三维不可压Navier-Stokes方程组轨道统计解的退化正则性

Abstract

Abstract:

In this article, the authors prove that if the (generalized) 3D Grashof number of the 3D autonomous incompressible Navier-Stokes equations is less than 2.057, then its weak trajectory statistical solutions possess (partial) degenerate regularity in the sense that they are supported by a set in which the weak solutions are in fact (partially) strong solutions. Also, they reveal that if the 3D Grashof number is less than 2.057, then the 3D incompressible Navier-Stokes equations possess only one complete and bounded strong solution which not only forward attracts but also pullback attracts its trajectories.

Key words: Trajectory statistical solution, Degenerate regularity, 3D incompressible Navier-Stokes equations, Trajectory attractor, Grashof number

CLC Number:

O.175.8

Mingyue Xu,Caidi Zhao,Caraballo Tomás. Degenerate Regularity of Trajectory Statistical Solutions for the 3D Incompressible Navier-Stokes Equations[J].Acta mathematica scientia,Series A, 2021, 41(2): 336-344.

Trendmd

References 34

1	Bronzi A , Mondaini C , Rosa R . Trajectory statistical solutions for three-dimensional Navier-Stokes-like systems. SIAM J Math Anal, 2014, 46, 1893- 1921 doi: 10.1137/130931631
2	Bronzi A , Rosa R . On the convergence of statistical solutions of the 3D Navier-Stokes-α model as α vanishes. Discrete Cont Dyn Syst, 2014, 34, 19- 49 doi: 10.3934/dcds.2014.34.19
3	Bronzi A , Mondaini C , Rosa R . Abstract framework for the theory of statistical solutions. J Differential Equations, 2016, 260, 8428- 8484 doi: 10.1016/j.jde.2016.02.027
4	Caraballo T , Kloeden P , Real J . Invariant measures and statistical solutions of the globally modified Navier-Stokes equations. Discrete Cont Dyn Syst-B, 2008, 10, 761- 781
5	Chepyzhov V, Vishik M. Attractors for Equations of Mathematical Physics. Providence, RI: AMS Colloquium Publications, 2002
6	Cheskidov A , Kavlie L . Degenerate pullback attractors for the 3D Navier-Stokes equations. J Math Fluid Mech, 2015, 17, 411- 421 doi: 10.1007/s00021-015-0214-9
7	Chekroun M , Glatt-Holtz N . Invariant measures for dissipative dynamical systems: abstract results and applications. Comm Math Phys, 2012, 316, 723- 761 doi: 10.1007/s00220-012-1515-y
8	Foias C , Prodi G . Sur les solutions statistiques des équations de Naiver-Stokes. Ann Mat Pura Appl, 1976, 111, 307- 330 doi: 10.1007/BF02411822
9	Foias C , Manley O , Rosa R , Temam R . Navier-Stokes Equations and Turbulence. Cambridge: Cambridge University Press, 2001
10	Foias C , Rosa R , Temam R . A note on statistical solutions of the three-dimensional Navier-Stokes equations: the stationary case. C R Math, 2010, 348, 235- 240 doi: 10.1016/j.crma.2009.12.017
11	Foias C , Rosa R , Temam R . A note on statistical solutions of the three-dimensional Navier-Stokes equations: the time-dependent case. C R Math, 2010, 348, 347- 353 doi: 10.1016/j.crma.2009.12.018
12	Foias C , Rosa R , Temam R . Properties of time-dependent statistical solutions of the three-dimensional Navier-Stokes equations. Annales de L'Institut Fourier, 2013, 63, 2515- 2573 doi: 10.5802/aif.2836
13	Foias C , Rosa R , Temam R . Convergence of time averages of weak solutions of the three-dimensional Navier-Stokes equations. J Stat Phys, 2015, 160, 519- 531 doi: 10.1007/s10955-015-1248-3
14	Foias C , Rosa R , Temam R . Properties of stationary statistical solutions of the three-dimensional Navier-Stokes equations. J Dyn Differential Equations, 2019, 31, 1689- 1741 doi: 10.1007/s10884-018-9719-2
15	Kloeden P , Marín-Rubio P , Real J . Equivalence of invariant measures and stationary statistical solutions for the autonomous globally modified Navier-Stokes equations. Comm Pure Appl Anal, 2009, 8, 785- 802 doi: 10.3934/cpaa.2009.8.785
