Space-time decay estimates of solutions to 3D incompressible viscous Camassa-Holm equations
DOI:
https://doi.org/10.12775/TMNA.2020.045Keywords
Space-time decay, viscous Camassa-Holm equations, weighted estimates, parabolic interpolation inequalityAbstract
In this paper, based on the parabolic interpolation inequality and inductive argument, we study the space-time decay estimates of higher-order time and spatial derivatives of strong solutions for the 3D incompressible viscous Camassa-Holm equations provided that the initial datum is well localized.References
C. Amrouche, V. Girault, M. Schonbek and T. Schonbek, Pointwidse decay of solutions and of higher derivatives to Navier–Stokes equations, SIAM J. Math. Anal. 31 (2000), 740–753.
C.T. Anh and P.T. Trang, Decay characterization of solutions to the viscous Camassa–Holm equations, Nonlineary 31 (2018), 621–650.
C. Bjorland, Decay asymptotics of the viscous Camassa–Holm equations in the plane, SIAM J. Math. Anal. 40 (2008), 516–539.
C. Bjorland and M.E. Schonbek, On questions of decay and existence for the viscous Camassa–Holm equations, Ann. Inst. H. Poincaré Anal. Non Linéaire 25 (2008), 907–936.
C. Bjorland and M. E. Schonbek, Poincaré’s inequality and diffusive evolution equations, Adv. Differential Equations 14 (2009), 241–260.
L. Brandolese, Characterization of solutions to dissipative systems with sharp algebraic decay, SIAM J. Math. Anal. 48 (2016), 1616–1633.
L. Brandolese, On a non-solenoidal approximation to the incompressible Navier–Stokes equations, J. London Math. Soc. 96 (2017), 326–344.
A.V. Busuioc, On the large time behavior of sluitons of the alpha Navier–Stokes equations, Phys. D 238 (2009), 2261–2272.
L. Caffarelli, R. Kohn and L. Nirenberg, First order interpolation inequalities with weights, Compos. Math. 53 (1984), 259–275.
S. Chen, C. Foias, D.D. Holm, E. Olson, E.S. Titi and S. Wynne, The Camassa–Holm equations as a closure model for turbulent channel flow, Phys. Rev. Lett. 81 (1998), 5338–5341.
S S. Chen, C. Foias, D.D. Holm, E. Olson, E.S. Titi and S. Wynne, A connection between the Camassa-Holm equations and turbulent flows in channels and pipes, Phys. Fluids 11 (1999), 2343–2353.
D. Coutand, J. Peirce and S. Shkoller, Global well-posedness of weak solutions for the Lagrangian averaged Navier–Stokes equations on bounded domains, Commun. Pure Appl. Anal. 2 (2002), 35–50.
H.-H. Dai, Model equations for nonlinear dispersive waves in a compressible Mooney–Rivlin rod, Acta Mech. 127 (1998), 193–207.
R. Danchin, A note on well-posedness for Camassa–Holm equation, J. Differential Equations 192 (2003), 429–444.
C. Foias, D.D. Holm and E.S. Titi, The Navier–Stokes-alpha model of fluid turbulence, Advances in Nonlinear Mathematics and Science, Phys. D, 152/153 (2001), 505–519.
C. Foias, D.D. Holm, E.S. Titi, The three dimensional viscous Camassa–Holm equations, and their relation to the Navier–Stokes equations and turbulence theory, J. Dynam. Differential Equations 14 (2002), 1–35.
C. Foias, J. Tian and B. Zhang, On the emergence of the Navier–Stokes-α model for turbulent channel flows, J. Math. Phys. 57 (2016), 081510, 14 pp.
A.A. Ilyin and E.S. Titi, Attractors for the 2D Navier–Stokes-α model: an α-dependence study, J. Dynam. Differential Equations 15 (2003), 751–778.
B.-S. Kim and B. Nicolaenko, Existence and continuity of exponential attractors of the 3D Navier–Stokes-α equations for uniformly rotating geophysical fluids, Commun. Math. Sci. 4 (2006), 399–452.
I. Kukavica, Space-time decay for solutions of the Navier–Stokes equations, Indiana Univ. Math. J. 50 (2001), 205–222.
I. Kukavica, On the weighted decay for solutions of the Navier–Stokes system, Nonlinear Anal. 70 (2009), 2466–2470.
I. Kukavica and J.J. Torres, Weighted bounds for the velocity and the vorticity for the Navier–Stokes equations, Nonlinearity 19 (2006), 293–303.
I. Kukavica and J.J. Torres, Weighted Lp decay for solutions of the Navier–Stokes equations, Comm. Partial Differential Equations 32 (2007), 819–831.
J.E. Marsden and S. Shkoller, Global well-posedness for the Lagrangian averaged Navier–Stokes (LANS-α) equations on bounded domains, Philos. Trans. Roy. Soc. A 359 (2001), 1449–1468.
T. Miyakawa, On space-time decay properties of nonstationary incompressible Navier–Stokes flows in Rn , Funkcial. Ekvac. 43 (2000), 541–557.
C.J. Niche, Decay characterization of solutions to Navier–Stokes–Voigt equations in terms of the initial datum, J. Differential Equations 260 (2016), 4440–4453.
C.J. Niche and M.E. Schonbek, Decay characterization of solutions to dissipative equations, J. London Math. Soc. 91 (2015), no. 2, 573–595.
M.E. Schonbek, L2 decay for weak solutions of the Navier–Stokes equations, Arch. Ration. Mech. Anal. 88 (1985), no. 2, 209–222.
M.E. Schonbek, Large time behaviour of solutions to the Navier–Stokes equations, Comm. Partial Differential Equations 11 (1986), no. 7, 733–763.
M. Schonbek and T. Schonbek, On the boundedness and decay of moments of solutions to the Navier–Stokes equations, Adv. Differential Equations 5 (2000), 861–898.
M. Stanislavova and A. Stefanov, Attractors for the viscous Camassa–Holm equation, Discrete Contin. Dyn. Syst. 18 (2007), 159–186.
S. Takahashi, A weighted equation approach to decay rate estimates for the Navier–Stokes equations, Nonlinear Anal. 37 (1999), 751–789.
M.I. Vishik, E.S. Titi and V.V. Chepyzhov, On the convergence of trajectory attractors of the 3D Navier–Stokes-α model as α → 0, Sb. Math. 198 (2007), 1703–1736.
S. Weng, Space-time decay estimates for the incompressible viscous resistive MHD and Hall-MHD equations, J. Funct. Anal. 270 (2016), 2168–2187.
S. Weng, Remarks on asymptotic behaviors of strong solutions to a viscous Boussinesq system, Math. Methods Appl. Sci. 39 (2016), 4398–4418.
X. Zhao, Decay of solutions to a new Hall-MHD equations in Rn , C.R. Acad. Sci. Paris Sér. I 355 (2017), 310–317.
X. Zhao, Space-time decay estimates of solutions to Liquid crystal system in R3 , Commun. Pure Appl. Anal. 18 (2019), 1–13.
X. Zhao and M. Zhu, Global well-posedness and asymptotic behavior of solutions for the three-dimensional MHD equations with Hall and ion-slip effects, Z. Angew. Math. Phys. 69 (2018), Art. 22, 13 pp.
Y. Zhou and J. Fan, Regularity criteria for the viscous Camassa–Holm equations, Int. Math. Res. Not. IMRN 2009 (2009), 2508–2518.
Published
How to Cite
Issue
Section
Stats
Number of views and downloads: 0
Number of citations: 0
