Global well-posedness of the Navier-Stokes equations with Navier-slip boundary conditions in a strip domain

Abstract

<p style='text-indent:20px;'>This paper is concerned with the existence and uniqueness of the strong solution to the incompressible Navier-Stokes equations with Navier-slip boundary conditions in a two-dimensional strip domain where the slip coefficients may not have defined sign. In the meantime, we also establish a number of Gagliardo-Nirenberg inequalities in the corresponding Sobolev spaces which will be applicable to other similar situations.</p>

Rayleigh–Taylor instability for nonhomogeneous incompressible fluids with Navier‐slip boundary conditions

“…It should be noted that, the noncompact infinite slab case is not included in any result mentioned above. In 2018, Ding and Li proved the existence and uniqueness of strong solution of incompressible fluid with Navier boundary conditions in 2D infinite slab [23]. For the references on the vanishing viscosity limit of Navier-Stokes equation with Navier boundary conditions, we refer the readers to [5,34,35] and the reference therein.…”

“…For convenience, we list a few lemmas that will be used in this paper, without the proofs. The readers interested in the proof could refer to [23] for the details.…”

“…Then, one obtains Compared to classical well-posedness results of nonhomogeneous Navier-Stokes equations in [4,29], the boundary integral terms produced by the Navier-slip boundary condition in this paper bring some new difficulties in the energy estimates. However, these difficulties can be overcome by using the technique applied in [23]. Thus, based on estimate (4.41), the global well-posedness of the nonlinear system (1.6)-(1.7) follows.…”

On Inhibition of Rayleigh--Taylor Instability by a Horizontal Magnetic Field in Non-resistive MHD Fluids: the Viscous Case

“…where n = ( n 1 , n 2 ) T denotes the outward normal unit vector on ∂Ω, Dv = (∇v + ∇v T )/2 the strain tensor, and the subscript "tan" the tangential component of a vector (for example [7,8,34,38]. Here and in what follows, we always use the superscript 0 to emphasize the initial data.…”