Navier Boundary Condition Research Articles

$ L^p $-strong solution for the stationary exterior Stokes equations with Navier boundary condition

<p style='text-indent:20px;'>This paper treats the stationary Stokes problem in exterior domain of <inline-formula><tex-math id="M2">\begin{document}$ {{\mathbb{R}}}^3 $\end{document}</tex-math></inline-formula> with Navier slip boundary condition. The behavior at infinity of the data and the solution are determined by setting the problem in <inline-formula><tex-math id="M3">\begin{document}$ L^p $\end{document}</tex-math></inline-formula>-spaces, for <inline-formula><tex-math id="M4">\begin{document}$ p> 2 $\end{document}</tex-math></inline-formula>, with weights. The main results are the existence and uniqueness of strong solutions of the corresponding system.</p>

On a p x -Biharmonic Kirchhoff Problem with Navier Boundary Conditions

In this article, we study the existence of solutions for nonlocal p x -biharmonic Kirchhoff-type problem with Navier boundary conditions. By different variational methods, we determine intervals of parameters for which this problem admits at least one nontrivial solution.

Remarks on the 3D Stokes eigenvalue problem under Navier boundary conditions

We study the Stokes eigenvalue problem under Navier boundary conditions in C^{1,1}-domains Omega subset mathbb {R}^3. Differently from the Dirichlet boundary conditions, zero may be the least eigenvalue. We fully characterize the domains where this happens and we show that the ball is the unique domain where the zero eigenvalue is not simple, it has multiplicity three. We apply these results to show the validity/failure of a suitable Poincaré-type inequality. The proofs are obtained by combining analytic and geometric arguments.

Liouville type theorem for critical order Hénon-Lane-Emden type equations on a half space and its applications

In this paper, we are concerned with the critical (i.e., n-th) order Hénon-Lane-Emden type equations with Navier boundary conditions on a half space R+n:(0.1){(−Δ)n2u(x)=f(x,u(x)),u(x)≥0,x∈R+n,u(x)=−Δu(x)=⋯=(−Δ)n2−1u(x)=0,x∈∂R+n, where u∈Cn(R+n)∩Cn−2(R+n‾) and n≥2 is even. We first consider the typical case f(x,u)=|x|aup with 0≤a<∞ and 1<p<∞. We prove the super poly-harmonic properties and establish the equivalence between (0.1) and the corresponding integral equations(0.2)u(x)=∫R+nG+(x,y)f(y,u(y))dy, where G+(x,y) denotes the Green function for (−Δ)n2 on R+n with Navier boundary conditions. Then, we establish Liouville theorem for (0.2) and hence obtain the Liouville theorem for (0.1) on R+n. As an application of the Liouville theorem on R+n (Theorem 1.6) and Liouville theorems in Rn, we derive a priori estimates via blowing-up methods for solutions (possibly change signs) to Navier problems involving critical order uniformly elliptic operators L. Consequently, by using the Leray-Schauder fixed point theorem, we derive existence of positive solutions to critical order Lane-Emden equations in bounded domains for all n≥2 and 1<p<∞. In contrast to the subcritical order cases, our results seem to be the first work on Navier problems for critical order equations on R+n, which is the critical-order counterpart to those results on subcritical order cases in [6,20,21]. Extensions to IEs and PDEs with general nonlinearities f(x,u) are also included. Surprisingly, there are no growth conditions on u and hence f(x,u) can grow exponentially (or even faster) on u.

A 3D Non-Stationary Micropolar Fluids Equations with Navier Slip Boundary Conditions

Micropolar fluids are fluids with microstructure and belong to a class of fluids with asymmetric stress tensor that called Polar fluids, and include, as a special case, the well-established Navier–Stokes model. In this work we study a 3D micropolar fluids model with Navier boundary conditions without friction for the velocity field and homogeneous Dirichlet boundary conditions for the angular velocity. Using the Galerkin method, we prove the existence of weak solutions and establish a Prodi–Serrin regularity type result which allow us to obtain global-in-time strong solutions at finite time.

Multiple Solutions for a Nonlocal Elliptic Problem Involving p x , q x -Biharmonic Operator

In this paper, using the variational principle, the existence and multiplicity of solutions for p x , q x -Kirchhoff type problem with Navier boundary conditions are proved. At the same time, the sufficient conditions for the multiplicity of solutions are obtained.

On the spectral properties of the differential operators with involution

In this paper we deal with the problems of the eigenfunction expansions related to the differential operators with involution. The mean value formula for the eigenfunction is obtained with application of the transformation methods of the operators in the symmetric regions. The obtained formula is applied to estimate the eigenfunctions of the given differential operator in the ball. For domains with smooth boundary, the solution to these differential operator problems involves eigenfunction expansions associated with biharmonic-type operator with Navier boundary conditions.

