Navier-type Boundary Condition Research Articles

Global classical solutions to the compressible Navier–Stokes equations with Navier-type slip boundary condition in 2D bounded domains

We study the barotropic compressible Navier–Stokes equations with Navier-type boundary condition in a two-dimensional simply connected bounded domain with C∞ boundary ∂Ω. By some new estimates on the boundary related to the Navier-type slip boundary condition, the classical solution to the initial-boundary-value problem of this system exists globally in time provided the initial energy is suitably small even if the density has large oscillations and contains vacuum states. Furthermore, we also prove that the oscillation of the density will grow unboundedly in the long run with an exponential rate provided vacuum (even a point) appears initially. As we known, this is the first result concerning the global existence of classical solutions to the compressible Navier–Stokes equations with Navier-type slip boundary condition and the density containing vacuum initially for general 2D bounded smooth domains.

Generalized and strong solutions for an elliptic system with Navier-type boundary condition

In this paper, we study a linear elliptic system involving Navier-type boundary conditions or tangential components in a three dimensional bounded domain, possibly multiply-connected and with boundary possibly non connected. We establish an appropriate Inf-Sup condition and then we prove the existence and uniqueness of the generalized solutions in Lp-Theory. We also investigate the case of strong solutions.

ВИКОРИСТАННЯ СИМЕТРІЇ ДЛЯ РОЗВ’ЯЗУВАННЯ ЗАДАЧ ДЕФОРМУВАННЯ ПЛИТ НА ЖОРСТКІЙ ОСНОВІ

The optimization of the design scheme of deformation of a rectangular multilayer plate with transversally isotropic layers resting on a rigid (non-deformable) foundation is proposed. The essence of optimization is to consider such a design diagram of the plate, in which the stress-strain state (SSS) of plate would be fully described by only one component, namely the unflexural component of symmetrical SSS relative to the middle surface of plate which is bilaterally symmetrically loaded. To do this, instead of the actual design of the multilayer plate, which is deformed without separation from the foundation, it is suggested to consider the design diagram of the plate, which is formed by supplementing it with a symmetric one about the contact surface of the foundation. In this case, the plate will be symmetrically loaded with respect to the middle surface of the plate, and the thickness of the plate will double. On the middle surface, the conditions of sliding contact of the upper and lower parts of the symmetrical plate, i.e. sliding contact of the plate with foundation are fulfilled. To model absolutely rigid contact at the border with the foundation, an additional thin layer of high rigidity ("non-deformable layer") is introduced into the supplemented plate. This does not change the essence of the calculation model of plate. The SSS of plate will be unflexural, which significantly simplifies its modeling. A two-dimensional model of deformation of multilayer rectangular plates on a rigid foundation with isotropic and transversally-isotropic layers is constructed in an elastic formulation for a unflexural SSS, with a high degree of iterative approximation, but three-dimensional by the nature display of the SSS. This model sufficiently takes into account transverse shear deformations and of transverse compression of the plate under transverse loading. The model is continuous, that is, the number of equations and the order of differentiation of the solving system of equations does not depend on the number of layers in the plate. This order of differentiation and the number of solving equations can depend only on the order of iterative approximation of the model. The derivation of the solving differential equations in the generalized forces and displacement functions is given, as well as the boundary conditions are obtained by the variational Lagrange method. The results of the analytical solution of the problem of deformation of homogeneous rectangular plate in sliding contact with a rigid foundation whith Navier-type boundary conditions under the action of a transverse sinusoidal load are given. A comparison of the calculation results with the exact three-dimensional solution ones was made. Keywords: multilayered plate, rigid foundation, transverse shear, transverse compression, continual model

Navier–Stokes equations: regularity criteria in terms of the derivatives of several fundamental quantities along the streamlines—the case of a bounded domain

In this paper we deal with the conditional regularity of the weak solutions of the Navier–Stokes equations on a bounded domain endowed with Navier boundary conditions, Navier-type boundary conditions or Dirichlet boundary conditions. We prove the regularity criteria which are based on the directional derivatives of several fundamental quantities along the streamlines, namely the velocity magnitude, the kinetic energy, the pressure, the velocity field and the Bernoulli pressure. In striking contrast to the known criteria in which the mentioned quantities were differentiated along a fixed vector, our criteria are mostly optimal for the whole range of parameters and have a clear physical meaning.

