# American Institute of Mathematical Sciences

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 and 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, Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-16830-7. [2] H. Berestycki, P. Constantin and L. Ryzhik, Non-planar fronts in Boussinesq reactive flows, Ann. Inst. H. Poincaré Anal. Non Linéaire, 23 (2006), 407-437. doi: 10.1016/j.anihpc.2004.10.010. [3] J. Bergh and J. Löfström, Interpolation Spaces, An Introduction, Springer-Verlag, Berlin-New York, 1976. [4] Y. Brenier, Optimal transport, convection, magnetic relaxation and generalized Boussinesq equations, J. Nonlinear Sci., 19 (2009), 547-570. doi: 10.1007/s00332-009-9044-3. [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), 55-80. [6] D. Chae and H. Nam, Local existence and blow-up criterion for the Boussinesq equations, Proc. Roy. Soc. Edinburgh Sect. A, 127 (1997), 935-946. doi: 10.1017/S0308210500026810. [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), 220-238. [8] W. E and C. Shu, Small scale structures on Boussinesq convection, Phys. Fluids, 6 (1994), 49-58. doi: 10.1063/1.868044. [9] B. Guo, Spectral method for solving two-dimensional Newton-Boussinesq equation, Acta Math. Appl. Sinica (English Ser.), 5 (1989), 201-218. doi: 10.1007/BF02006004. [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), 959-1014. doi: 10.1080/03605309408821042. [11] P. G. Lemarié-Rieusset, Recent Progress in the Navier-Stokes Problem, Chapman and Hall/CRC, Boca Raton, FL, 2002. doi: 10.1201/9781420035674. [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), 280-292. doi: 10.1007/s00021-008-0286-x. [13] A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow, Cambridge University Press, 2002. [14] L. Tang, A remark on the well-posedness of the Euler equation in the Besov-Morrey space,, Available from: , (). [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), 131-140. [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), Art. ID 573087, 17 pp. [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), 1333-1347. doi: 10.3934/dcds.2009.25.1333. [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, (). [19] B. Yuan, Local existence and continuity conditions of solutions to the Boussinesq equations in Besov spaces, Acta Math. Sinica (Chin. Ser.), 53 (2010), 455-468.

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##### References:
 [1] H. Bahouri, J. Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der mathematischen Wissenschaften, Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-16830-7. [2] H. Berestycki, P. Constantin and L. Ryzhik, Non-planar fronts in Boussinesq reactive flows, Ann. Inst. H. Poincaré Anal. Non Linéaire, 23 (2006), 407-437. doi: 10.1016/j.anihpc.2004.10.010. [3] J. Bergh and J. Löfström, Interpolation Spaces, An Introduction, Springer-Verlag, Berlin-New York, 1976. [4] Y. Brenier, Optimal transport, convection, magnetic relaxation and generalized Boussinesq equations, J. Nonlinear Sci., 19 (2009), 547-570. doi: 10.1007/s00332-009-9044-3. [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), 55-80. [6] D. Chae and H. Nam, Local existence and blow-up criterion for the Boussinesq equations, Proc. Roy. Soc. Edinburgh Sect. A, 127 (1997), 935-946. doi: 10.1017/S0308210500026810. [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), 220-238. [8] W. E and C. Shu, Small scale structures on Boussinesq convection, Phys. Fluids, 6 (1994), 49-58. doi: 10.1063/1.868044. [9] B. Guo, Spectral method for solving two-dimensional Newton-Boussinesq equation, Acta Math. Appl. Sinica (English Ser.), 5 (1989), 201-218. doi: 10.1007/BF02006004. [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), 959-1014. doi: 10.1080/03605309408821042. [11] P. G. Lemarié-Rieusset, Recent Progress in the Navier-Stokes Problem, Chapman and Hall/CRC, Boca Raton, FL, 2002. doi: 10.1201/9781420035674. [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), 280-292. doi: 10.1007/s00021-008-0286-x. [13] A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow, Cambridge University Press, 2002. [14] L. Tang, A remark on the well-posedness of the Euler equation in the Besov-Morrey space,, Available from: , (). [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), 131-140. [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), Art. ID 573087, 17 pp. [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), 1333-1347. doi: 10.3934/dcds.2009.25.1333. [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, (). [19] B. Yuan, Local existence and continuity conditions of solutions to the Boussinesq equations in Besov spaces, Acta Math. Sinica (Chin. Ser.), 53 (2010), 455-468.
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