×

zbMATH — the first resource for mathematics

Global smooth solution to the 2D Boussinesq equations with fractional dissipation. (English) Zbl 1369.35072
Summary: In this paper, we consider the 2D incompressible Boussinesq system with fractional Laplacian dissipation and thermal diffusion. On the basis of the previous works and some new observations, we show that the condition \(1-\alpha < \beta < \min \left\{3-3 \alpha, \frac {\alpha}{2}, \frac {3 \alpha^2+4 \alpha-4}{8(1-\alpha)}\right\}\) with \(0.7351 \approx \frac {10-2\sqrt{10}}{5} < \alpha < 1\) suffices in order for the solution pair of velocity and temperature to remain smooth for all time.

MSC:
35Q35 PDEs in connection with fluid mechanics
35B65 Smoothness and regularity of solutions to PDEs
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Clavin, Nonlinear PDEs in Condensed Matter and Reactive Flows pp 49– (2002) · doi:10.1007/978-94-010-0307-0_3
[2] Droniou, Fractal first-order partial differential equations, Archive for Rational Mechanics and Analysis 182 pp 299– (2006) · Zbl 1111.35144 · doi:10.1007/s00205-006-0429-2
[3] Majda, Vorticity and Incompressible Flow (2001) · Zbl 0983.76001 · doi:10.1017/CBO9780511613203
[4] Pedlosky, Geophysical Fluid Dynamics (1987) · doi:10.1007/978-1-4612-4650-3
[5] Caputo, Linear models of dissipation whose Q is almost frequency independent-II, Geophysical Journal of the Royal Astronomical Society 13 pp 529– (1967) · Zbl 1210.65130 · doi:10.1111/j.1365-246X.1967.tb02303.x
[6] Constantin, Formation of strong fronts in the 2-D quasi-geostrophic thermal active scalar, Nonlinearity 7 pp 1495– (1994) · Zbl 0809.35057 · doi:10.1088/0951-7715/7/6/001
[7] Cannon, Lecture Notes in Mathematics, in: The Initial Value Problem for the Boussinesq Equation with Data in lp pp 129– (1980) · doi:10.1007/BFb0086903
[8] Chae, Global regularity for the 2D Boussinesq equations with partial viscosity terms, Advances in Mathematics 203 pp 497– (2006) · Zbl 1100.35084 · doi:10.1016/j.aim.2005.05.001
[9] Hou, Global well-posedness of the viscous Boussinesq equations, Discrete and Continuous Dynamical Systems 12 pp 1– (2005)
[10] Xu, Global regularity of solutions of 2D Boussinesq equations with fractional diffusion, Nonlinear Analysis 72 pp 677– (2010) · Zbl 1177.76024 · doi:10.1016/j.na.2009.07.008
[11] Hmidi, On the global well-posedness of the Euler-Boussinesq system with fractional dissipation, Physica D 239 pp 1387– (2010) · Zbl 1194.35329 · doi:10.1016/j.physd.2009.12.009
[12] Hmidi, Global well-posedness for a Boussinesq-Navier-Stokes system with critical dissipation, Journal Differential Equations 249 pp 2147– (2010) · Zbl 1200.35228 · doi:10.1016/j.jde.2010.07.008
[13] Hmidi, Global well-posedness for Euler-Boussinesq system with critical dissipation, Communications in Partial Differential Equations 36 pp 420– (2011) · Zbl 1284.76089 · doi:10.1080/03605302.2010.518657
[14] Jiu, The 2D incompressible Boussinesq equations with general critical dissipation, SIAM Journal on Mathematical Analysis 46 pp 3426– (2014) · Zbl 1319.35193 · doi:10.1137/140958256
[15] Stefanov A Wu J A global regularity result for the 2D Boussinesq equations with critical dissipation
[16] Wu, Regularity results for the 2D Boussinesq equations with critical and supercritical dissipation, Communications in Mathematical Sciences 14 pp 1963– (2016) · Zbl 1358.35136 · doi:10.4310/CMS.2016.v14.n7.a9
[17] Constantin, Nonlinear maximum principles for dissipative linear nonlocal operators and applications, Geometric and Functional Analysis 22 pp 1289– (2012) · Zbl 1256.35078 · doi:10.1007/s00039-012-0172-9
[18] Miao, On the global well-posedness of a class of Boussinesq-Navier-Stokes systems, NoDEA Nonlinear Differential Equations Applications 18 pp 707– (2011) · Zbl 1235.76020 · doi:10.1007/s00030-011-0114-5
[19] Yang, Global well-posedness for a class of 2D Boussinesq systems with fractional dissipation, Journal Differential Equations 257 pp 4188– (2014) · Zbl 1300.35108 · doi:10.1016/j.jde.2014.08.006
[20] Ye, A note on global regularity results for 2D Boussinesq equations with fractional dissipation, Annales Polonici Mathematici 117 pp 231– (2016)
[21] Ye, Remarks on global regularity of the 2D Boussinesq equations with fractional dissipation, Nonlinear Analysis 125 pp 715– (2015) · Zbl 1405.35171 · doi:10.1016/j.na.2015.06.021
[22] Ye, Global well-posedness of the 2D Boussinesq equations with fractional Laplacian dissipation, Journal Differential Equations 260 pp 6716– (2016) · Zbl 1341.35135 · doi:10.1016/j.jde.2016.01.014
