Abstract
A uniqueness result of weak solution for the 3D viscous magneto-hydrodynamics equations in \({B^1_{\infty,\infty}}\) is proved by means of the Fourier localization technique and the losing derivative estimates.
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Beirão da Veiga H.: A new regularity class for the Navier–Stokes equations in \({{\mathbb R}^n}\). Chin. Ann. Math. 16B, 407–412 (1995)
Bony J.M.: Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires. Ann. Sci. École Norm. Sup. 14, 209–246 (1981)
Chen Q., Miao C., Zhang Z.: The Beale–Kato–Majda criterion to the 3D magneto-hydrodynamics equations. Commun. Math. Phys. 275, 861–872 (2007)
Chen Q., Miao C., Zhang Z.: On the regularity criterion of weak solution for the 3D viscous magneto-hydrodynamics equations. Commun. Math. Phys. 284, 919–930 (2008)
Chen Q., Miao C., Zhang Z.: On the uniqueness of weak solutions for the 3D Navier–Stokes equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 26, 2165–2180 (2009)
Chemin J.Y., Lerner N.: Flot de champs de vecteurs non lipschitziens et équations de Navier–Stokes. J. Differ. Equ. 121, 314–328 (1995)
Danchin R.: Estimates in Besov spaces for transport and transport-diffusion equations with almost Lipschitz coefficients. Rev. Mat. Iberoamericana 21, 863–888 (2005)
Escauriaza L., Seregin G., S̆verák V.: \({L_{3,\infty}}\)-solutions to the Navier–Stokes equations and backward uniqueness. Russian Math. Surveys 58, 211–250 (2003)
Giga Y.: Solutions for semilinear parabolic equations in L p and regularity of weak solutions of the Navier–Stokes system. J. Differ. Equ. 62, 186–212 (1986)
He C., Xin Z.: On the regularity of weak solutions to the magnetohydrodynamic equations. J. Differ. Equ. 213, 235–254 (2005)
Kozono H., Sohr H.: Remark on uniqueness of weak solutions to the Navier–Stokes equations. Analysis 16, 255–271 (1996)
Miao, C., Wu, J., Zhang, Z.: Littlewood–Paley Theory and Applications to Fluid Dynamics Equations, Monogr. Modern Pure Math. 142. Science Press, Beijing (2012)
Politano H., Pouquet A., Sulem P.L.: Current and vorticity dynamics in three-dimensional magnetohydrodynamic turbulence. Phys. Plasmas 2, 2931–2939 (1995)
Ribaud F.: A remark on the uniqueness problem for the weak solutions of Navier–Stokes equations. Ann. Fac. Sci. Toulouse Math. 11, 225–238 (2002)
Serrin J.: The initial value problem for the Navier–Stokes equations. In: Langer, R.E. (ed.) Nonlinear Problems, pp. 69–98. University of Wisconsin Press, Madison (1963)
Sermange M., Teman R.: Some mathematical questions related to the MHD equations. Comm. Pure Appl. Math. 36, 635–664 (1983)
Triebel, H.: Theory of Function Spaces. In: Monograph in Mathematics, vol.78. Birkhauser Verlag, Basel (1983)
Wu J.: Bounds and new approaches for the 3D MHD equations. J. Nonlinear Sci. 12, 395–413 (2002)
Wu J.: Regularity results for weak solutions of the 3D MHD equations. Discrete Contin. Dyn. Syst. 10, 543–556 (2004)
Zhang Q.: On the uniqueness of weak solutions for the 3D viscous Magneto-hydrodynamics equations. Nonlinear Anal. 74, 5000–5007 (2011)
Zhou Y.: Remarks on regularities for the 3D MHD equations. Discrete Contin. Dyn. Syst. 12, 881–886 (2005)
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Shi, J., Zhang, Q. Remarks on the uniqueness of weak solution for the 3D viscous magneto-hydrodynamics equations in \({B^{1}_{\infty,\infty}}\) . Z. Angew. Math. Phys. 67, 7 (2016). https://doi.org/10.1007/s00033-015-0594-y
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DOI: https://doi.org/10.1007/s00033-015-0594-y