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Optimal decay estimates for 2D Boussinesq equations with partial dissipation. (English) Zbl 1462.35407

Summary: Buoyancy-driven fluids such as many atmospheric and oceanic flows and the Rayleigh-Bénard convection are modeled by the Boussinesq systems. By rigorously estimating the large-time behavior of solutions to a special Boussinesq system, this paper reveals a fascinating phenomenon on buoyancy-driven fluids that the temperature can actually stabilize the fluids. The Boussinesq system concerned here governs the motion of perturbations near the hydrostatic equilibrium. When the buoyancy forcing is not present, the velocity of the fluid obeys the 2D Navier-Stokes equation with only vertical dissipation and its Sobolev norm could potentially grow even though its precise large-time behavior remains open. This paper shows that the temperature through the coupling and interaction tames and regularizes the fluids, and causes the velocity (measured in Sobolev norms) to decay in time. Optimal decay rates are obtained.

MSC:

35Q86 PDEs in connection with geophysics
35Q35 PDEs in connection with fluid mechanics
35B40 Asymptotic behavior of solutions to PDEs
42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
76D50 Stratification effects in viscous fluids
86A05 Hydrology, hydrography, oceanography
86A10 Meteorology and atmospheric physics
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[1] Adhikari, D.; Cao, C.; Wu, J., The 2D Boussinesq equations with vertical viscosity and vertical diffusivity, J. Differ. Equ., 249, 1078-1088 (2010) · Zbl 1193.35144 · doi:10.1016/j.jde.2010.03.021
[2] Adhikari, D.; Cao, C.; Wu, J., Global regularity results for the 2D Boussinesq equations with vertical dissipation, J. Differ. Equ., 251, 1637-1655 (2011) · Zbl 1232.35111 · doi:10.1016/j.jde.2011.05.027
[3] Adhikari, D.; Cao, C.; Wu, J.; Xu, X., Small global solutions to the damped two-dimensional Boussinesq equations, J. Differ. Equ., 256, 3594-3613 (2014) · Zbl 1290.35193 · doi:10.1016/j.jde.2014.02.012
[4] Adhikari, D.; Cao, C.; Shang, H.; Wu, J.; Xu, X.; Ye, Z., Global regularity results for the 2D Boussinesq equations with partial dissipation, J. Differ. Equ., 260, 1893-1917 (2016) · Zbl 1328.35161 · doi:10.1016/j.jde.2015.09.049
[5] Bahouri, H.; Chemin, J-Y; Danchin, R., Fourier Analysis and Nonlinear Partial Differential Equations (2011), Berlin: Springer, Berlin · Zbl 1227.35004 · doi:10.1007/978-3-642-16830-7
[6] Ben Said, O., Pandey, U., Wu, J.: The Stabilizing Effect of the Temperature on Buoyancy-driven Fluids. arXiv:2005.11661v2 [math.AP] (26 May 2020)
[7] Bianchini, R., Coti Zelati, M., Dolce, M.: Linear Inviscid Damping for Shear Flows Near Couette in the 2D Stably Stratified Regime . arXiv:2005.09058v1 [math.AP] (18 May 2020).
