Blowup criteria in terms of pressure for the 3D nonlinear dissipative system modeling electro-diffusion

Global regularity criterion for the dissipative systems modelling electrohydrodynamics involving the middle eigenvalue of the strain tensor

“…Remark 2.6. In fact, since L 2p/(p−2),2 → L 2p/(p−2),2p/(p−2) for 2 < p < ∞, we have similar result for estimate (2.3) in anisotropic Lebesgue space, for more detail refer to [15]. However, we should point out that lemma 2.5 that holds in Lorentz space mainly depends on the Sobolev's embedding in lemma 2.4.…”

“…In particular, the generalized Sobolev inequality is a strengthening of a similar result in [9,15]. The result shows that the Serrin type criteria via only middle eigenvalue of strain tensor control the breakdown of local smooth solutions of the system (1.1) in the framework of anisotropic Lorentz space.…”

“…for local smooth solutions have been established by Zhao and Bai [33], which coincide with the results for conventional Navier-Stokes equations. Later, Zhao and Bai's work has been improved by some other authors (see, e.g., [14,15,26,32]). Inspired by the above mentioned works on global regularity criteria of Navier-Stokes equations and system (1.1), the purpose of the present paper is to generalize regularity criteria (1.5) for classical Navier-Stokes equations to the system (1.1), moreover, we will consider it in more general anisotropic Lorentz space.…”

“…For the system (1.1), after a careful exploration of the nonlinear coupling between the velocity field and charge transport, some regularity criteria for solutions to (1.1) that only involve the velocity field in the three-dimensional case, the Serrin type blowup criteria (1.2)–(1.3) and the Beale-Kato-Majda type criteria for local smooth solutions have been established by Zhao and Bai [33], which coincide with the results for conventional Navier–Stokes equations. Later, Zhao and Bai's work has been improved by some other authors (see, e.g., [14, 15, 26, 32]).…”