[23] Ye, On the global regularity of the 2D Boussinesq equations with fractional dissipation, Mathematische Nachrichten pp 1– (2016) · Zbl 1346.35168
[24] Jiu, Eventual regularity of the two-dimensional Boussinesq equations with supercritical dissipation, Journal Nonlinear Science 25 pp 37– (2015) · Zbl 1311.35221 · doi:10.1007/s00332-014-9220-y
[25] Adhikari, Global regularity results for the 2D Boussinesq equations with partial dissipation, Journal Differential Equations 260 pp 1893– (2016) · Zbl 1328.35161 · doi:10.1016/j.jde.2015.09.049
[26] Adhikari, Small global solutions to the damped two-dimensional Boussinesq equations, Journal Differential Equations 256 pp 3594– (2014) · Zbl 1290.35193 · doi:10.1016/j.jde.2014.02.012
[27] Cao, Global regularity for the 2D anisotropic Boussinesq equations with vertical dissipation, Archive for Rational Mechanics and Analysis 208 pp 985– (2013) · Zbl 1284.35140 · doi:10.1007/s00205-013-0610-3
[28] Chae, The 2D Boussinesq equations with logarithmically supercritical velocities, Advances in Mathematics 230 pp 1618– (2012) · Zbl 1248.35156 · doi:10.1016/j.aim.2012.04.004
[29] Chae, An incompressible 2D didactic model with singularity and explicit solutions of the 2D Boussinesq equations, Journal of Mathematical Fluid Mechanics 16 pp 473– (2014) · Zbl 1307.35213 · doi:10.1007/s00021-014-0166-5
[30] Danchin, Remarks on the lifespan of the solutions to some models of incompressible fluid mechanics, Proceedings of the American Mathematical Society 141 pp 1979– (2013) · Zbl 1283.35080 · doi:10.1090/S0002-9939-2012-11591-6
[31] Danchin, Global existence results for the anisotropic Boussinesq system in dimension two, Mathematical Models and Methods in Applied Sciences 21 pp 421– (2011) · Zbl 1223.35249 · doi:10.1142/S0218202511005106
[32] Hmidi, On a maximum principle and its application to the logarithmically critical Boussinesq system, Analysis & PDE 4 pp 247– (2011) · Zbl 1264.35173 · doi:10.2140/apde.2011.4.247
[33] Hmidi, On the global well-posedness of the two-dimensional Boussinesq system with a zero diffusivity, Advances in Difference Equations 12 pp 461– (2007) · Zbl 1154.35073
[34] Jia, On the global well-posedness of a generalized 2D Boussinesq equations, NoDEA Nonlinear Differential Equations Applications 22 pp 911– (2015) · Zbl 1327.76048 · doi:10.1007/s00030-014-0309-7
[35] Kc, The 2D Euler-Boussinesq equations with a singular velocity, Journal Differential Equations 257 pp 82– (2014) · Zbl 1291.35221 · doi:10.1016/j.jde.2014.03.012
[36] Lai, Initial boundary value problem for two-dimensional viscous Boussinesq equations, Archive for Rational Mechanics and Analysis 199 pp 739– (2011) · Zbl 1231.35171 · doi:10.1007/s00205-010-0357-z
[37] Larios, Global well-posedness for the 2D Boussinesq system with anisotropic viscosity and without heat diffusion, Journal Differential Equations 255 pp 2636– (2013) · Zbl 1284.35343 · doi:10.1016/j.jde.2013.07.011
[38] Wu, Well-posedness and inviscid limits of the Boussinesq equations with fractional Laplacian dissipation, Nonlinearity pp 2215– (2014) · Zbl 1301.35115 · doi:10.1088/0951-7715/27/9/2215
[39] Wu, Global smooth solutions to the n-dimensional damped models of incompressible fluid mechanics with small initial datum, Journal Nonlinear Science 25 pp 157– (2015) · Zbl 1311.35236 · doi:10.1007/s00332-014-9224-7
[40] Ye, Blow-up criterion of smooth solutions for the Boussinesq equations, Nonlinear Analysis 110 pp 97– (2014) · Zbl 1300.35109 · doi:10.1016/j.na.2014.07.022
[41] Bahouri, Grundlehren der mathematischen Wissenschaften, in: Fourier Analysis and Nonlinear Partial Differential Equations (2011) · Zbl 1227.35004 · doi:10.1007/978-3-642-16830-7
[42] Miao, Littlewood-Paley Theory and Its Applications in Partial Differential Equations of Fluid Dynamics (2012)
[43] Runst, Nemytskij Operators and Nonlinear Partial Differential Equations, in: Sobolev Spaces of Fractional Order (1996) · Zbl 0873.35001
[44] Triebel, Theory of Function Spaces II (1992) · Zbl 1235.46003 · doi:10.1007/978-3-0346-0419-2
[45] Brezis, Gagliardo-Nirenberg, composition and products in fractional Sobolev spaces, Journal of Evolution Equations 1 pp 387– (2001) · Zbl 1023.46031 · doi:10.1007/PL00001378
[46] Chen, A new Bernstein inequality and the 2D dissipative quasigeostrophic equation, Communications in Mathematical Physics 271 pp 821– (2007) · Zbl 1142.35069 · doi:10.1007/s00220-007-0193-7
[47] Hajaiej, RIMS Kokyuroku Bessatsu B26: Harmonic Analysis and Nonlinear Partial Differential Equations pp 159– (2011)
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.