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Local existence and blow-up criterion of Hölder continuous solutions of the Boussinesq equations

  • Dongho Chae (a1), Sung-Ki Kim (a2) and Hee-Seok Nam (a3)

Abstract

In this paper we prove the local existence and uniqueness of C 1+γ solutions of the Boussinesq equations with initial data υ0, θ0 C1+γ , ω 0 , ∇θ0Lq for 0 < γ < 1 and 1 < q < 2. We also obtain a blow-up criterion for this local solutions. More precisely we show that the gradient of the passive scalar θ controls the breakdown of C1+γ solutions of the Boussinesq equations.

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References

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[1] Bahouri, H. et Dehman, B., Remarques sur l’apparition de singularités dans les écoulements Eulériens incompressibles à donnée initiale Höldérienne, J. Math. Pures Appl., 73 (1994), 335346.
[2] Beale, J. T., Kato, T. and Majda, A., Remarks on the breakdown of smooth solutions for the 3-D Euler equations, Commun. Math. Phys., 94, 6166.
[3] Bergers, M.S., Nonlinearity and functional analysis, Academic Press, INC, 1977.
[4] Chae, Dongho and Nam, Hee-Seok, Local existence and blow-up criterion for the Boussinesq equations, Proc. Royal Soc. Edinburgh, A, 127 (1997), no. 5, 935946.
[5] Chemin, J.-Y., Régularité de la trajectoire des particules d’un fluide parfait incompressible remplissant l’espace, J. Math. Pures Appl., 71 (1992), 407417.
[6] Kato, T., Remarks on the Euler and Navier-Stokes equations in R2, Nonlinear functional analysis and its applications, 45 (Part 2), 17.
[7] Majda, A., Vorticity and the mathematical theory of incompressible fluid flow, Princeton University graduate course lecture note (19861987).
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