Well-posedness and decay for the dissipative system modeling electro-hydrodynamics in negative Besov spaces

Abstract

In [10] (Y. Guo, Y. Wang, Decay of dissipative equations and negative Sobolev spaces, Commun. Partial Differ. Equ. 37 (2012) 2165-2208), Y. Guo and Y. Wang developed a general new energy method for proving the optimal time decay rates of the solutions to dissipative equations. In this paper, we generalize this method in the framework of homogeneous Besov spaces. Moreover, we apply this method to a model arising from electro-hydrodynamics, which is a strongly coupled system of the Navier-Stokes equations and the Poisson-Nernst-Planck equations through charge transport and external forcing terms. We show that the negative Besov norms are preserved along time evolution, and obtain the optimal time decay rates of the higher-order spatial derivatives of solutions by the Fourier splitting approach and the interpolation techniques.

The optimal temporal decay estimates for the fractional power dissipative equation in negative Besov spaces

“…Remark 1.1 Theorem 1.1 is essentially inspired by Y. Guo and Y. Wang [27], where they developed a new energy approach in the framework of Sobolev spaces for proving the optimal time decay rates of the solutions to the dissipative equation (α = 2). Remark 1.2 Theorem 1.1 generalizes the corresponding result in [53], which we relax the regularity of the initial data in a wider range of Besov spaces. Moreover, the restrictive condition p ≥ 2 is due to Bernstein's inequality, which we don't know whether or not it is true for 1 ≤ p < 2.…”

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

“…Based on Kato's semigroup framework, the local smooth theory of the system (1.1) has been established by Jerome [14]. In more general situation, the global existence of strong solutions with small initial data and the local existence of strong solutions with any initial data in various scaling invariant spaces have been studied by Zhang and Yin [29], Zhao, Deng and Cui [31], Zhao and Liu [32], and Zhao, Zhang and Liu [33]. The Prodi-Serrin type blow up criterion and the Beale-Kato-Majda type blow-up criterion for local smooth solutions have been established by Zhao and Bai [30].…”

Global existence and temporal decay of large solutions for the Poisson–Nernst–Planck equations in low regularity spaces

“…Now we are in a position to drop the smallness assumption imposed on $$$ {w}_0 $$$ to still ensure the global existence of solutions, which need us to employ the following product estimates in Besov spaces; for details, see earlier studies 28,30 …”