September  2015, 8(3): 395-411. doi: 10.3934/krm.2015.8.395

On the well-posedness of the inviscid Boussinesq equations in the Besov-Morrey spaces

1. 

School of Mathematics and Computational Science, Sun Yat-Sen University, Guangzhou, 510275, China, China

2. 

Department of Mathematics and Computational Science, Sun Yat-Sen University, Guangzhou 510275, China

Received  March 2014 Revised  December 2014 Published  June 2015

This paper is devoted to the study of the inviscid Boussinesq equations. We establish the local well-posedness and blow-up criteria in Besov-Morrey spaces $N_{p,q,r}^s(\mathbb{R}^n)$ for super critical case $s > 1 + \frac{n}{p}, 1 < q \leq p < \infty, 1 \leq r\leq \infty$, and critical case $s=1+\frac{n}{p}, 1 < q \leq p < \infty, r=1$. Main analysis tools are Littlewood-Paley decomposition and the paradifferential calculus.
Citation: Qunyi Bie, Qiru Wang, Zheng-An Yao. On the well-posedness of the inviscid Boussinesq equations in the Besov-Morrey spaces. Kinetic & Related Models, 2015, 8 (3) : 395-411. doi: 10.3934/krm.2015.8.395
References:
[1]

H. Bahouri, J. Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations,, Grundlehren der mathematischen Wissenschaften, (2011).  doi: 10.1007/978-3-642-16830-7.  Google Scholar

[2]

H. Berestycki, P. Constantin and L. Ryzhik, Non-planar fronts in Boussinesq reactive flows,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 23 (2006), 407.  doi: 10.1016/j.anihpc.2004.10.010.  Google Scholar

[3]

J. Bergh and J. Löfström, Interpolation Spaces, An Introduction,, Springer-Verlag, (1976).   Google Scholar

[4]

Y. Brenier, Optimal transport, convection, magnetic relaxation and generalized Boussinesq equations,, J. Nonlinear Sci., 19 (2009), 547.  doi: 10.1007/s00332-009-9044-3.  Google Scholar

[5]

D. Chae, S.-K. Kim and H.-S. Nam, Local existence and blow-up criterion of Hölder continuous solutions of the Boussinesq equations,, Nagoya Math. J., 155 (1999), 55.   Google Scholar

[6]

D. Chae and H. Nam, Local existence and blow-up criterion for the Boussinesq equations,, Proc. Roy. Soc. Edinburgh Sect. A, 127 (1997), 935.  doi: 10.1017/S0308210500026810.  Google Scholar

[7]

X. Cui, C. Dou and Q. Jiu, Local well-posedness and blow up criterion for the inviscid Boussinesq system in Hölder spaces,, J. Partial Differ. Equ., 25 (2012), 220.   Google Scholar

[8]

W. E and C. Shu, Small scale structures on Boussinesq convection,, Phys. Fluids, 6 (1994), 49.  doi: 10.1063/1.868044.  Google Scholar

[9]

B. Guo, Spectral method for solving two-dimensional Newton-Boussinesq equation,, Acta Math. Appl. Sinica (English Ser.), 5 (1989), 201.  doi: 10.1007/BF02006004.  Google Scholar

[10]

H. Kozoho and M. Yamazaki, Semilinear heat equations and the Navier-Stokes equations with distributions in new function spaces as initial data,, Comm. Partial Differential Equations, 19 (1994), 959.  doi: 10.1080/03605309408821042.  Google Scholar

[11]

P. G. Lemarié-Rieusset, Recent Progress in the Navier-Stokes Problem,, Chapman and Hall/CRC, (2002).  doi: 10.1201/9781420035674.  Google Scholar

[12]

X. Liu, M. Wang and Z. Zhang, Local well-posedness and blowup criterion of the Boussinesq equations in critical Besov spaces,, J. Math. Fluid Mech., 12 (2010), 280.  doi: 10.1007/s00021-008-0286-x.  Google Scholar

[13]

A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow,, Cambridge University Press, (2002).   Google Scholar

[14]

L. Tang, A remark on the well-posedness of the Euler equation in the Besov-Morrey space,, Available from: , ().   Google Scholar

[15]