[8] Boardman, N.; Ji, R.; Qiu, H.; Wu, J., Global existence and uniqueness of weak solutions to the Boussinesq equations without thermal diffusion, Commun. Math. Sci., 17, 1595-1624 (2019) · Zbl 1433.35260 · doi:10.4310/CMS.2019.v17.n6.a5
[9] Cao, C.; Wu, J., Global regularity for the 2D anisotropic Boussinesq equations with vertical dissipation, Arch. Ration. Mech. Anal., 208, 985-1004 (2013) · Zbl 1284.35140 · doi:10.1007/s00205-013-0610-3
[10] Castro, A.; Córdoba, D.; Lear, D., On the asymptotic stability of stratified solutions for the 2D Boussinesq equations with a velocity damping term, Math. Models Methods Appl. Sci., 29, 1227-1277 (2019) · Zbl 1425.35149 · doi:10.1142/S0218202519500210
[11] Chae, D., Global regularity for the 2D Boussinesq equations with partial viscosity terms, Adv. Math., 203, 497-513 (2006) · Zbl 1100.35084 · doi:10.1016/j.aim.2005.05.001
[12] Chae, D.; Wu, J., The 2D Boussinesq equations with logarithmically supercritical velocities, Adv. Math., 230, 1618-1645 (2012) · Zbl 1248.35156 · doi:10.1016/j.aim.2012.04.004
[13] Chae, D.; Constantin, P.; Wu, J., An incompressible 2D didactic model with singularity and explicit solutions of the 2D Boussinesq equations, J. Math. Fluid Mech., 16, 473-480 (2014) · Zbl 1307.35213 · doi:10.1007/s00021-014-0166-5
[14] Choi, K.; Kiselev, A.; Yao, Y., Finite time blow up for a 1D model of 2D Boussinesq system, Commun. Math. Phys., 334, 1667-1679 (2015) · Zbl 1309.35072 · doi:10.1007/s00220-014-2146-2
[15] Constantin, P.; Doering, C., Heat transfer in convective turbulence, Nonlinearity, 9, 1049-1060 (1996) · Zbl 0899.35078 · doi:10.1088/0951-7715/9/4/013
[16] Constantin, P.; Vicol, V.; Wu, J., Analyticity of Lagrangian trajectories for well posed inviscid incompressible fluid models, Adv. Math., 285, 352-393 (2015) · Zbl 1422.35135 · doi:10.1016/j.aim.2015.05.019
[17] Danchin, R.; Paicu, M., Global well-posedness issues for the inviscid Boussinesq system with Yudovich’s type data, Commun. Math. Phys., 290, 1-14 (2009) · Zbl 1186.35157 · doi:10.1007/s00220-009-0821-5
[18] Danchin, R.; Paicu, M., Global existence results for the anisotropic Boussinesq system in dimension two, Math. Models Methods Appl. Sci., 21, 421-457 (2011) · Zbl 1223.35249 · doi:10.1142/S0218202511005106
[19] Deng, W., Wu, J., Zhang, P.: Stability of Couette Flow for 2D Boussinesq System with Vertical Dissipation. arXiv:2004.09292v1. [math.AP] (20 Apr 2020)
[20] Denisov, S., Double exponential growth of the vorticity gradient for the two-dimensional Euler equation, Proc. Am. Math. Soc., 143, 1199-1210 (2015) · Zbl 1315.35150 · doi:10.1090/S0002-9939-2014-12286-6
[21] Doering, C.; Gibbon, J., Applied Analysis of the Navier-Stokes Equations. Cambridge Texts in Applied Mathematics (1995), Cambridge: Cambridge University Press, Cambridge · Zbl 0838.76016 · doi:10.1017/CBO9780511608803
[22] Doering, CR; Wu, J.; Zhao, K.; Zheng, X., Long time behavior of the two-dimensional Boussinesq equations without buoyancy diffusion, Physica D, 376, 377, 144-159 (2018) · Zbl 1398.35164 · doi:10.1016/j.physd.2017.12.013
[23] Elgindi, T., Jeong, I.: Finite-time singularity formation for strong solutions to the Boussinesq system. Ann. PDE 6, Paper No. 5 (2020) · Zbl 1462.35287
