doi: 10.3934/dcdsb.2022057
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Smoothing effect and well-posedness for 2D Boussinesq equations in critical Sobolev space

Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China

* Corresponding author: Chenyin Qian

Received  November 2021 Revised  February 2022 Early access March 2022

Fund Project: The second author is supported by Natural Science Foundation of Zhejiang Province, No. LY20A010017

In this paper, we investigate the fractional dissipation 2D Boussinesq equations with initial data in the critical space $ u_0\in H^{2-2\alpha}(\mathbb{R}^2) $ and $ \theta_0\in H^{2-2\beta}(\mathbb{R}^2) $. The local well-posedness for the equations is firstly established by using some a priori estimates for the solution in $ L^{p}(0, T;{H}^{2-\frac{p-1}{p} 2\alpha}(\mathbb{R}^2))\times L^{p}(0, T;{H}^{2-\frac{p-1}{p} 2\beta}(\mathbb{R}^2)) $ with some suitable $ p $. And then the generalized blow-up criterion and smoothing effect are obtained in turn, which improves some of the previous results for (critical, subcritcial or supcritical) Boussnesq equations. The results of the present paper are based on the Littlewood-Paley theory and the nonlinear lower bounds estimates for the fractional Laplacian, and can be treated as a generalization of results for 2D quasi-geostrophic equation.

Citation: Aiting Le, Chenyin Qian. Smoothing effect and well-posedness for 2D Boussinesq equations in critical Sobolev space. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022057
References:
[1]

H. Abidi and T. Hmidi, On the global well-posedness for Boussinesq equations, J. Differ. Equ., 233 (2007), 199-220.  doi: 10.1016/j.jde.2006.10.008.

[2]

H. Bahouri, J.-Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, volume 343., Springer Science & Business Media, 2011. doi: 10.1007/978-3-642-16830-7.

[3]

D. Chae, Global regularity for the 2D Boussinesq equations with partial viscosity terms, Adv. Math, 203 (2006), 497-513.  doi: 10.1016/j.aim.2005.05.001.

[4]

D. ChaeS. K. Kim and H. S. Nam, Local existence and blow-up criterion of Hölder continuous solutions of the Boussinesq equation, Nagoya Math. J, 155 (1999), 55-80.  doi: 10.1017/S0027763000006991.

[5]

P. Constantin and V. Vicol, Nonlinear maximum principles for dissipative linear nonlocal operators and applications, Geom. Funct. Anal., 22 (2012), 1289-1321.  doi: 10.1007/s00039-012-0172-9.

[6]

A. Cordoba and D. Cordoba, A maximum principle applied to quasi-geostrophic equations, Commun.Math.Phys, 249 (2004), 511-528.  doi: 10.1007/s00220-004-1055-1.

[7]

R. Danchin, Density-dependent incompressible viscous fluids in critical spaces, Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 133 (2003), 1311-1334.  doi: 10.1017/S030821050000295X.

[8]

R. Danchin and M. Paicu, Existence and uniqueness results for the Boussinesq equations with data in Lorentz spaces, Phys. D, Nonlinear Phenom, 237 (2008), 1444-1460.  doi: 10.1016/j.physd.2008.03.034.

[9]

H. Dong, Dissipative Quasi-Geostrophic equation in critical Sobolev space: Smoothing effect and global well-posedness, Discrete and Continuous Dynamical Equationss, 24 (2010), 1197-1211.  doi: 10.3934/dcds.2010.26.1197.

[10]

H. DongD. Du and F. Lin, Global well-posedness and a decay estimate for the critical dissipative Quasi-Geostrophic equation in the whole space, Discrete and Continuous Dynamical Equationss, 21 (2008), 1095-1011.  doi: 10.3934/dcds.2008.21.1095.

[11]

D. FangC. Qian and T. Zhang, Global well-posedness for 2D Boussinesq equations with general supercritical dissipation, Nonlinear Analysis: Real Word Applications, 27 (2016), 326-349.  doi: 10.1016/j.nonrwa.2015.08.004.