Zero Kinematic Viscosity-Magnetic Diffusion Limit of the Incompressible Viscous Magnetohydrodynamic Equations with Navier Boundary Conditions

We investigate the uniform regularity and zero kinematic viscosity-magnetic diffusion limit for the incompressible viscous magnetohydrodynamic equations with the Navier boundary conditions on the velocity and perfectly conducting conditions on the magnetic field in a smooth bounded domain Ω ⊂ ℝ3. It is shown that there exists a unique strong solution to the incompressible viscous magnetohydrodynamic equations in a finite time interval which is independent of the viscosity coefficient and the magnetic diffusivity coefficient. The solution is uniformly bounded in a conormal Sobolev space and W1,∞ (Ω) which allows us to take the zero kinematic viscosity-magnetic diffusion limit. Moreover, we also get the rates of convergence in L∞(0, T; L2), L∞(0, T; W1, p) (2 ≤ p < ∞), and L∞((0, T) × Ω) for some T > 0.

The extended discontinuous Galerkin method adapted for moving contact line problems via the generalized Navier boundary condition

AbstractIn this work, an extended discontinuous Galerkin (extended DG/XDG also called unfitted DG) solver for two‐dimensional flow problems exhibiting moving contact lines is presented. The generalized Navier boundary condition is employed within the XDG discretization for the handling of the moving contact lines. The spatial discretization is based on a symmetric interior penalty method and the numerical treatment of the surface tension force is done via the Laplace–Beltrami formulation. The XDG method adapts the approximation space conformal to the position of the interface and allows a sub‐cell accurate representation within the sharp interface formulation. The interface is described as the zero set of a signed‐distance level‐set function and discretized by a standard DG method. No adaption of the level‐set evolution algorithm is needed for the extension to moving contact line problems. The developed solver is validated against typical two‐dimensional contact line driven flow phenomena including droplet simulations on a wall and the two‐phase Couette flow.

A Hopf Bifurcation in the Planar Navier\u2013Stokes Equations

We consider the Navier–Stokes equation for an incompressible viscous fluid on a square, satisfying Navier boundary conditions and being subjected to a time-independent force. As the kinematic viscosity is varied, a branch of stationary solutions is shown to undergo a Hopf bifurcation, where a periodic cycle branches from the stationary solution. Our proof is constructive and uses computer-assisted estimates.

Global regularity and stability of solutions to the 3D double-diffusive convection system with Navier boundary conditions

This paper is concerned with the initial-boundary value problem to 3D double-diffusive convection system. First, we establish the global weak solutions for the double-diffusive convection system in a domain $\Omega$ with Navier boundary condition by Galerkin approximation method. Then by making higher-order estimates of the approximation function with higher regularity assumption on the initial data, combined with compactness argument we obtain the local strong solution with uniqueness. Secondly, we obtain the classical Serrin-type blow-up criterion for the local strong solution in the Lorentz space. Lastly, we focus on global stability of strong large solutions for the double- diffusive convection system with Navier boundary conditions, it is shown that the large solution is stable with the additional assumption on some suitable integrable property of solution.

An isoparametric mixed finite element method for approximating a class of fourth-order elliptic problem

A Ciarlet-Raviart type isoparametric mixed finite element method (MFEM) is constructed and analyzed for solving a class of fourth-order elliptic equation with Navier boundary condition defined on a curved domain in R2, and numerical quadrature is also considered in the scheme. The existence and uniqueness of the numerical solutions are proved under certain numerical quadrature. With the help of the special technically analysis, the optimal error estimates with H1 norm are obtained in Ωh, which yields better accuracy than using a convex polygonal domain to approximate the curved domain. For either constant coefficients or nonconstant coefficients problem, numerical examples are listed to confirm theoretical analysis respectively.

Higher order approximation of biharmonic problem using the WEB-Spline based mesh-free method

Abstract In this paper, we consider a biharmonic equation with Navier boundary conditions and we decoupled this fourth order biharmonic equation into a system of second order equations. In numerics, there is an advantage because of this decoupling, i.e., one can use C 0 finite elements to solve the decoupled system instead of C 1 finite elements. For solving this decoupled system of second order equations, we use the weighted extended b-spline (WEB-Spline) based mesh-free finite element method. The WEB-Spline method does not require any mesh generation and eliminates the difficult, time-consuming preprocessing step. Also, by the WEB-Spline method higher order approximations are possible with relatively low dimensional spaces. Numerical results based on the WEB-Spline method are compared with the usual finite element method in order to demonstrate the efficiency of the proposed method.

