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

Qunyi  Bie; Qiru  Wang; Zheng-An Yao

doi:10.3934/krm.2015.8.395

Article Contents

2015, Volume 8, Issue 3: 395-411. Doi: 10.3934/krm.2015.8.395

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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
2.
Department of Mathematics and Computational Science, Sun Yat-Sen University, Guangzhou 510275

Received: March 2014

Revised: December 2014

Published: September 2015

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Abstract

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.

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Mathematics Subject Classification: Primary: 35Q35; Secondary: 76B03, 35E15.

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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: http://162.105.204.96/var/preprint/572.pdf.
[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, arXiv:1303.6005v1.
[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.