Y. Taniuchi, A note on the blow-up criterion for the inviscid 2D Boussinesq equations,, Lecture Notes in Pure and Appl. Math., 223 (2002), 131.   Google Scholar

[16]

Z. Xiang and W. Yan, On the well-posedness of the Boussinesq equation in the Triebel-Lizorkin-Lorentz spaces,, Abstr. Appl. Anal., 2012 (2012).   Google Scholar

[17]

X. Xu, Local existence and blow-up criterion of the 2-D compressible Boussinesq equations without dissipation terms,, Discrete Contin. Dyn. Syst., 25 (2009), 1333.  doi: 10.3934/dcds.2009.25.1333.  Google Scholar

[18]

J. Xu and Y. Zhou, Commutator estimates in Besov-Morrey spaces with applications to the well-posedness of the Euler equations and ideal MHD system,, preprint, ().   Google Scholar

[19]

B. Yuan, Local existence and continuity conditions of solutions to the Boussinesq equations in Besov spaces,, Acta Math. Sinica (Chin. Ser.), 53 (2010), 455.   Google Scholar

show all references

References:
[1]

H. Bahouri, J. Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations,, Grundlehren der mathematischen Wissenschaften, (2011).  doi: 10.1007/978-3-642-16830-7.  Google Scholar

[2]

H. Berestycki, P. Constantin and L. Ryzhik, Non-planar fronts in Boussinesq reactive flows,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 23 (2006), 407.  doi: 10.1016/j.anihpc.2004.10.010.  Google Scholar

[3]

J. Bergh and J. Löfström, Interpolation Spaces, An Introduction,, Springer-Verlag, (1976).   Google Scholar

[4]

Y. Brenier, Optimal transport, convection, magnetic relaxation and generalized Boussinesq equations,, J. Nonlinear Sci., 19 (2009), 547.  doi: 10.1007/s00332-009-9044-3.  Google Scholar

[5]

D. Chae, S.-K. Kim and H.-S. Nam, Local existence and blow-up criterion of Hölder continuous solutions of the Boussinesq equations,, Nagoya Math. J., 155 (1999), 55.   Google Scholar

[6]

D. Chae and H. Nam, Local existence and blow-up criterion for the Boussinesq equations,, Proc. Roy. Soc. Edinburgh Sect. A, 127 (1997), 935.  doi: 10.1017/S0308210500026810.  Google Scholar

[7]

X. Cui, C. Dou and Q. Jiu, Local well-posedness and blow up criterion for the inviscid Boussinesq system in Hölder spaces,, J. Partial Differ. Equ., 25 (2012), 220.   Google Scholar

[8]

W. E and C. Shu, Small scale structures on Boussinesq convection,, Phys. Fluids, 6 (1994), 49.  doi: 10.1063/1.868044.  Google Scholar

[9]

B. Guo, Spectral method for solving two-dimensional Newton-Boussinesq equation,, Acta Math. Appl. Sinica (English Ser.), 5 (1989), 201.  doi: 10.1007/BF02006004.  Google Scholar

[10]

H. Kozoho and M. Yamazaki, Semilinear heat equations and the Navier-Stokes equations with distributions in new function spaces as initial data,, Comm. Partial Differential Equations, 19 (1994), 959.  doi: 10.1080/03605309408821042.  Google Scholar

[11]

P. G. Lemarié-Rieusset, Recent Progress in the Navier-Stokes Problem,, Chapman and Hall/CRC, (2002).  doi: 10.1201/9781420035674.  Google Scholar

[12]

X. Liu, M. Wang and Z. Zhang, Local well-posedness and blowup criterion of the Boussinesq equations in critical Besov spaces,, J. Math. Fluid Mech., 12 (2010), 280.  doi: 10.1007/s00021-008-0286-x.  Google Scholar

[13]

A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow,, Cambridge University Press, (2002).   Google Scholar

[14]

L. Tang, A remark on the well-posedness of the Euler equation in the Besov-Morrey space,, Available from: , ().   Google Scholar

[15]

Y. Taniuchi, A note on the blow-up criterion for the inviscid 2D Boussinesq equations,, Lecture Notes in Pure and Appl. Math., 223 (2002), 131.   Google Scholar

[16]

