scholarly journals On Regularity of a Weak Solution to the Navier–Stokes Equations with the Generalized Navier Slip Boundary Conditions

2018 ◽  
Vol 2018 ◽  
pp. 1-7
Author(s):  
Jiří Neustupa ◽  
Patrick Penel
Keyword(s):  
Weak Solution ◽  
Boundary Conditions ◽  
Stokes Equations ◽  
Regularity Criterion ◽  
Navier Stokes ◽  
Deformation Tensor ◽  
Slip Boundary Conditions ◽  
Navier Stokes Equation ◽  
Navier Slip ◽  
Slip Boundary

The paper shows that the regularity up to the boundary of a weak solution v of the Navier–Stokes equation with generalized Navier’s slip boundary conditions follows from certain rate of integrability of at least one of the functions ζ1, (ζ2)+ (the positive part of ζ2), and ζ3, where ζ1≤ζ2≤ζ3 are the eigenvalues of the rate of deformation tensor D(v). A regularity criterion in terms of the principal invariants of tensor D(v) is also formulated.

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

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Quanrong Li ◽  
Shijin Ding
Keyword(s):  
Boundary Conditions ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Slip Boundary Conditions ◽  
Navier Stokes Equations ◽  
Navier Slip ◽  
Slip Boundary ◽  
Strip Domain ◽  
Navier Slip Boundary Conditions ◽  
Slip Coefficients

<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>

scholarly journals Asymptotic analysis of an optimal control problem for a viscous incompressible fluid with Navier slip boundary conditions

2021 ◽  
pp. 1-21
Author(s):  
Claudia Gariboldi ◽  
Takéo Takahashi
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Boundary Conditions ◽  
Control Problem ◽  
Stokes System ◽  
Navier Stokes ◽  
Slip Boundary Conditions ◽  
Navier Slip ◽  
Slip Boundary ◽  
Navier Slip Boundary Conditions

We consider an optimal control problem for the Navier–Stokes system with Navier slip boundary conditions. We denote by α the friction coefficient and we analyze the asymptotic behavior of such a problem as α → ∞. More precisely, we prove that if we take an optimal control for each α, then there exists a sequence of optimal controls converging to an optimal control of the same optimal control problem for the Navier–Stokes system with the Dirichlet boundary condition. We also show the convergence of the corresponding direct and adjoint states.

Parallel iterative finite-element algorithms for the Navier–Stokes equations with nonlinear slip boundary conditions

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Kangrui Zhou ◽  
Yueqiang Shang
Keyword(s):  
Finite Element ◽  
Boundary Conditions ◽  
Parallel Algorithms ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Slip Boundary Conditions ◽  
Navier Stokes Equations ◽  
Slip Boundary ◽  
Parallel Iterative ◽  
Nonlinear Slip Boundary Conditions

AbstractBased on full domain partition, three parallel iterative finite-element algorithms are proposed and analyzed for the Navier–Stokes equations with nonlinear slip boundary conditions. Since the nonlinear slip boundary conditions include the subdifferential property, the variational formulation of these equations is variational inequalities of the second kind. In these parallel algorithms, each subproblem is defined on a global composite mesh that is fine with size h on its subdomain and coarse with size H (H ≫ h) far away from the subdomain, and then we can solve it in parallel with other subproblems by using an existing sequential solver without extensive recoding. All of the subproblems are nonlinear and are independently solved by three kinds of iterative methods. Compared with the corresponding serial iterative finite-element algorithms, the parallel algorithms proposed in this paper can yield an approximate solution with a comparable accuracy and a substantial decrease in computational time. Contributions of this paper are as follows: (1) new parallel algorithms based on full domain partition are proposed for the Navier–Stokes equations with nonlinear slip boundary conditions; (2) nonlinear iterative methods are studied in the parallel algorithms; (3) new theoretical results about the stability, convergence and error estimates of the developed algorithms are obtained; (4) some numerical results are given to illustrate the promise of the developed algorithms.

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