[24] Elgindi, T.; Widmayer, K., Sharp decay estimates for an anisotropic linear semigroup and applications to the surface quasi-geostrophic and inviscid Boussinesq systems, SIAM J. Math. Anal., 47, 4672-4684 (2015) · Zbl 1326.35266 · doi:10.1137/14099036X
[25] Hmidi, T.; Keraani, S.; Rousset, F., Global well-posedness for a Boussinesq-Navier-Stokes system with critical dissipation, J. Differ. Equ., 249, 2147-2174 (2010) · Zbl 1200.35228 · doi:10.1016/j.jde.2010.07.008
[26] Hmidi, T.; Keraani, S.; Rousset, F., Global well-posedness for Euler-Boussinesq system with critical dissipation, Commun. Partial Differ. Equ., 36, 420-445 (2011) · Zbl 1284.76089 · doi:10.1080/03605302.2010.518657
[27] Hou, T.; Li, C., Global well-posedness of the viscous Boussinesq equations, Discr. Cont. Dyn. Syst. Ser. A, 12, 1-12 (2005) · Zbl 1274.76185 · doi:10.3934/dcds.2005.12.1
[28] Hu, W.; Kukavica, I.; Ziane, M., Persistence of regularity for the viscous Boussinesq equations with zero diffusivity, Asymptot. Anal., 91, 111-124 (2015) · Zbl 1319.35192 · doi:10.3233/ASY-141261
[29] Hu, W.; Wang, Y.; Wu, J.; Xiao, B.; Yuan, J., Partially dissipative 2D Boussinesq equations with Navier type boundary conditions, Physica D, 376, 377, 39-48 (2018) · Zbl 1398.35007 · doi:10.1016/j.physd.2017.07.003
[30] Jiu, Q.; Miao, C.; Wu, J.; Zhang, Z., The two-dimensional incompressible Boussinesq equations with general critical dissipation, SIAM J. Math. Anal., 46, 3426-3454 (2014) · Zbl 1319.35193 · doi:10.1137/140958256
[31] Jiu, Q.; Wu, J.; Yang, W., Eventual regularity of the two-dimensional Boussinesq equations with supercritical dissipation, J. Nonlinear Sci., 25, 37-58 (2015) · Zbl 1311.35221 · doi:10.1007/s00332-014-9220-y
[32] Kc, D.; Regmi, D.; Tao, L.; Wu, J., Generalized 2D Euler-Boussinesq equations with a singular velocity, J. Differ. Equ., 257, 82-108 (2014) · Zbl 1291.35221 · doi:10.1016/j.jde.2014.03.012
[33] Kiselev, A.; Sverak, V., Small scale creation for solutions of the incompressible two-dimensional Euler equation, Ann. Math., 180, 1205-1220 (2014) · Zbl 1304.35521 · doi:10.4007/annals.2014.180.3.9
[34] Kiselev, A.; Tan, C., Finite time blow up in the hyperbolic Boussinesq system, Adv. Math., 325, 34-55 (2018) · Zbl 1382.35054 · doi:10.1016/j.aim.2017.11.019
[35] Lai, M.; Pan, R.; Zhao, K., Initial boundary value problem for two-dimensional viscous Boussinesq equations, Arch. Ration. Mech. Anal., 199, 739-760 (2011) · Zbl 1231.35171 · doi:10.1007/s00205-010-0357-z
[36] Lai, S., Xu, X., Zhang, J.: On the Cauchy problem of compressible full Hall-MHD equations. Z. Angew. Math. Phys. 70, Paper No. 139 (2019) · Zbl 1423.76493
[37] Lai, S.; Wu, J.; Zhong, Y., Stability and large-time behavior of the 2D Boussinesq equations with partial dissipation, J. Differ. Equ., 271, 764-796 (2021) · Zbl 1454.35288 · doi:10.1016/j.jde.2020.09.022
[38] Larios, A.; Lunasin, E.; Titi, ES, Global well-posedness for the 2D Boussinesq system with anisotropic viscosity and without heat diffusion, J. Differ. Equ., 255, 2636-2654 (2013) · Zbl 1284.35343 · doi:10.1016/j.jde.2013.07.011