MHD flow with Navier type boundary conditions

We are concerned here with results of existence and regularity of solutions to a stationary MHD problem, with a possibly multi-connected 2D domain associated with Navier-type boundary conditions. The RHS f, corresponding to the external forces, is in Lp(Ω), with p≥2, without any restriction on its norm. Two main difficulties are encountered in solving these problems: the geometry of the domain and the non-linear term (u⋅∇)(u−αΔu). We prove the existence of a solution (u,B) in H2(Ω)×W2,r(Ω) for any finite r.

$$L^2$$-Decay of Solutions to the Navier–Stokes System with Navier-Type Boundary Conditions

$$L^2$$-Decay of Solutions to the Navier–Stokes System with Navier-Type Boundary Conditions

The response of a 2D droplet on a wall executing small sinusoidal vibrations

This study concerns a two-dimensional liquid drop surrounded by gas and attached to a sinusoidally vibrating wall. Gravity is neglected and the moving contact lines are modelled using a Navier-type boundary condition at the wall and a prescribed contact angle, θ¯, which can take any value in the range 0<θ¯<π. The vibration amplitude, and hence the departure from equilibrium of the drop, is assumed sufficiently small that the problem can be linearized. Wall vibration can have components both normal and tangential to the wall. The solution of the linear problem can be expressed as the sum of two decoupled components corresponding to the response to purely normal and purely tangential vibration, which are respectively symmetric and antisymmetric with respect to reflection in the symmetry plane of the equilibrium drop. Asymptotic analysis of the drop oscillations for small Ohnesorge number, Oh, brings out two distinct damping mechanisms, both of which are accounted for. One, arising from viscous dissipation in the regions near the contact lines, is characterized by a parameter β. The other comes from the boundary layer at the wall and is of order Oh1/2. The small-Oh problem has been implemented numerically. As expected, lightly damped normal modes are found to have resonant response close to their inviscid oscillation frequencies. Damping coefficients for each of the two damping mechanisms and lightly damped modes are determined as a function of contact angle. The relative importance of boundary-layer and contact-line damping is quantified and found to depend both on the contact angle and on β/Oh1/2. Cases can be found in which the two damping mechanisms have comparable effects, as well as others for which one or other of the mechanisms is dominant. Comparison with DNS, which allows for nonlinear effects and has the same contact-line model, shows agreement for a particular case having small Oh and small wall displacement amplitude.

English

We consider a biharmonic problem Δ2uω = fω with Navier type boundary conditions uω = Δuω = 0, on a family of truncated sectors Ωω in ℝ2 of radius r, 0 < r < 1 and opening angle ω, ω ∈ (2π/3, π] when ω is close to π. The family of right-hand sides (fω)ω∈(2π/3, π] is assumed to depend smoothly on ω in L2(Ωω). The main result is that uω converges to uπ when ω → π with respect to the H2-norm. We can also show that the H2-topology is optimal for such a convergence result.

Global solutions to the incompressible magneto-micropolar system in a bounded domain in 2D

We prove the existence and uniqueness of global strong solutions to the incompressible magneto-micropolar system in a two dimensional bounded domain with Navier type boundary condition for the velocity.

On the role of pressure in the theory of MHD equations

We consider the system of MHD equations in Ω×(0,T), where Ω is a domain in R3 and T>0, with the no slip boundary condition for the velocity u and the Navier-type boundary condition for the magnetic induction b. We show that an associated pressure p, as a distribution with a certain structure, can be always assigned to a weak solution (u,b). The pressure is a function with some rate of integrability if the domain Ω is “smooth”, see section 3. In section 4, we study the regularity of p in a sub-domain Ω1×(t1,t2) of Ω×(0,T), where u (or, alternatively, both u and b) satisfies Serrin’s integrability conditions. Regularity criteria for weak solutions to the MHD equations in terms of π≔p+12|b|2 are studied in section 5. Finally, section 6 contains remarks on analogous results in the case of Navier’s or Navier-type boundary conditions for the velocity u.

Vector Potentials with Mixed Boundary Conditions. Application to the Stokes Problem with Pressure and Navier-type Boundary Conditions

In a three-dimensional bounded possibly multiply connected domain, we prove the existence, uniqueness, and regularity of some vector potentials, associated with a divergence-free function and satis...