[12]

R. Farwig, C. Qian and P. Zhang, Incompressible inhomogeneous fluids in bounded domain of $\mathbb{R}^3$ with bounded density, Journal of Functional Analysis, 278 (2020), 108394, 36pp. doi: 10.1016/j.jfa.2019.108394.

[13]

F. Hadadifard and A. Stefanov, On the global regularity of the 2D critical Boussinesq system with $\alpha>\frac{2}{3}$, Comm. Math. Sci., 15 (2017), 1325-1351.  doi: 10.4310/CMS.2017.v15.n5.a6.

[14]

Z. Hassainia and T. Hmidi, On the inviscid Boussinesq equations with rough initial data, J. Math. Anal. Appl, 430 (2015), 777-809.  doi: 10.1016/j.jmaa.2015.04.087.

[15]

T. Hmidi and S. Keraani, On the global well-posedness of the Boussinesq equations with zero viscosity, Indiana Univ. Math. J, 58 (2009), 1591-1618.  doi: 10.1512/iumj.2009.58.3590.

[16]

T. HmidiS. Keraani and F. Rousset, Global well-posedness of Euler-Boussinesq equations with critical dissipation, Commun. Partial Differ. Equ, 36 (2011), 420-445.  doi: 10.1080/03605302.2010.518657.

[17]

T. Y. Hou and C. Li, Global well-posedness of the viscous Boussinesq equations, Discrete Contin. Dyn. Syst, 12 (2005), 1-12.  doi: 10.3934/dcds.2005.12.1.

[18]

W. HuI. Kukavica and M. Ziane, Persistence of regularity for the viscous Boussinesq equation with zero diffusivity, Asymptot. Anal., 91 (2015), 111-124.  doi: 10.3233/ASY-141261.

[19]

Q. JiuC. MiaoJ. Wu and Z. Zhang, The 2D incompressible Boussinesq equations with general critical dissipation, SIAM J. Math. Anal, 46 (2014), 3426-3454.  doi: 10.1137/140958256.

[20]

H. Kozono and Y. Taniuchi, Limiting case of the sobolev inequality in BMO, with application to the Euler equality, Commun.Math.Phys, 214 (2000), 191-200.  doi: 10.1007/s002200000267.

[21]

I. KukavicaF. Wang and M. Ziane, Persistence of regularity for solutions of the Boussinesq equations in Sobolev spaces, Adv. Differ. Equ, 21 (2016), 85-108. 

[22]

M. LaiR. Pan and K. Zhao, Initial boundary value problem for two-dimensional viscous Boussinesq equations, Arch. Ration. Mech. Anal., 199 (2011), 739-760.  doi: 10.1007/s00205-010-0357-z.

[23]

C. Miao and L. Xue, On the global well-posedness of a class of Boussinesq-Navier-Stokes equations, NoDEA Nonlinear Differential Equations Appl, 18 (2011), 707-735.  doi: 10.1007/s00030-011-0114-5.

[24]

H. Miura, Dissipative Quasi-Geostrophic equation for large initial data in the critical Sobolev space, Commun.Math.Phys, 267 (2006), 141-157.  doi: 10.1007/s00220-006-0023-3.

[25]

A. Stefanov and J. Wu, A global regularity result for the Boussinesq equations with critical dissipation, Journal d'Analyse Mathématique, 137 (2019), 269-290.  doi: 10.1007/s11854-018-0073-4.

[26]

J. WuX. XuL. Xue and Z. Ye, Regularity results for the 2D Boussinesq equations with critical and supercritical dissipation, Commun. Math. Sci, 14 (2016), 1963-1997.  doi: 10.4310/CMS.2016.v14.n7.a9.

[27]

F. Xu and J. Yuan, On the global well-posedness for the 2D Euler-Boussinesq equations, Nonlinear Anal, Real World Appl, 17 (2014), 137-146.  doi: 10.1016/j.nonrwa.2013.11.001.

