×

zbMATH — the first resource for mathematics

Global well-posedness for the 2D Boussinesq system with anisotropic viscosity and without heat diffusion. (English) Zbl 1284.35343
Summary: We establish global existence and uniqueness theorems for the two-dimensional non-diffusive Boussinesq system with anisotropic viscosity acting only in the horizontal direction, which arises in ocean dynamics models. Global well-posedness for this system was proven by R. Danchin and M. Paicu [Math. Models Methods Appl. Sci. 21, No. 3, 421–457 (2011; Zbl 1223.35249)]; however, an additional smoothness assumption on the initial density was needed to prove uniqueness. They stated that it is not clear whether uniqueness holds without this additional assumption. The present work resolves this question and we establish uniqueness without this additional assumption. Furthermore, the proof provided here is more elementary; we use only tools available in the standard theory of Sobolev spaces, and without resorting to para-product calculus. We use a new approach by defining an auxiliary “stream-function” associated with the density, analogous to the stream-function associated with the vorticity in 2D incompressible Euler equations, then we adapt some of the ideas of V. I. Yudovich [Zh. Vychisl. Mat. Mat. Fiz. 3, 1032–1066 (1963; Zbl 0129.19402)] for proving uniqueness for 2D Euler equations.

MSC:
35Q35 PDEs in connection with fluid mechanics
76B03 Existence, uniqueness, and regularity theory for incompressible inviscid fluids
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
76D09 Viscous-inviscid interaction
35Q31 Euler equations
35A01 Existence problems for PDEs: global existence, local existence, non-existence
86A05 Hydrology, hydrography, oceanography
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Adams, R. A.; Fournier, J. J.F., Sobolev spaces, Pure Appl. Math. (Amst.), vol. 140, (2003), Elsevier/Academic Press Amsterdam, MR 2424078 · Zbl 1098.46001
[2] Adhikari, D.; Cao, C.; Wu, J., Global regularity for the 2D anisotropic Boussinesq equations with vertical dissipation, J. Differential Equations, 251, 6, 1637-1655, (2011), MR 2813893 (2012h:35271) · Zbl 1232.35111
[3] Agmon, S., Lectures on elliptic boundary value problems, Van Nostrand Math. Stud., vol. 2, (1965), D. Van Nostrand Co., Inc. Princeton, NJ-Toronto-London, prepared for publication by B. Frank Jones, Jr. with the assistance of George W. Batten, Jr. MR 0178246 (31 #2504)
[4] Bony, J.-M., Calcul symbolique et propagation des singularités pour LES équations aux dérivées partielles non linéaires, Ann. Sci. École Norm. Sup. (4), 14, 2, 209-246, (1981), MR 631751 (84h:35177) · Zbl 0495.35024
[5] Cannon, J. R.; DiBenedetto, E., 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, (1980), Springer Berlin), 129-144, MR 565993 (81f:35101)
[6] Chae, D., Global regularity for the 2D Boussinesq equations with partial viscosity terms, Adv. Math., 203, 2, 497-513, (2006), MR 2227730 (2007e:35223) · Zbl 1100.35084
[7] Chae, D.; Nam, H.-S., Local existence and blow-up criterion for the Boussinesq equations, Proc. Roy. Soc. Edinburgh Sect. A, 127, 5, 935-946, (1997), MR 1475638 (98e:35133) · Zbl 0882.35096
[8] Constantin, P.; Foias, C., Navier-Stokes equations, Chicago Lectures in Math., (1988), University of Chicago Press Chicago, IL, MR 972259 (90b:35190) · Zbl 0687.35071
