonomous incompressible Navier-Stokes equations with nonhomogeneous boundary on Lipschitz domain, submitted, 2016.

show all references

References:
[1]

R. M. BrownP. A. Perry and Z. Shen, On the dimension of the attractor of the non-homogeneous Navier-Stokes equations in non-smooth domains, Indiana Univ. Math. J., 49 (2000), 1-34. 

[2]

Y. P. CaoE. M. Lunasin and E. S. Titi, Global well-posedness of the three dimensional viscous and inviscid simplified Bardina turbulence models, Comm. Math. Sci., 4 (2006), 823-848.  doi: 10.4310/CMS.2006.v4.n4.a8.

[3]

A. O. ÇelebiV. K. Kalantarov and M. Polat, Global attractors for 2D Navier-Stokes-Voight equations in an unbounded domain, Appl. Anal., 88 (2009), 381-392.  doi: 10.1080/00036810902766682.

[4]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, Providence, RI: American Mathematical Society, 2002.

[5]

V. K. Kalantarov, Attractors for some nonlinear problems of mathematical physics, Zap. Nauchn. Sem. LOMI, 152 (1986), 50-54. 

[6]

V. K. Kalantarov, Global Behavior of Solutions of Nonlinear Equations of Mathematical Physics of Classical and Non-Classical Type, Postdoctoral Thesis, St. Petersburg, 1988.

[7]

V. K. KalantarovB. Levant and E. S. Titi, Gevrey regularity for the attractor of the 3D Navier-Stokes-Voight equations, J. Nonlinear Sci., 19 (2009), 133-152.  doi: 10.1007/s00332-008-9029-7.

[8]

V. K. Kalantarov and E. S. Titi, Global attractors and determining models for the 3D Navier-Stokes-Voight equations, Chinese Ann. Math. B, 30 (2009), 697-714.  doi: 10.1007/s11401-009-0205-3.

[9]

O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, New York: Gordon and Breach Science Publishers, 1969.

[10]

W. J. Layton and L. G. Rebholz, On relaxation times in the Navier-Stokes-Voigt model, Inter. J. Comp. Fluid Dyn., 27 (2013), 184-187.  doi: 10.1080/10618562.2013.766328.

[11]

S. Lu, Attractors for nonautonomous 2D Navier-Stokes equations with less regular normal forces, J. Diff. Equ., 230 (2006), 196-212.  doi: 10.1016/j.jde.2006.07.009.

[12]

S. LuH. Wu and C. Zhong, Attractors for nonautonomous $2D$ Navier-Stokes equations with normal external forces, Disc. Cont. Dyn. Syst., 13 (2005), 701-719.  doi: 10.3934/dcds.2005.13.701.

[13]

J. G. LuengoP. Marín-Rubio and J. Real, Pullback attractors for three-dimensional non-autonomous Navier-Stokes-Voigt equations, Nonlinearity, 25 (2012), 905-930.  doi: 10.1088/0951-7715/25/4/905.

[14]

Q. MaS. Wang and C. K. Zhong, Necessary and sufficient conditions for the existence of global attractors for semigroups and applications, Indiana Univ. Math. J., 51 (2002), 1542-1559.  doi: 10.1512/iumj.2002.51.2255.

[15]

A. Miranville and X. Wang, Upper bounded on the dimension of the attractor for nonhomogeneous Navier-Stokes equations, Disc. Cont. Dyn. Syst., 2 (1996), 95-110. 

[16]

A. Miranville and X. Wang, Attractors for non-autonomous nonhomogenerous Navier-Stokes equations, Nonlinearity, 10 (1997), 1047-1061.  doi: 10.1088/0951-7715/10/5/003.

[17]

A. P. Oskolkov, The uniqueness and solvability in the large of boundary value problems for the equations of motion of aqueous solutions of polymers, Zap. Nauchn. Sem., LOMI, 38 (1973), 98-136. 

[18]

Y. QinX. Yang and X. Liu, Averaging of 3D Navier-Stokes-Voigt equations with singularly oscillating force, Nonlinear Anal., RWA, 13 (2012), 893-904.  doi: 10.1016/j.nonrwa.2011.08.025.

[19]

J. C. Robinson, Infinite-dimensional Dynamical Systems-An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors, Cambridge Univ. Press, Cambridge, 2001.

[20]

J. C. Robinson, Attractors and finite-dimensional behaviour in the 2D Navier-Stokes equations, ISRN Math Anal., 203 (2013), Article ID 291823, 29pp.

[21]

R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Berlin: Springer, the 2nd editon, 1997.

[22]

R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, Studies in Mathematics and its Applications, Vol. 2. North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977.

[23]

W. Voigt, Ueber innere Reibung fester Krper, insbesondere der Metalle, Annalen der Physik, 283 (1892), 671-693. 

[24]

D. Wu and C. Zhong, The attractors for the nonhomogeneous nonautonomous Navier-Stokes equations, J. Math. Anal. Appl., 321 (2006), 426-444.  doi: 10.1016/j.jmaa.2005.08.044.

[25]

X. Yang, Y. Qin and T. F. Ma, Pullback attractors for the 2D non-autonomous incompressible Navier-Stokes equations with nonhomogeneous boundary on Lipschitz domain, submitted, 2016.

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