Quasi-neutral limit and the boundary layer problem of Planck-Nernst-Poisson-Navier-Stokes equations for electro-hydrodynamics

Boundary layer analysis for a 2-D Keller-Segel model

“…The above inequality is an ε-weighted generalized entropy-production inequality. By using a Gronwalltype estimate in [29]…”

“…The boundary layer phenomenon in fluid mechanics is well-known and has been studied extensively [23][24][25][26][27][28][29][30][31][32][33][34][35]. In contrast, there are much fewer studies available for the boundary layer problems for the Keller-Segel model.…”

Stability of the nonconstant stationary solution to the damped Poisson–Nernst–Planck–Euler equations

“…1,2 There are a lot of research works on the incompressible version of (1.1), which mainly focus on two systems: one is called the Poisson-Nernst-Planck-Euler equations (namely, 𝛼 = 0 in (1.1)); compare earlier research [3][4][5] ; the other is called the Poisson-Nernst-Planck-Navier-Stokes (PNP-NS) equations (namely, the damping term 𝛼𝜌u in (1.1) being replaced with the viscosity term, like −Δu); compare previous studies. [6][7][8][9][10][11][12][13][14] Among their results of the PNP-NS equations, Jerome 6 proved the local unique smooth solution of the Cauchy problem; Schmuck 7 showed the global weak solutions for the initial boundary value problem (IBVP); Fan-Gao 9 proved the uniqueness of weak solutions in critical spaces; Bothe-Fischer-Saal 10 obtained the global unique strong solution for the 2-D IBVP; Zhang-Yin 11 showed the global unique strong solution for the 2-D Cauchy problem; Constantin-Ignatova 12 proved the global existence of strong solutions to the 2-D IBVP for arbitrary large initial data; Liu-Wang 14 proved the global existence of weak solutions for the 3-D Cauchy problem; Li 8 and Wang-Jiang-Liu 13 studied the 3-D quasi-neutral limit problem in the periodic domain and a bounded domain, respectively. From a modeling perspective, the volume-preserving in nanofluid flow with charged particles may fail and thus the compressibility of fluids has to be considered; compare Puliti et al 15 However, there are few results for the compressible system (1.1).…”

“…System () models the transport of the microscopic charged particles under the influence of the self‐consistent electrostatic field in a macroscopic compressible fluid, which can be similarly derived by using an energetic variational approach as done in previous studies 1,2 . There are a lot of research works on the incompressible version of (), which mainly focus on two systems: one is called the Poisson–Nernst–Planck–Euler equations (namely, $$$ \alpha =0 $$$ in ()); compare earlier research 3–5 ; the other is called the Poisson–Nernst–Planck‐Navier–Stokes (PNP‐NS) equations (namely, the damping term $$$ \alpha \rho u $$$ in () being replaced with the viscosity term, like $$$ -\Delta u $$$); compare previous studies 6–14 . Among their results of the PNP‐NS equations, Jerome 6 proved the local unique smooth solution of the Cauchy problem; Schmuck 7 showed the global weak solutions for the initial boundary value problem (IBVP); Fan‐Gao 9 proved the uniqueness of weak solutions in critical spaces; Bothe‐Fischer‐Saal 10 obtained the global unique strong solution for the 2‐D IBVP; Zhang‐Yin 11 showed the global unique strong solution for the 2‐D Cauchy problem; Constantin‐Ignatova 12 proved the global existence of strong solutions to the 2‐D IBVP for arbitrary large initial data; Liu‐Wang 14 proved the global existence of weak solutions for the 3‐D Cauchy problem; Li 8 and Wang‐Jiang‐Liu 13 studied the 3‐D quasi‐neutral limit problem in the periodic domain and a bounded domain, respectively.…”

Charged Fluids in Porous Media

“…With periodic boundary conditions, two dimensional strong solutions exist globally in time, and long time behaviors of the solutions are studied in [1] under the influence of body forces or body charges. Some zero Debye length limit (ε → 0 in (1.2)) results for the NPNS system are proved in [7,18,23,24]. In the limit of zero viscosity in the Navier-Stokes equations, the solutions of NPNS system in two dimensions converges to the solutions of the corresponding Nernst-Planck-Euler (NPE) system, whose solutions exist and are global [14,25,27].…”