Z. Xiang and W. Yan, On the well-posedness of the Boussinesq equation in the Triebel-Lizorkin-Lorentz spaces,, Abstr. Appl. Anal., 2012 (2012).   Google Scholar

[17]

X. Xu, Local existence and blow-up criterion of the 2-D compressible Boussinesq equations without dissipation terms,, Discrete Contin. Dyn. Syst., 25 (2009), 1333.  doi: 10.3934/dcds.2009.25.1333.  Google Scholar

[18]

J. Xu and Y. Zhou, Commutator estimates in Besov-Morrey spaces with applications to the well-posedness of the Euler equations and ideal MHD system,, preprint, ().   Google Scholar

[19]

B. Yuan, Local existence and continuity conditions of solutions to the Boussinesq equations in Besov spaces,, Acta Math. Sinica (Chin. Ser.), 53 (2010), 455.   Google Scholar

[1]

Xavier Carvajal, Liliana Esquivel, Raphael Santos. On local well-posedness and ill-posedness results for a coupled system of mkdv type equations. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020382

[2]

Justin Holmer, Chang Liu. Blow-up for the 1D nonlinear Schrödinger equation with point nonlinearity II: Supercritical blow-up profiles. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020264

[3]

Antoine Benoit. Weak well-posedness of hyperbolic boundary value problems in a strip: when instabilities do not reflect the geometry. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5475-5486. doi: 10.3934/cpaa.2020248

[4]

Youshan Tao, Michael Winkler. Critical mass for infinite-time blow-up in a haptotaxis system with nonlinear zero-order interaction. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 439-454. doi: 10.3934/dcds.2020216

[5]

Hua Qiu, Zheng-An Yao. The regularized Boussinesq equations with partial dissipations in dimension two. Electronic Research Archive, 2020, 28 (4) : 1375-1393. doi: 10.3934/era.2020073

[6]

Bahaaeldin Abdalla, Thabet Abdeljawad. Oscillation criteria for kernel function dependent fractional dynamic equations. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020443

[7]

Thabet Abdeljawad, Mohammad Esmael Samei. Applying quantum calculus for the existence of solution of $ q $-integro-differential equations with three criteria. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020440

[8]

Mostafa Mbekhta. Representation and approximation of the polar factor of an operator on a Hilbert space. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020463

[9]

Abdelghafour Atlas, Mostafa Bendahmane, Fahd Karami, Driss Meskine, Omar Oubbih. A nonlinear fractional reaction-diffusion system applied to image denoising and decomposition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020321

[10]

Hirokazu Ninomiya. Entire solutions of the Allen–Cahn–Nagumo equation in a multi-dimensional space. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 395-412. doi: 10.3934/dcds.2020364

[11]

Thierry Cazenave, Ivan Naumkin. Local smooth solutions of the nonlinear Klein-gordon equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020448

[12]

Gongbao Li, Tao Yang. Improved Sobolev inequalities involving weighted Morrey norms and the existence of nontrivial solutions to doubly critical elliptic systems involving fractional Laplacian and Hardy terms. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020469

[13]

Dan Zhu, Rosemary A. Renaut, Hongwei Li, Tianyou Liu. Fast non-convex low-rank matrix decomposition for separation of potential field data using minimal memory. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020076

[14]

Leilei Wei, Yinnian He. A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020319

[15]

Gunther Uhlmann, Jian Zhai. Inverse problems for nonlinear hyperbolic equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 455-469. doi: 10.3934/dcds.2020380

[16]

Thomas Bartsch, Tian Xu. Strongly localized semiclassical states for nonlinear Dirac equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 29-60. doi: 10.3934/dcds.2020297

[17]

Lorenzo Zambotti. A brief and personal history of stochastic partial differential equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 471-487. doi: 10.3934/dcds.2020264

[18]

Hua Chen, Yawei Wei. Multiple solutions for nonlinear cone degenerate elliptic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020272

[19]

Fabio Camilli, Giulia Cavagnari, Raul De Maio, Benedetto Piccoli. Superposition principle and schemes for measure differential equations. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020050

[20]

Zhenzhen Wang, Tianshou Zhou. Asymptotic behaviors and stochastic traveling waves in stochastic Fisher-KPP equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020323

2019 Impact Factor: 1.311

Metrics

  • PDF downloads (43)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]