Large time decay and growth for solutions of a viscous Boussinesq system
HTML articles powered by AMS MathViewer
- by Lorenzo Brandolese and Maria E. Schonbek PDF
- Trans. Amer. Math. Soc. 364 (2012), 5057-5090 Request permission
Abstract:
In this paper we analyze the decay and the growth for large time of weak and strong solutions to the three-dimensional viscous Boussinesq system. We show that generic solutions blow up as $t\to \infty$ in the sense that the energy and the $L^p$-norms of the velocity field grow to infinity for large time for $1\le p<3$. In the case of strong solutions we provide sharp estimates, both from above and from below, and explicit asymptotic profiles. We also show that solutions arising from $(u_0,\theta _0)$ with zero-mean for the initial temperature $\theta _0$ have a special behavior as $|x|$ or $t$ tends to infinity: contrary to the generic case, their energy dissipates to zero for large time.References
- H. Abidi and T. Hmidi, On the global well-posedness for Boussinesq system, J. Differential Equations 233 (2007), no. 1, 199–220. MR 2290277, DOI 10.1016/j.jde.2006.10.008
- Hammadi Abidi, Taoufik Hmidi, and Sahbi Keraani, On the global regularity of axisymmetric Navier-Stokes-Boussinesq system, Discrete Contin. Dyn. Syst. 29 (2011), no. 3, 737–756. MR 2773149, DOI 10.3934/dcds.2011.29.737
- M. Abounouh, A. Atlas, and O. Goubet, Large-time behavior of solutions to a dissipative Boussinesq system, Differential Integral Equations 20 (2007), no. 7, 755–768. MR 2333655
- Lorenzo Brandolese, Asymptotic behavior of the energy and pointwise estimates for solutions to the Navier-Stokes equations, Rev. Mat. Iberoamericana 20 (2004), no. 1, 223–256. MR 2076779, DOI 10.4171/RMI/387
- Lorenzo Brandolese, Space-time decay of Navier-Stokes flows invariant under rotations, Math. Ann. 329 (2004), no. 4, 685–706. MR 2076682, DOI 10.1007/s00208-004-0533-2
- Lorenzo Brandolese and François Vigneron, New asymptotic profiles of nonstationary solutions of the Navier-Stokes system, J. Math. Pures Appl. (9) 88 (2007), no. 1, 64–86 (English, with English and French summaries). MR 2334773, DOI 10.1016/j.matpur.2007.04.007
- Lorenzo Brandolese, Fine properties of self-similar solutions of the Navier-Stokes equations, Arch. Ration. Mech. Anal. 192 (2009), no. 3, 375–401. MR 2505358, DOI 10.1007/s00205-008-0149-x
- L. Caffarelli, R. Kohn, and L. Nirenberg, Partial regularity of suitable weak solutions of the Navier-Stokes equations, Comm. Pure Appl. Math. 35 (1982), no. 6, 771–831. MR 673830, DOI 10.1002/cpa.3160350604
- J. R. Cannon and Emmanuele DiBenedetto, The initial value problem for the Boussinesq equations with data in $L^{p}$, Approximation methods for Navier-Stokes problems (Proc. Sympos., Univ. Paderborn, Paderborn, 1979) Lecture Notes in Math., vol. 771, Springer, Berlin, 1980, pp. 129–144. MR 565993
- Thierry Cazenave, Flávio Dickstein, and Fred B. Weissler, Chaotic behavior of solutions of the Navier-Stokes system in $\Bbb R^N$, Adv. Differential Equations 10 (2005), no. 4, 361–398. MR 2122695
- Dongho Chae, Global regularity for the 2D Boussinesq equations with partial viscosity terms, Adv. Math. 203 (2006), no. 2, 497–513. MR 2227730, DOI 10.1016/j.aim.2005.05.001