[28]

Z. Ye and X. Xu, Global well-posedness of the 2D Boussinesq equations with fractional Laplacian dissipation, Journal of Differential Equations, 260 (2016), 6716-6744.  doi: 10.1016/j.jde.2016.01.014.

[29]

D. Zhou and Z. Li, Global well-posedness for the 2D Boussinesq equation with zero viscosity, J. Math. Anal. Appl., 447 (2017), 1072–1079, arXiv: 1603.08301v2 [math.AP]. doi: 10.1016/j.jmaa.2016.10.058.

[30]

D. ZhouZ. LiH. ShangJ. WuB. Yuan and J. Zhao, Global well-posedness for the 2D fractional Boussinesq equations in the subcritical case, Pacific Journal of Mathematics, 298 (2019), 233-255.  doi: 10.2140/pjm.2019.298.233.

show all references

References:
[1]

H. Abidi and T. Hmidi, On the global well-posedness for Boussinesq equations, J. Differ. Equ., 233 (2007), 199-220.  doi: 10.1016/j.jde.2006.10.008.

[2]

H. Bahouri, J.-Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, volume 343., Springer Science & Business Media, 2011. doi: 10.1007/978-3-642-16830-7.

[3]

D. Chae, Global regularity for the 2D Boussinesq equations with partial viscosity terms, Adv. Math, 203 (2006), 497-513.  doi: 10.1016/j.aim.2005.05.001.

[4]

D. ChaeS. K. Kim and H. S. Nam, Local existence and blow-up criterion of Hölder continuous solutions of the Boussinesq equation, Nagoya Math. J, 155 (1999), 55-80.  doi: 10.1017/S0027763000006991.

[5]

P. Constantin and V. Vicol, Nonlinear maximum principles for dissipative linear nonlocal operators and applications, Geom. Funct. Anal., 22 (2012), 1289-1321.  doi: 10.1007/s00039-012-0172-9.

[6]

A. Cordoba and D. Cordoba, A maximum principle applied to quasi-geostrophic equations, Commun.Math.Phys, 249 (2004), 511-528.  doi: 10.1007/s00220-004-1055-1.

[7]

R. Danchin, Density-dependent incompressible viscous fluids in critical spaces, Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 133 (2003), 1311-1334.  doi: 10.1017/S030821050000295X.

[8]

R. Danchin and M. Paicu, Existence and uniqueness results for the Boussinesq equations with data in Lorentz spaces, Phys. D, Nonlinear Phenom, 237 (2008), 1444-1460.  doi: 10.1016/j.physd.2008.03.034.

[9]

H. Dong, Dissipative Quasi-Geostrophic equation in critical Sobolev space: Smoothing effect and global well-posedness, Discrete and Continuous Dynamical Equationss, 24 (2010), 1197-1211.  doi: 10.3934/dcds.2010.26.1197.

[10]

H. DongD. Du and F. Lin, Global well-posedness and a decay estimate for the critical dissipative Quasi-Geostrophic equation in the whole space, Discrete and Continuous Dynamical Equationss, 21 (2008), 1095-1011.  doi: 10.3934/dcds.2008.21.1095.

[11]

D. FangC. Qian and T. Zhang, Global well-posedness for 2D Boussinesq equations with general supercritical dissipation, Nonlinear Analysis: Real Word Applications, 27 (2016), 326-349.  doi: 10.1016/j.nonrwa.2015.08.004.

[12]

R. Farwig, C. Qian and P. Zhang, Incompressible inhomogeneous fluids in bounded domain of $\mathbb{R}^3$ with bounded density, Journal of Functional Analysis, 278 (2020), 108394, 36pp. doi: 10.1016/j.jfa.2019.108394.

[13]

F. Hadadifard and A. Stefanov, On the global regularity of the 2D critical Boussinesq system with $\alpha>\frac{2}{3}$, Comm. Math. Sci., 15 (2017), 1325-1351.  doi: 10.4310/CMS.2017.v15.n5.a6.

[14]

Z. Hassainia and T. Hmidi, On the inviscid Boussinesq equations with rough initial data, J. Math. Anal. Appl, 430 (2015), 777-809.  doi: 10.1016/j.jmaa.2015.04.087.