[39] Li, J.; Titi, ES, Global well-posedness of the 2D Boussinesq equations with vertical dissipation, Arch. Rational Mech. Anal., 220, 983-1001 (2016) · Zbl 1336.35297 · doi:10.1007/s00205-015-0946-y
[40] Li, J.; Shang, H.; Wu, J.; Xu, X.; Ye, Z., Regularity criteria for the 2D Boussinesq equations with supercritical dissipation, Commun. Math. Sci., 14, 1999-2022 (2016) · Zbl 1355.35152 · doi:10.4310/CMS.2016.v14.n7.a10
[41] Lieb, EH; Loss, M., Analysis (2001), Providence: American Mathematical Society, Providence · Zbl 0966.26002
[42] Majda, A., Introduction to PDEs and Waves for the Atmosphere and Ocean, Courant Lecture Notes 9 (2003), Providence: Courant Institute of Mathematical Sciences and American Mathematical Society, Providence · Zbl 1278.76004
[43] Majda, A.; Bertozzi, A., Vorticity and Incompressible Flow (2002), Cambridge: Cambridge University Press, Cambridge · Zbl 0983.76001
[44] Miao, C.; Xue, L., On the global well-posedness of a class of Boussinesq-Navier-Stokes systems, NoDEA Nonlinear Differ. Equ. Appl., 18, 707-735 (2011) · Zbl 1235.76020 · doi:10.1007/s00030-011-0114-5
[45] Pedlosky, J., Geophysical Fluid Dynamics (1987), New York: Springer, New York · Zbl 0713.76005 · doi:10.1007/978-1-4612-4650-3
[46] Sarria, A.; Wu, J., Blowup in stagnation-point form solutions of the inviscid 2d Boussinesq equations, J. Differ. Equ., 259, 3559-3576 (2015) · Zbl 1327.35044 · doi:10.1016/j.jde.2015.04.029
[47] Schonbek, M., \(L^2\) decay for weak solutions of the Navier-Stokes equations, Arch. Rational Mech. Anal., 88, 209-222 (1985) · Zbl 0602.76031 · doi:10.1007/BF00752111
[48] Schonbek, ME; Schonbek, T., Moments and lower bounds in the far-field of solutions to quasi-geostrophic flows, Discr. Contin. Dyn. Syst., 13, 1277-1304 (2005) · Zbl 1091.35070 · doi:10.3934/dcds.2005.13.1277
[49] Schonbek, M.; Wiegner, M., On the decay of higher-order norms of the solutions of Navier-Stokes equations, Proc. R. Soc. Edinb. Sect. A, 126, 677-685 (1996) · Zbl 0862.35086 · doi:10.1017/S0308210500022976
[50] Stefanov, A.; Wu, J., A global regularity result for the 2D Boussinesq equations with critical dissipation, J. Anal. Math., 137, 269-290 (2019) · Zbl 1420.35263 · doi:10.1007/s11854-018-0073-4
[51] Tao, T.: https://terrytao.wordpress.com/2007/03/18/why-global-regularity-for-navier-stokes-is-hard/
[52] Tao, T.: Nonlinear dispersive equations: local and global analysis. In: CBMS Regional Conference Series in Mathematics, vol. 106. Amercian Mathematical Society, Providence, RI (2006) · Zbl 1106.35001
[53] Tao, L.; Wu, J., The 2D Boussinesq equations with vertical dissipation and linear stability of shear flows, J. Differ. Equ., 267, 1731-1747 (2019) · Zbl 1416.35218 · doi:10.1016/j.jde.2019.02.020
[54] Tao, L.; Wu, J.; Zhao, K.; Zheng, X., Stability near hydrostatic equilibrium to the 2D Boussinesq equations without thermal diffusion, Arch. Ration. Mech. Anal., 237, 585-630 (2020) · Zbl 1437.35549 · doi:10.1007/s00205-020-01515-5
[55] Wan, R., Global well-posedness for the 2D Boussinesq equations with a velocity damping term, Discr. Contin. Dyn. Syst., 39, 2709-2730 (2019) · Zbl 1412.35269 · doi:10.3934/dcds.2019113
