On attractor’s dimensions of the modified Leray-alpha equation. (English) Zbl 1509.35241
Summary: The primary objective of this paper is to investigate the modified Leray-alpha equation on the two-dimensional sphere \(\mathbb{S}^2\), the square torus \(\mathbb{T}^2\) and the three-torus \(\mathbb{T}^3\). In the strategy, we prove the existence and the uniqueness of the weak solutions and also the existence of the global attractor for the equation. Then we establish the upper and lower bounds of the Hausdorff and fractal dimensions of the global attractor on both \(\mathbb{S}^2\) and \(\mathbb{T}^2\). Our method is based on the estimates for the vorticity scalar equations and the stationary solutions around the invariant manifold that are constructed by using the Kolmogorov flows. Finally, we will use the results on \(\mathbb{T}^2\) to study the lower bound for attractor’s dimensions in the case of \(\mathbb{T}^3\).
MSC:
| 35Q35 | PDEs in connection with fluid mechanics |
| 76D05 | Navier-Stokes equations for incompressible viscous fluids |
| 76F99 | Turbulence |
| 35B41 | Attractors |
| 35D30 | Weak solutions to PDEs |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 35A02 | Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 37L30 | Attractors and their dimensions, Lyapunov exponents for infinite-dimensional dissipative dynamical systems |
| 28A80 | Fractals |
| 35R01 | PDEs on manifolds |
Keywords:
modified Leray-alpha equation; 2-dimensional sphere; square torus; three-torus; global attractor; Hausdorff (fractal) dimension; Kolmogorov flowsReferences:
| [1] | H.T. Banks, Applied Functional Analysis, Lecture Notes, Spring, 2010, https://projects.ncsu.edu/crsc//htbanks/FA2010-3. pdf. · Zbl 1218.46002 |
| [2] | C. Bjorland and M.E. Schonbek, On questions of decay and existence for the viscous Camassa-Holm equations, Ann. I. H. Poincaré -AN 25 (2008), 907-936. · Zbl 1156.35323 |
| [3] | Y. Cao, E.M. Lunasin and E.S. Titi, Global well-posdness of the three-dimensional viscous and inviscid viscous Camassa-Holm turbulence models, Comm. Math. Sci. 4 (2006), 823-848. doi:10.4310/CMS.2006.v4.n4.a8. · Zbl 1127.35034 · doi:10.4310/CMS.2006.v4.n4.a8 |
| [4] | V.V. Chepyzhov and A.A. Ilyin, A note on the fractal dimension of attractors of dissipative dynamical systems, Nonlinear Anal. 44(6) (2001), 811-819. doi:10.1016/S0362-546X(99)00309-0. · Zbl 1153.37438 · doi:10.1016/S0362-546X(99)00309-0 |
| [5] | V.V. Chepyzhov and A.A. Ilyin, On the fractal dimension of invariant sets: Applications to Navier-Stokes equations, Discrete Contin. Dyn. Syst. 10(1-2) (2004), 117-135. doi:10.3934/dcds.2004.10.117. · Zbl 1049.37047 · doi:10.3934/dcds.2004.10.117 |
| [6] | A. Cheskidov, D.D. Holm, E. Olson and E.S. Titi, On a Leray-α model of turbulence, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 461 (2005), 629-649. · Zbl 1145.76386 |
| [7] | P. Constantin, C. Foias and R. Temam, On the dimension of the attractors in two-dimensional turbulence, Physical D 30 (1988), 284-296. doi:10.1016/0167-2789(88)90022-X. · Zbl 0658.58030 · doi:10.1016/0167-2789(88)90022-X |
| [8] | C. Foias, D.D. Holm and E.S. Titi, The three dimensional viscous Camassa-Holm equations, and their relation to the Navier-Stokes equations and turbulence theory, J. Dyn. Dif. Equ. 14 (2002), 1-35. doi:10.1023/A:1012984210582. · Zbl 0995.35051 · doi:10.1023/A:1012984210582 |
| [9] | M.A. Hamed, Y. Guo and E.S. Titi, Inertial manifolds for certian sub-grid scale alpha models of turbulence, SIAM Journal on Applied Dynamical Systems (2014). |