[9] Danchin, R.; Paicu, M., Global existence results for the anisotropic Boussinesq system in dimension two, Math. Models Methods Appl. Sci., 21, 3, 421-457, (2011), MR 2782720 (2012g:35254) · Zbl 1223.35249
[10] Danchin, R.; Paicu, M., LES théorèmes de Leray et de Fujita-Kato pour le système de Boussinesq partiellement visqueux, Bull. Soc. Math. France, 136, 2, 261-309, (2008), MR 2415344 (2009k:35230) · Zbl 1162.35063
[11] Evans, L. C., Partial differential equations, Grad. Stud. Math., vol. 19, (1998), American Mathematical Society Providence, RI, MR 1625845 (99e:35001)
[12] Foias, C.; Manley, O.; Rosa, R.; Temam, R., Navier-Stokes equations and turbulence, Encyclopedia Math. Appl., vol. 83, (2001), Cambridge University Press Cambridge, MR 1855030 (2003a:76001) · Zbl 0994.35002
[13] Foias, C.; Manley, O.; Temam, R., Attractors for the Bénard problem: existence and physical bounds on their fractal dimension, Nonlinear Anal., 11, 8, 939-967, (1987), MR 903787 (89f:35166) · Zbl 0646.76098
[14] He, L., Smoothing estimates of 2d incompressible Navier-Stokes equations in bounded domains with applications, J. Funct. Anal., 262, 7, 3430-3464, (2012), MR MR2885958 · Zbl 1234.35180
[15] Hmidi, T.; Keraani, S., On the global well-posedness of the two-dimensional Boussinesq system with a zero diffusivity, Adv. Differential Equations, 12, 4, 461-480, (2007), MR 2305876 (2009c:35404) · Zbl 1154.35073
[16] Hoff, D., Uniqueness of weak solutions of the Navier-Stokes equations of multidimensional, compressible flow, SIAM J. Math. Anal., 37, 6, 1742-1760, (2006), MR 2213392 (2006m:35280) · Zbl 1100.76052
[17] Hou, T. Y.; Li, C., Global well-posedness of the viscous Boussinesq equations, Discrete Contin. Dyn. Syst., 12, 1, 1-12, (2005), MR 2121245 (2005j:76026) · Zbl 1274.76185
[18] Larios, A.; Lunasin, E.; Titi, E. S., Global well-posedness for the 2D Boussinesq system without heat diffusion and with either anisotropic viscosity or inviscid Voigt-α regularization, (2010), 38 pp
[19] Lieb, E. H.; Loss, M., Analysis, Grad. Stud. Math., vol. 14, (2001), American Mathematical Society Providence, RI, MR 1817225 (2001i:00001) · Zbl 0966.26002
[20] Majda, A. J.; Bertozzi, A. L., Vorticity and incompressible flow, Cambridge Texts Appl. Math., vol. 27, (2002), Cambridge University Press Cambridge, MR 1867882 (2003a:76002) · Zbl 0983.76001
[21] Temam, R., Navier-Stokes equations and nonlinear functional analysis, CBMS-NSF Regional Conf. Ser. in Appl. Math., vol. 66, (1995), Society for Industrial and Applied Mathematics (SIAM) Philadelphia, PA, MR 1318914 (96e:35136) · Zbl 0833.35110
[22] Temam, R., Infinite-dimensional dynamical systems in mechanics and physics, Appl. Math. Sci., vol. 68, (1997), Springer-Verlag New York, MR 1441312 (98b:58056) · Zbl 0871.35001
[23] Temam, R., Navier-Stokes equations: theory and numerical analysis, (2001), AMS Chelsea Publishing Providence, RI, reprint of the 1984 edition, MR 1846644 (2002j:76001) · Zbl 0981.35001
[24] Wang, X. M., A remark on the characterization of the gradient of a distribution, Appl. Anal., 51, 1-4, 35-40, (1993), MR 1278991 (95k:46064) · Zbl 0797.46032
[25] Yudovich, V. I., Non-stationary flows of an ideal incompressible fluid, Z̆. Vyčisl. Mat. i Mat. Fiz., 3, 1032-1066, (1963), MR 0158189 (28 #1415)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.