Blow-up criteria for the three dimensional nonlinear dissipative system modeling electro-hydrodynamics

Abstract

In this paper, logarithmically improved regularity criteria for the Navier-Stokes/Poisson-Nernst-Planck system are established in terms of both the pressure and the gradient of pressure in the homogeneous Besov space.2010 Mathematics Subject Classification. 35B45, 35B65, 35Q35.

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

“…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]).…”

“…Combining the classical Beale-Kato-Majda's type global regularity criteria (1.6), as in [33], we complete the proof of theorem 1.1.…”

“…We first recall from [24,33] that the fundamental energy inequalities have been established, i.e., for any 0 < t < T ,…”

“…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]).…”

Existence of $C^{1,α}$ Singular Solutions to Euler-Nernst-Planck-Poisson System on $\mathbb{R}^3$ with Free-Moving Charges

“…As for the blow up aspects, there has been several blowup criterions for (1.2) obtained in [26] and [18]. However, the existence of blowup solutions still remains open, let alone the blowup dynamics for some appropriately chosen initial data.…”