| [10] | A.A. Ilyin, The Navier-Stokes and Euler equations on two-dimensional closed manifolds, Mat. Sb. 181 (1990), 521-539; · Zbl 0713.35074 |
| [11] | English transl. in Math. USSR Sb. 69 (1991). |
| [12] | A.A. Ilyin, On the dimension of attractors for Navier-Stokes equations on two-dimensional compact manifolds, Differen-tial and Integral Equations 6(1) (1993), 183-214. · Zbl 0771.35045 |
| [13] | A.A. Ilyin, Navier-Stokes equations on the rotating sphere. A simple proof of the attractor dimension estimate, Nonlin-earity 7 (1999), 31-39. doi:10.1088/0951-7715/7/1/002. · Zbl 0840.35077 · doi:10.1088/0951-7715/7/1/002 |
| [14] | A.A. Ilyin, E.M. Lunasin and E.S. Titi, A modified-Leray-α subgrid scale model of turbulence, Nonlinearity 19 (2006), 879-897. doi:10.1088/0951-7715/19/4/006. · Zbl 1106.35050 · doi:10.1088/0951-7715/19/4/006 |
| [15] | A.A. Ilyin, A. Miranville and E.S. Titi, Small viscosity sharp estimates for the global attractor of the 2-D damped-driven Navier-Stokes equations, Comm. Math. Sci 2(3) (2004), 403-426. doi:10.4310/CMS.2004.v2.n3.a4. · Zbl 1084.35058 · doi:10.4310/CMS.2004.v2.n3.a4 |
| [16] | A.A. Ilyin and E.S. Titi, Attractors for two-dimensional Navier-Stokes-α model: An α-dependence study, Journal of Dyn. dif. equ. 15(4) (2004), 751-778. doi:10.1023/B:JODY.0000010064.06851.ff. · Zbl 1039.35078 · doi:10.1023/B:JODY.0000010064.06851.ff |
| [17] | A.A. Ilyin and S.V. Zelik, Sharp dimension estimates of the attractor of the damped 2-D Euler-Bardina equations, in: Partial Differential Equation, Spectral Theory and Mathematical Physics, 2020, arXiv:2011.00607. |
| [18] | A.A. Ilyin, S.V. Zelik and A. Kostiano, Sharp upper and lower bounds of the attractor dimension for 3D damped Euler-Bardina equations, Physica D: Nonlinear Phenomena 432 (2022), 133156. doi:10.1016/j.physd.2022.133156. · Zbl 1490.35273 · doi:10.1016/j.physd.2022.133156 |
| [19] | A. Kostianko, Inertial manifolds for the 3D modified-Leray-α model with periodic boundary conditions, Journal of Dy-namics and Differential Equations 30 (2018), 1-24. doi:10.1007/s10884-017-9635-x. · Zbl 1390.35023 · doi:10.1007/s10884-017-9635-x |
| [20] | W. Layton and R. Lewandowski, On a well-posed turbulence model, Dicrete and Continuous Dyn. Sys. B 6 (2006), 111-128. · Zbl 1089.76028 |
| [21] | X. Li and C. Sun, Inertial manifolds for the 3D modified-Leray-α model, Journal of Differential Equations 268(4) (2020), 1532-1569. doi:10.1016/j.jde.2019.09.001. · Zbl 1433.35239 · doi:10.1016/j.jde.2019.09.001 |
| [22] | V.X. Liu, A sharp lower bound for the Hausdorff dimension of the global attractors of the 2D Navier-Stokes equations, Comm. Math. Phys. 158 (1993), 327-339. doi:10.1007/BF02108078. · Zbl 0790.35085 · doi:10.1007/BF02108078 |
| [23] | J.E. Marsden and S. Shkoller, Global well-posedness for the Lagrangian averaged Navier-Stokes (LANS-α) equations on bounded domains, Philos. Trans. R. Soc. Lond., Ser. A 359 (2001), 1449-1468. doi:10.1098/rsta.2001.0852. · Zbl 1006.35074 · doi:10.1098/rsta.2001.0852 |
| [24] | R. Teman, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer, New York, 1988. · Zbl 0662.35001 |
| [25] | P.T. Xuan, The simplified Bardina equation on two-dimensional closed manifolds, Dynamics of PDE 18(4) (2021), 293-326. doi:10.4310/dpde.2021.v18.n4.a3. · Zbl 1481.35315 · doi:10.4310/dpde.2021.v18.n4.a3 |
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