scholarly journals On the Stokes and Navier-Stokes flows in a perturbed half-space

2005 ◽  
Author(s):  
Takayuki Kubo ◽  
Yoshihiro Shibata
Keyword(s):  
Half Space ◽  
Navier Stokes ◽  
Stokes Flows

Decay properties for the incompressible Navier-Stokes flows in a half space

2021 ◽  
pp. 1-24
Author(s):  
Pigong Han
Keyword(s):  
Half Space ◽  
Neumann Boundary ◽  
Navier Stokes ◽  
Pressure Function ◽  
Stokes Flows ◽  
Decay Properties ◽  
Homogeneous Neumann Boundary Condition ◽  
Pressure Term ◽  
Comprehensive Characterization

In this article, we give a comprehensive characterization of $L^1$ -summability for the Navier-Stokes flows in the half space, which is a long-standing problem. The main difficulties are that $L^q-L^r$ estimates for the Stokes flow don't work in this end-point case: $q=r=1$ ; the projection operator $P: L^1\longrightarrow L^1_\sigma$ is not bounded any more; useful information on the pressure function is missing, which arises in the net force exerted by the fluid on the noncompact boundary. In order to achieve our aims, by making full use of the special structure of the half space, we decompose the pressure function into two parts. Then the knotty problem of handling the pressure term can be transformed into establishing a crucial and new weighted $L^1$ -estimate, which plays a fundamental role. In addition, we overcome the unboundedness of the projection $P$ by solving an elliptic problem with homogeneous Neumann boundary condition.

Weighted spatial decay rates for the Navier–Stokes flows in a half-space

2014 ◽  
Vol 144 (3) ◽  
pp. 491-510 ◽  
Author(s):  
Pigong Han
Keyword(s):  
Half Space ◽  
Stokes Flow ◽  
Projection Operator ◽  
Decay Rates ◽  
Navier Stokes ◽  
Convection Term ◽  
Spatial Decay ◽  
Stokes Flows ◽  
Spatial Derivatives ◽  
Derivatives Of

The weighted Lq − Lr-estimates for the Stokes flow are given in half-spaces. Furthermore, the weighted decays for the first and second spatial derivatives of the Navier-Stokes flows are also established, where the unboundedness of the projection operator is overcome by employing a decomposition for the convection term. The main results in this paper are inspired by the works of Bae and Jin.

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