Geodesic
An improved blow-up criterion for the magnetohydrodynamics with the Hall and ion-slip effects
Contemporary Mathematics. Fundamental Directions, Dedicated to 70th anniversary of the President of the RUDN University V. M. Filippov, Tome 67 (2021) no. 3, pp. 526-534.

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In this work, we consider the magnetohydrodynamics system with the Hall and ion-slip effects in $\mathbb{R}^{3}$. The main result is a sufficient condition for regularity on a time interval $[0,T]$ expressed in terms of the norm of the homogeneous Besov space $\dot{B}_{\infty ,\infty }^{0}$ with respect to the pressure and the $BMO-$norm with respect to the gradient of the magnetic field, respectively \begin{equation*} \int_{0}^{T}\left( \left\Vert \nabla \pi (t)\right\Vert _{\dot{B}_{\infty ,\infty }^{0}}^{\frac{2}{3}}+\left\Vert \nabla B(t)\right\Vert _{BMO}^{2}\right) dt\infty , \end{equation*} which can be regarded as improvement of the result in [3]. 
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S. Gala; M. A. Ragusa. An improved blow-up criterion for the magnetohydrodynamics with the Hall and ion-slip effects. Contemporary Mathematics. Fundamental Directions, Dedicated to 70th anniversary of the President of the RUDN University V. M. Filippov, Tome 67 (2021) no. 3, pp. 526-534. http://geodesic.mathdoc.fr/item/CMFD_2021_67_3_a7/

[1] Beale J., Kato T., Majda A., “Remarks on breakdown of smooth solutions for the three-dimensional Euler equations”, Commun. Math. Phys., 94 (1984), 61–66 | DOI | Zbl

[2] Chemin J.-Y., Perfect incompressible fluids, Clarendon Press Oxford University Press, New York, 1998 | Zbl

[3] Fan J., Jia X., Nakamura G., Zhou Y., “On well-posedness and blowup criteria for the magnetohydrodynamics with the Hall and ion-slip effects”, Z. Angew. Math. Phys., 66 (2015), 1695–1706 | DOI | Zbl

[4] Gala S., Ragusa M. A., “On the blow-up criterion of strong solutions for the MHD equations with the Hall and ion-slip effects in $\mathbb{R}^{3}$”, Z. Angew. Math. Phys., 67 (2016), 18 | DOI

[5] Kozono H., Taniuchi Y., “Bilinear estimates in $BMO$ and the Navier—Stokes equations”, Math. Z., 235 (2000), 173–194 | DOI | Zbl

[6] Maiellaro M., “Uniqueness of MHD thermodiffusive mixture flows with Hall and ion-slip effects”, Meccanica, 12 (1977), 9–14 | DOI | Zbl

[7] Mulone G., Salemi F., “Some continuous dependence theorems in MHD with Hall and ion-slip currents in unbounded domains”, Rend. Accad. Sci. Fis. Mat. Napoli., 55 (1988), 139–152 | Zbl

[8] Mulone G., Solonnikov V. A., “On an initial boundary-value problem for the equation of magnetohydrodynamics with the Hall and ion-slip effects”, J. Math. Sci. (N.Y.), 87 (1997), 3381–3392 | DOI

[9] Triebel H., Theory of function spaces, Birkhäuser, Basel, 1983 | Zbl

[10] Zhou Y., “Regularity criteria for the 3D MHD equations in terms of the pressure”, Int. J. Nonlinear Mech., 41 (2006), 1174–1180 | DOI | Zbl