Uniqueness and Bifurcation Branches for Planar Steady Navier\u2013Stokes Equations Under Navier Boundary Conditions

The stationary Navier–Stokes equations under Navier boundary conditions are considered in a square. The uniqueness of solutions is studied in dependence of the Reynolds number and of the strength of the external force. For some particular forcing, it is shown that uniqueness persists on some continuous branch of solutions, when these quantities become arbitrarily large. On the other hand, for a different forcing, a branch of symmetric solutions is shown to bifurcate, giving rise to a secondary branch of nonsymmetric solutions. This proof is computer-assisted, based on a local representation of branches as analytic arcs.

Approximate analytical solution for the flow of a Phan-Thien–Tanner fluid through an axisymmetric hyperbolic contraction with slip boundary condition

An analytic approximation for the flow of a linear Phan-Thien–Tanner model fluid through an axisymmetric semi-hyperbolic contraction is presented. Such an approximation allows us to compute velocity and pressure response for the flow through axisymmetric contraction geometries; in particular, we have considered here the semi-hyperbolic contraction, which is a geometry where an almost constant extension-rate is reached at different radial positions. In addition, we present a semi-analytic solution capable of representing the exponential version of the selected viscoelastic model; this solution was compared to the results of commercial software, demonstrating the excellent approximation level of the semi-analytic model proposed. Alternatively, for both approaches (linear and exponential Phan-Thien–Tanner), the flow model equations are solved by considering the Navier boundary condition, which allows these models to represent flows with some degree of slip at the geometry wall.

Matrix-free TriGlobal adjoint stability analysis of compressible Navier–Stokes equations

A numerical method for TriGlobal (i.e., fully three-dimensional) adjoint stability analysis for compressible flows was developed and is presented in this paper. The developed method solves the adjoint stability problem using a matrix-free method based on Krylov-Schur method and a time-stepping approach. Because of the low memory (RAM) requirement of the matrix-free approach, the developed method can analyze fully three-dimensional flows that are difficult to analyze with conventional matrix-based methods. To perform time-stepping on the adjoint variables, the adjoint equations, including appropriate boundary conditions for the compressible Navier–Stokes equations, were derived. The equations were discretized using a finite compact difference method, and time integration was conducted using the three-step third-order Runge–Kutta method. A flow field around a two-dimensional square cylinder and a cubic cavity flow were analyzed using the developed method, and it was confirmed that the method reproduces the dominant instabilities reported in the literature. In addition, in the square cylinder flow analysis, the receptivity and sensitivity regions of the secondary wake mode, which corresponds to far wake instability, were clarified. Finally, the TriGlobal direct and adjoint stabilities of compressible flows over a finite width cavity were analyzed for the first time, and it was proven that large-scale adjoint stability analysis can be performed with the developed method. The results also show that instability phenomena similar to those obtained with BiGlobal stability analysis appear, but sidewall effects exist.

A Kirchhoff $p(x)$-Biharmonic Problem Involving Singular Nonlinearities and Navier Boundary Conditions

The aim of this work is to study the existence of weak solutions for a nonhomogeneous singular p(x) -Kirchhoff problem of the following form (\textbf{P}_{\pm \lambda}) \quad \left\{ \begin{aligned} M(t)\Delta(|\Delta u|^{p(x)-2}\Delta u) &amp;=a(x) u^{-\gamma (x)}\pm \lambda u^{q(x)-2}u, &amp;\ &amp;\quad \mathrm{in} \quad \Omega, \\ \Delta u&amp;=u=0, &amp;\ &amp;\quad \mathrm{on} \quad\partial\Omega, \end{aligned} \right. by using variational techniques and monotonicity arguments combined with the theory of the generalized Lebesgue Sobolev spaces.

Stokes and Navier-Stokes equations with Navier boundary conditions

We study the stationary Stokes and Navier-Stokes equations with nonhomogeneous Navier boundary conditions in a bounded domain Ω⊂R3 of class C1,1. We prove the existence and uniqueness of weak and strong solutions in W1,p(Ω) and W2,p(Ω) for all 1<p<∞, considering minimal regularity on the friction coefficient α. Moreover, we deduce uniform estimates for the solution with respect to α which enables us to analyze the behavior of the solution when α→∞.

Fourth-order elliptic problems with critical nonlinearities by a sublinear perturbation

In this paper, we get the existence of two positive solutions for a fourth-order problem with Navier boundary condition. Our nonlinearity has a critical growth, and the method is a local minimum theorem obtained by Bonanno.

On a bi-nonlocal fourth order elliptic problem

This paper is aiming at obtaining weak solution for a bi-nonlocal fourth order elliptic problem with Navier boundary condition. Our approach is based on variational methods and critical point theory.