Artículos de revistas
On The Stability Problem For The Boussinesq Equations In Weak-lp Spaces
Registro en:
Communications On Pure And Applied Analysis. , v. 9, n. 3, p. 667 - 684, 2010.
15340392
10.3934/cpaa.2010.9.667
2-s2.0-77249129417
Autor
Ferreira L.C.F.
Villamizar-Roa E.J.
Institución
Resumen
We consider the Boussinesq equations in either an exterior domain in ℝn, the whole space ℝn, the half space ℝn + or a bounded domain in ℝn, where the dimension n satisfies n ≥ 3. We give a class of stable steady solutions, which improves and complements the previous stability results. Our results give a complete answer to the stability problem for the Boussinesq equations in weak-Lp spaces, in the sense that we only assume that the stable steady solution belongs to scaling invariant class L σ (n,∞) x L(n,∞). Moreover, some considerations about the exponential decay (in bounded domains) and the uniqueness of the disturbance are done. 9 3 667 684 Bergh, J., Löfström, J., (1976) Interpolation Spaces, , Springer-Verlag, Berlin Barraza, O., Regularity and stability for the solutions of the Navier-Stokes equations in Lorentz spaces (1999) Nonlinear Analysis, 35, pp. 747-764 Borchers, W., Miyakawa, T., On stability of exterior stationary Navier-Stokes flows (1995) Acta Math, 174, pp. 311-382 Biler, P., Cannone, M., Karch, G., Asymptotic stability of Navier-Stokes flow past an ob-stacle (2004) Nonlocal elliptic and parabolic problems, Banach Center Publ, 66, pp. 47-59 Cannone, M., Karch, G., About the regularized Navier-Stokes equations (2005) J. Math. Fluid Mechanics, 7, pp. 1-28 Karch, G., Prioux, N., Self-similarity in viscous Boussinesq equations (2008) Proc. Amer. Math. Soc, 136, pp. 879-888 Cannone, M., Planchon, F., Self-similar solutions for Navier-Stokes equations in ℝ3 (1996) Comm. Partial Differential Equations, 21, pp. 179-193 Chandrasekhar, S., (1981) Hydrodynamic and Hydromagnetic Stability, , Dover, New York Chen, Z., Kagei, Y., Miyakawa, T., Remarks on stability of purely conductive steady states to the exterior Boussinesq problem (1992) Adv. Math. Sci. Appl, 1, pp. 411-430 Galdi, G., Padula, M., A new approach to energy theory in the stability of fluid motion (1990) Arch. Rational Mech. Anal, 110, pp. 187-286 D. Fujiwara and H. Morimoto, An Lr-Theorem of the Helmholtz decomposition of vector fields, Fac. Sci. Univ. Tokio, sec. IA, 24 (1977), 685-700Hishida, T., Asymptotic behavior and stability of solutions to the exterior convection problem (1994) Nonlinear Anal, 22, pp. 895-925 Hishida, T., Global existence and exponential stability of convection (1995) J. Math. Anal. Appl, 196, pp. 699-721 Hishida, T., On a class of stable steady flow to the exterior convection problem (1997) J. Diff. Eq, 141, pp. 54-85 Joseph, D., (1976) Stability of Fluid Motion, , Springer-Verlag, Berlin Kozono, H., Yamazaki, M., On a larger class of stable solutions to the Navier-Stokes equations in exterior domains (1998) Math. Z, 228, pp. 751-785 Landau, L., Lifshitz, E., Theorical Physics: Fluid Mechanics (1987) 2nd edition, , Pergamon Press, Oxford Ferreira, L.C.F., Villamizar-Roa, E.J., Well-posedness and asymptotic behaviour for the convection problem in ℝ n (2006) Nonlinearity, 19, pp. 2169-2191 Ferreira, L.C.F., Villamizar-Roa, E.J., Existence of solutions to the convection problem in a pseudomeasure-type space (2008) Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci, 464, pp. 1983-1999 Lemarié-Rieusset, P., (2002) Recent Developments in the Navier-Stokes Problem, , Chapman & Hall/ CRC Press, Boca Raton Meyer, Y., (1997) Wavelets, paraproducts and Navier-Stokes equations, current developments in Mathematics 1996, pp. 105-212. , International Press, Cambridge O'Neil, R., Convolution operators and L(p, q) spaces (1963) Duke Math. J, 30, pp. 129-142 Yamazaki, M., The Navier-Stokes equations in the weak-Ln spaces with time-dependent ex-ternal force (2000) Math. Ann, 317, pp. 635-675