- Min Chen and Olivier Goubet, Long-time asymptotic behavior of two-dimensional dissipative Boussinesq systems, Discrete Contin. Dyn. Syst. Ser. S 2 (2009), no. 1, 37–53. MR 2481579, DOI 10.3934/dcdss.2009.2.37
- Hi Jun Choe and Bum Ja Jin, Weighted estimate of the asymptotic profiles of the Navier-Stokes flow in $\Bbb R^n$, J. Math. Anal. Appl. 344 (2008), no. 1, 353–366. MR 2416311, DOI 10.1016/j.jmaa.2008.02.040
- Antonio Córdoba and Diego Córdoba, A maximum principle applied to quasi-geostrophic equations, Comm. Math. Phys. 249 (2004), no. 3, 511–528. MR 2084005, DOI 10.1007/s00220-004-1055-1
- Diego Córdoba, Charles Fefferman, and Rafael de la Llave, On squirt singularities in hydrodynamics, SIAM J. Math. Anal. 36 (2004), no. 1, 204–213. MR 2083858, DOI 10.1137/S0036141003424095
- Raphaël Danchin and Marius Paicu, Les théorèmes de Leray et de Fujita-Kato pour le système de Boussinesq partiellement visqueux, Bull. Soc. Math. France 136 (2008), no. 2, 261–309 (French, with English and French summaries). MR 2415344, DOI 10.24033/bsmf.2557
- Raphaël Danchin and Marius Paicu, Existence and uniqueness results for the Boussinesq system with data in Lorentz spaces, Phys. D 237 (2008), no. 10-12, 1444–1460. MR 2454598, DOI 10.1016/j.physd.2008.03.034
- Raphaël Danchin and Marius Paicu, Global well-posedness issues for the inviscid Boussinesq system with Yudovich’s type data, Comm. Math. Phys. 290 (2009), no. 1, 1–14. MR 2520505, DOI 10.1007/s00220-009-0821-5
- Miguel Escobedo and Enrike Zuazua, Large time behavior for convection-diffusion equations in $\textbf {R}^N$, J. Funct. Anal. 100 (1991), no. 1, 119–161. MR 1124296, DOI 10.1016/0022-1236(91)90105-E
- Jishan Fan and Yong Zhou, A note on regularity criterion for the 3D Boussinesq system with partial viscosity, Appl. Math. Lett. 22 (2009), no. 5, 802–805. MR 2514915, DOI 10.1016/j.aml.2008.06.041
- L. C. F. Ferreira and E. J. Villamizar Roa, Well-posedness and asymptotic behaviour for the convection problem in $\Bbb R^n$, Nonlinearity 19 (2006), no. 9, 2169–2191. MR 2256658, DOI 10.1088/0951-7715/19/9/011
- Thierry Gallay and C. Eugene Wayne, Long-time asymptotics of the Navier-Stokes and vorticity equations on ${\Bbb R}^3$, R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci. 360 (2002), no. 1799, 2155–2188. Recent developments in the mathematical theory of water waves (Oberwolfach, 2001). MR 1949968, DOI 10.1098/rsta.2002.1068
- Boling Guo and Guangwei Yuan, On the suitable weak solutions for the Cauchy problem of the Boussinesq equations, Nonlinear Anal. 26 (1996), no. 8, 1367–1385. MR 1377668, DOI 10.1016/0362-546X(94)00350-Q
- Taoufik Hmidi and Frédéric Rousset, Global well-posedness for the Navier-Stokes-Boussinesq system with axisymmetric data, Ann. Inst. H. Poincaré C Anal. Non Linéaire 27 (2010), no. 5, 1227–1246. MR 2683758, DOI 10.1016/j.anihpc.2010.06.001
- Ryuji Kajikiya and Tetsuro Miyakawa, On $L^2$ decay of weak solutions of the Navier-Stokes equations in $\textbf {R}^n$, Math. Z. 192 (1986), no. 1, 135–148. MR 835398, DOI 10.1007/BF01162027
- Grzegorz Karch and Nicolas Prioux, Self-similarity in viscous Boussinesq equations, Proc. Amer. Math. Soc. 136 (2008), no. 3, 879–888. MR 2361860, DOI 10.1090/S0002-9939-07-09063-6
- P. G. Lemarié-Rieusset, Recent developments in the Navier-Stokes problem, Chapman & Hall/CRC Research Notes in Mathematics, vol. 431, Chapman & Hall/CRC, Boca Raton, FL, 2002. MR 1938147, DOI 10.1201/9781420035674