[15]

T. Hmidi and S. Keraani, On the global well-posedness of the Boussinesq equations with zero viscosity, Indiana Univ. Math. J, 58 (2009), 1591-1618.  doi: 10.1512/iumj.2009.58.3590.

[16]

T. HmidiS. Keraani and F. Rousset, Global well-posedness of Euler-Boussinesq equations with critical dissipation, Commun. Partial Differ. Equ, 36 (2011), 420-445.  doi: 10.1080/03605302.2010.518657.

[17]

T. Y. Hou and C. Li, Global well-posedness of the viscous Boussinesq equations, Discrete Contin. Dyn. Syst, 12 (2005), 1-12.  doi: 10.3934/dcds.2005.12.1.

[18]

W. HuI. Kukavica and M. Ziane, Persistence of regularity for the viscous Boussinesq equation with zero diffusivity, Asymptot. Anal., 91 (2015), 111-124.  doi: 10.3233/ASY-141261.

[19]

Q. JiuC. MiaoJ. Wu and Z. Zhang, The 2D incompressible Boussinesq equations with general critical dissipation, SIAM J. Math. Anal, 46 (2014), 3426-3454.  doi: 10.1137/140958256.

[20]

H. Kozono and Y. Taniuchi, Limiting case of the sobolev inequality in BMO, with application to the Euler equality, Commun.Math.Phys, 214 (2000), 191-200.  doi: 10.1007/s002200000267.

[21]

I. KukavicaF. Wang and M. Ziane, Persistence of regularity for solutions of the Boussinesq equations in Sobolev spaces, Adv. Differ. Equ, 21 (2016), 85-108. 

[22]

M. LaiR. Pan and K. Zhao, Initial boundary value problem for two-dimensional viscous Boussinesq equations, Arch. Ration. Mech. Anal., 199 (2011), 739-760.  doi: 10.1007/s00205-010-0357-z.

[23]

C. Miao and L. Xue, On the global well-posedness of a class of Boussinesq-Navier-Stokes equations, NoDEA Nonlinear Differential Equations Appl, 18 (2011), 707-735.  doi: 10.1007/s00030-011-0114-5.

[24]

H. Miura, Dissipative Quasi-Geostrophic equation for large initial data in the critical Sobolev space, Commun.Math.Phys, 267 (2006), 141-157.  doi: 10.1007/s00220-006-0023-3.

[25]

A. Stefanov and J. Wu, A global regularity result for the Boussinesq equations with critical dissipation, Journal d'Analyse Mathématique, 137 (2019), 269-290.  doi: 10.1007/s11854-018-0073-4.

[26]

J. WuX. XuL. Xue and Z. Ye, Regularity results for the 2D Boussinesq equations with critical and supercritical dissipation, Commun. Math. Sci, 14 (2016), 1963-1997.  doi: 10.4310/CMS.2016.v14.n7.a9.

[27]

F. Xu and J. Yuan, On the global well-posedness for the 2D Euler-Boussinesq equations, Nonlinear Anal, Real World Appl, 17 (2014), 137-146.  doi: 10.1016/j.nonrwa.2013.11.001.

[28]

Z. Ye and X. Xu, Global well-posedness of the 2D Boussinesq equations with fractional Laplacian dissipation, Journal of Differential Equations, 260 (2016), 6716-6744.  doi: 10.1016/j.jde.2016.01.014.

[29]

D. Zhou and Z. Li, Global well-posedness for the 2D Boussinesq equation with zero viscosity, J. Math. Anal. Appl., 447 (2017), 1072–1079, arXiv: 1603.08301v2 [math.AP]. doi: 10.1016/j.jmaa.2016.10.058.

[30]

D. ZhouZ. LiH. ShangJ. WuB. Yuan and J. Zhao, Global well-posedness for the 2D fractional Boussinesq equations in the subcritical case, Pacific Journal of Mathematics, 298 (2019), 233-255.  doi: 10.2140/pjm.2019.298.233.

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