[56] Wen, B.; Dianati, N.; Lunasin, E.; Chini, GP; Doering, CR, New upper bounds and reduced dynamical modeling for Rayleigh-Bénard convection in a fluid saturated porous layer, Commun. Nonlinear Sci. Numer. Simul., 17, 2191-2199 (2012) · doi:10.1016/j.cnsns.2011.06.039
[57] Wu, J., Dissipative quasi-geostrophic equations with \(L^p\) data, Electron. J. Differ. Equ., 2001, 1-13 (2001) · Zbl 0987.35127
[58] Wu, J., The 2D Boussinesq Equations with Partial or Fractional Dissipation, Lectures on the Analysis of Nonlinear Partial Differential Equations, Morningside Lectures in Mathematics, Part 4, 223-269 (2016), Somerville: International Press, Somerville · Zbl 1348.35204
[59] Wu, J.; Xu, X., Well-posedness and inviscid limits of the Boussinesq equations with fractional Laplacian dissipation, Nonlinearity, 27, 2215-2232 (2014) · Zbl 1301.35115 · doi:10.1088/0951-7715/27/9/2215
[60] Wu, J.; Xu, X.; Xue, L.; Ye, Z., Regularity results for the 2d Boussinesq equations with critical and supercritical dissipation, Commun. Math. Sci., 14, 1963-1997 (2016) · Zbl 1358.35136 · doi:10.4310/CMS.2016.v14.n7.a9
[61] Wu, J.; Xu, X.; Ye, Z., The 2D Boussinesq equations with fractional horizontal dissipation and thermal diffusion, J. Math. Pures Appl., 115, 187-217 (2018) · Zbl 1392.35240 · doi:10.1016/j.matpur.2018.01.006
[62] Wu, J.; Xu, X.; Zhu, N., Stability and decay rates for a variant of the 2D Boussinesq-Bénard system, Commun. Math. Sci., 17, 2325-2352 (2019) · Zbl 1434.35128 · doi:10.4310/CMS.2019.v17.n8.a11
[63] Xu, X., Fast growth of the vorticity gradient in symmetric smooth domains for 2D incompressible ideal flow, J. Math. Anal. Appl., 439, 594-607 (2016) · Zbl 1339.35251 · doi:10.1016/j.jmaa.2016.02.066
[64] Yang, J.; Lin, Z., Linear inviscid damping for Couette flow in stratified fluid, J. Math. Fluid Mech., 20, 445-472 (2018) · Zbl 1460.76298 · doi:10.1007/s00021-017-0328-3
[65] Yang, W.; Jiu, Q.; Wu, J., Global well-posedness for a class of 2D Boussinesq systems with fractional dissipation, J. Differ. Equ., 257, 4188-4213 (2014) · Zbl 1300.35108 · doi:10.1016/j.jde.2014.08.006
[66] Yang, W.; Jiu, Q.; Wu, J., The 3D incompressible Boussinesq equations with fractional partial dissipation, Commun. Math. Sci., 16, 617-633 (2018) · Zbl 1404.35373 · doi:10.4310/CMS.2018.v16.n3.a2
[67] Ye, Z.; Xu, X., Global well-posedness of the 2D Boussinesq equations with fractional Laplacian dissipation, J. Differ. Equ., 260, 6716-6744 (2016) · Zbl 1341.35135 · doi:10.1016/j.jde.2016.01.014
[68] Zhang, J.; Zhao, J., Some decay estimates of solutions for the 3-D compressible isentropic magnetohydrodynamics, Commun. Math. Sci., 8, 835-850 (2010) · Zbl 1372.76115 · doi:10.4310/CMS.2010.v8.n4.a2
[69] Zhao, K., 2D inviscid heat conductive Boussinesq system on a bounded domain, Michigan Math. J., 59, 329-352 (2010) · Zbl 1205.35048 · doi:10.1307/mmj/1281531460
[70] Zillinger, C.: On Enhanced Dissipation for the Boussinesq Equations. arXiv: 2004.08125v1 [math.AP] (17 Apr 2020)
[71] Zlatoš, A., Exponential growth of the vorticity gradient for the Euler equation on the torus, Adv. Math., 268, 396-403 (2015) · Zbl 1308.35194 · doi:10.1016/j.aim.2014.08.012
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