16	Ladyzhenskaya O A . The Mathematical Theory of Viscous Incompressible Flow. New York: Gordon and Breach, 1969
17	Lukaszewicz G . Pullback attractors and statistical solutions for 2-D Navier-Stokes equations. Discrete Cont Dyn Syst-B, 2008, 9, 643- 659 doi: 10.3934/dcdsb.2008.9.643
18	Lukaszewicz G , Robinson J C . Invariant measures for non-autonomous dissipative dynamical systems. Discrete Cont Dyn Syst, 2014, 34, 4211- 4222 doi: 10.3934/dcds.2014.34.4211
19	Rosa R. Theory and Applications of Statistical Solutions of the Navier-Stokes Equations//Robinson J C, Rodrigo J L. London Mathematical Society Lecture Note Series. Cambridge: Cambridge University Press, 2009: 228-257
20	Rudin W . Real and Complex Analysis. New York: McGraw-Hill Company, 1987
21	Temam R. On the Theory and Numerical Analysis of the Navier-Stokes Equations. Providence, RI: AMS Colloquium Publications, 2001
22	Vishik M , Fursikov A . Translationally homogeneous statistical solutions and individual solutions with infinite energy of a system of Navier-Stokes equations. Siberian Math J, 1978, 19, 710- 729
23	Zhao C , Li Y , Zhou S . Regularity of trajectory attractor and upper semicontinuity of global attractor for a 2D non-Newtonian fluid. J Differential Equations, 2009, 247, 2331- 2363 doi: 10.1016/j.jde.2009.07.031
24	Zhao C , Jia X , Yang X . Uniform attractors for non-autonomous incompressible non-Newtonian fluid with a new class of external forces. Acta Math Sci, 2011, 31, 1803- 1812 doi: 10.1016/S0252-9602(11)60362-7
25	Zhao C , Liu G , Wang W . Smooth pullback attractors for a non-autonomous 2D non-Newtonian fluid and their tempered behaviors. J Math Fluids Mech, 2014, 16, 243- 262 doi: 10.1007/s00021-013-0153-2
26	Zhao C , Kong L , Liu G W . The trajectory attractor and its limiting behavior for the convective Brinkman-Forchheimer equations. Topological Meth in Nonlinear Anal, 2014, 44, 413- 433
27	Zhao C , Li B . Analyticity of the global attractor for the 3D regularized MHD equations. E J Differential Equations, 2016, 179, 1- 20
28	Zhao C , Caraballo T . Asymptotic regularity of trajectory attractor and trajectory statistical solution for the 3D globally modified Navier-Stokes equations. J Differential Equations, 2019, 266, 7205- 7229 doi: 10.1016/j.jde.2018.11.032
29	Zhao C , Li Y , Caraballo T . Trajectory statistical solutions and Liouville type equations for evolution equations: Abstract results and applications. J Differential Equations, 2020, 269, 467- 494 doi: 10.1016/j.jde.2019.12.011
30	Zhao C , Li Y , Lukaszewicz G . Statistical solution and partial degenerate regularity for the 2D non-autonomous magneto-micropolar fluids. Z Angew Math Phys, 2020, 71, 1- 24 doi: 10.1007/s00033-019-1224-x
31	Zhao C , Song Z , Caraballo T . Strong trajectory statistical solutions and Liouville type equations for dissipative Euler equations. Appl Math Lett, 2020, 99, 105981 doi: 10.1016/j.aml.2019.07.012
32	Zhao C , Li Y , Song Z . Trajectory statistical solutions for the 3D Navier-Stokes equations: The trajectory attractor approach. Nonlinear Anal-RWA, 2020, 53, 103077 doi: 10.1016/j.nonrwa.2019.103077
33	Zhao C , Li Y , Sang Y . Using trajectory attractor to construct trajectory statistical solutions for 3D incompressible micropolar flows. Z Angew Math Mech, 2020, 100, e201800197
34	Zhu Z , Zhao C . Pullback attractor and invariant measures for the three-dimensional regularized MHD equations. Discrete Cont Dyn Syst, 2018, 38, 1461- 1477 doi: 10.3934/dcds.2018060

Degenerate Regularity of Trajectory Statistical Solutions for the 3D Incompressible Navier-Stokes Equations

RichHTML

PDF (PC)

Abstract

Cite this article

share this article

References 34

Related Articles 1

Metrics

Comments

Recommended 3