- Kyūya Masuda, Weak solutions of Navier-Stokes equations, Tohoku Math. J. (2) 36 (1984), no. 4, 623–646. MR 767409, DOI 10.2748/tmj/1178228767
- Tetsuro Miyakawa, On space-time decay properties of nonstationary incompressible Navier-Stokes flows in $\textbf {R}^n$, Funkcial. Ekvac. 43 (2000), no. 3, 541–557. MR 1815476
- Tetsuro Miyakawa and Maria Elena Schonbek, On optimal decay rates for weak solutions to the Navier-Stokes equations in $\Bbb R^n$, Proceedings of Partial Differential Equations and Applications (Olomouc, 1999), 2001, pp. 443–455. MR 1844282
- César J. Niche and María E. Schonbek, Decay of weak solutions to the 2D dissipative quasi-geostrophic equation, Comm. Math. Phys. 276 (2007), no. 1, 93–115. MR 2342289, DOI 10.1007/s00220-007-0327-y
- Takayoshi Ogawa, Shubha V. Rajopadhye, and Maria E. Schonbek, Energy decay for a weak solution of the Navier-Stokes equation with slowly varying external forces, J. Funct. Anal. 144 (1997), no. 2, 325–358. MR 1432588, DOI 10.1006/jfan.1996.3011
- Nicolas Prioux, Asymptotic stability results for systems of nonlinear evolution equations, Adv. Math. Sci. Appl. 17 (2007), no. 1, 37–65. MR 2337369
- Okihiro Sawada and Yasushi Taniuchi, On the Boussinesq flow with nondecaying initial data, Funkcial. Ekvac. 47 (2004), no. 2, 225–250. MR 2108674, DOI 10.1619/fesi.47.225
- Maria Elena Schonbek, $L^2$ decay for weak solutions of the Navier-Stokes equations, Arch. Rational Mech. Anal. 88 (1985), no. 3, 209–222. MR 775190, DOI 10.1007/BF00752111
- Maria E. Schonbek, Lower bounds of rates of decay for solutions to the Navier-Stokes equations, J. Amer. Math. Soc. 4 (1991), no. 3, 423–449. MR 1103459, DOI 10.1090/S0894-0347-1991-1103459-2
- James Serrin, On the interior regularity of weak solutions of the Navier-Stokes equations, Arch. Rational Mech. Anal. 9 (1962), 187–195. MR 136885, DOI 10.1007/BF00253344
- Roger Temam, Navier-Stokes equations, 3rd ed., Studies in Mathematics and its Applications, vol. 2, North-Holland Publishing Co., Amsterdam, 1984. Theory and numerical analysis; With an appendix by F. Thomasset. MR 769654
- Michael Wiegner, Decay results for weak solutions of the Navier-Stokes equations on $\textbf {R}^n$, J. London Math. Soc. (2) 35 (1987), no. 2, 303–313. MR 881519, DOI 10.1112/jlms/s2-35.2.303
Additional Information
- Lorenzo Brandolese
- Affiliation: Université de Lyon, CNRS UMR 5208 Institut Camille Jordan, Université Lyon 1, 43 bd. du 11 novembre, Villeurbanne Cedex F-69622, France
- Email: brandolese{@}math.univ-lyon1.fr
- Maria E. Schonbek
- Affiliation: Department of Mathematics, University of California Santa Cruz, Santa Cruz, California 95064
- MR Author ID: 156790
- ORCID: 0000-0002-9917-8495
- Email: schonbek@math.ucsc.edu
- Received by editor(s): March 18, 2010
- Received by editor(s) in revised form: July 27, 2010
- Published electronically: May 30, 2012
- Additional Notes: The work of both authors was partially supported by FBF Grant SC-08-34
The work of the second author was partially supported by NSF Grants DMS-0900909 and grant FRG-09523-503114. - © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 364 (2012), 5057-5090
- MSC (2010): Primary 76D05; Secondary 35Q35, 35B40
- DOI: https://doi.org/10.1090/S0002-9947-2012-05432-8
- MathSciNet review: 2931322