A free boundary problem driven by the biharmonic operator

In this paper we consider the minimization of the functional \[ J[u]:=\int_\Omega |\Delta u|^2+\chi_{\{u>0\}} \] in the admissible class of functions \[ \mathcal A:= \left\{u\in W^{2, 2}(\Omega) {\mbox{ s.t. }} u-u_0\in W^{1,2}_0(\Omega) \right\}. \] Here, $\Omega$ is a smooth and bounded domain and $u_0\in W^{2,2}(\Omega)$ is a given function defining the Navier type boundary condition. The scale invariance of the problem suggests that, at the singular points of the free boundary, quadratic growth of $u$ is expected. We prove that $u$ is quadratically nondegenerate at the singular free boundary points using a refinement of Whitney's cube decomposition, which applies, if, for instance, the set $\{ u>0\}$ is a John domain. The optimal growth is linked with the approximate symmetries of the free boundary. More precisely, if at small scales the free boundary can be approximated by zero level sets of a quadratic degree two homogeneous polynomial, then we say that $\partial\{ u>0\}$ is rank-2 flat. Using a dichotomy method for nonlinear free boundary problems, we also show that, at the free boundary points $x\in \Omega$ where $\nabla u(x)=0$, the free boundary is either well approximated by zero sets of quadratic polynomials, i.e. $\partial\{ u>0\}$ is rank-2 flat, or $u$ has quadratic growth. Differently from the classical free boundary problems driven by the Laplacian operator, the one-phase minimizers present structural differences with respect to the minimizers, and one notion is not included into the other. In addition, one-phase minimizers arise from the combination of a volume type free boundary problem and an obstacle type problem, hence their growth condition is influenced in a non-standard way by these two ingredients.

Global solutions to the Maxwell–Navier–Stokes system in a bounded domain in 2D

We prove the existence and uniqueness of global strong solutions to the Maxwell–Navier–Stokes system in a two dimensional bounded domain with Navier-type boundary condition for the velocity.

Order Reduced Methods for Quad-Curl Equations with Navier Type Boundary Conditions

Order Reduced Methods for Quad-Curl Equations with Navier Type Boundary Conditions

Global well-posedness for the 2D incompressible magneto-micropolar fluid system with partial viscosity

In this paper, we consider an initial-boundary value problem for the 2D incompressible magneto-micropolar fluid equations with zero magnetic diffusion and zero spin viscosity in the horizontally infinite flat layer with Navier-type boundary conditions. We establish the global well-posedness of strong solutions around the equilibrium $(0,e_1,0)$.

English

This paper proves a logarithmic regularity criterion for 3D Navier-Stokes system in a bounded domain with the Navier-type boundary condition.

Stokes and Navier-Stokes problems with Navier-type boundary condition in L^p-spaces

Stokes and Navier-Stokes problems with Navier-type boundary condition in L^p-spaces

The interior regularity of pressure associated with a weak solution to the Navier–Stokes equations with the Navier-type boundary conditions

We prove that if u is a weak solution to the Navier–Stokes system with the Navier-type boundary conditions in Ω×(0,T), satisfying the strong energy inequality in Ω×(0,T) and Serrin's integrability conditions in Ω′×(t1,t2) (where Ω′ is a sub-domain of Ω and 0≤t1<t2≤T) then p and ∂tu have spatial derivatives of all orders essentially bounded in Ω″×(t1+ϵ,t2−ϵ) for any bounded sub-domain Ω″⊂Ω″‾⊂Ω′ and ϵ>0 so small that t1+ϵ<t2−ϵ. (See Theorem 1.) We show an application of Theorem 1 to the procedure of localization.

Partially dissipative 2D Boussinesq equations with Navier type boundary conditions

This paper concerns itself with two systems of the 2D Boussinesq equations with partial dissipation in bounded domains with the Navier type boundary conditions. We attempt to achieve two main goals: first, to prove the global existence and uniqueness under minimal regularity assumptions on the initial data; and second, to provide a direct and transparent approach that explicitly reveals the impacts of the Navier boundary conditions. The 2D Boussinesq equations with partial dissipation have attracted considerable interests in the last few years, although most of the results are aimed at sufficiently regular solutions in the whole space or periodic domains. Larios et al. (2013) made serious efforts to minimize the regularity assumptions necessary for the uniqueness of solutions in the spatially periodic setting. In contrast to the whole space and the periodic domains, the Navier boundary conditions generate boundary terms and require compatibility conditions. In addition, due to the lack of boundary conditions for the pressure, we resort to the existence and regularity result on the associated Stokes problem with Navier boundary conditions. The uniqueness relies on the Yudovich techniques and the introduction of a lower regularity counterpart of the temperature.

Semi-Group Theory for the Stokes Operator with Navier-Type Boundary Conditions on L p -Spaces

In this article we consider the Stokes problem with Navier-type boundary conditions on a domain $${\Omega}$$ , not necessarily simply connected. Since, under these conditions, the Stokes problem has a non trivial kernel, we also study the solutions lying in the orthogonal of that kernel. We prove the analyticity of several semigroups generated by the Stokes operator considered in different functional spaces. We obtain strong, weak and very weak solutions for the time dependent Stokes problem with the Navier-type boundary condition under different hypotheses on the initial data u 0 and external force f. Then, we study the fractional and pure imaginary powers of several operators related with our Stokes operators. Using the fractional powers, we prove maximal regularity results for the homogeneous Stokes problem. On the other hand, using the boundedness of the pure imaginary powers, we deduce maximal $${L^{p}-L^{q}}$$ regularity for the inhomogeneous Stokes problem.