Stokes System Research Articles

Long-Time Behavior of Global Weak Solutions for a Beris-Edwards Type Model of Nematic Liquid Crystals

We consider a Beris-Edwards system modeling incompressible liquid crystal flows of nematic type. This system couples a Navier–Stokes system for the fluid velocity with a time-dependent system for the Q-tensor variable, whose spectral decomposition is related to the directors of liquid crystal molecules. The long-time behavior for global weak solutions is studied, proving that each whole trajectory converges to a single equilibrium whenever a regularity hypothesis is satisfied by the energy of the weak solution.

Remote trajectory tracking of a rigid body in an incompressible fluid at low Reynolds number

In this paper we study the motion of a rigid body driven by Newton’s law immersed in a stationary incompressible Stokes flow occupying a bounded simply connected domain. The aim is that of trajectory tracking of the solid by the means of a control in the form of Dirichlet boundary data on the outside boundary of the fluid domain. We show that it is possible to exactly achieve any smooth trajectory for the solid that stays away from the external boundary, by the means of such a remote control. The proof relies on some density methods for the Stokes system, as well as a reformulation of the solid equations into an ODE.

Efficient IMEX and consistently energy-stable methods of diffuse-interface models for incompressible three-component flows

In this study, we consider the numerical approximation of incompressible three-component fluids, in which the fluid interfaces are captured by ternary Cahn–Hilliard equations and the fluid flows are governed by Navier–Stokes equations. This system includes not only nonlinear effects but also coupling among phase-field variables, velocities, and pressure. The ternary Cahn–Hilliard–Navier–Stokes system also satisfies the energy dissipation law, which is a basic physical property. For the appropriate treatment of the nonlinear and coupling terms and preservation of the energy dissipation law in a discrete version, we develop second-order time-accurate, linearly implicit-explicit (IMEX) methods using a variation of the scalar auxiliary variable (SAV) method. To improve the consistency between the original and modified energies, a simple and effective energy relaxation technique is considered. We analytically proved the unique solvability and the relaxed energy dissipation law. The proposed schemes are highly efficient for implementation because only linear elliptic equations need to be solved separately. Extensive computational experiments are performed to validate the accuracy, energy stability, and performance of the proposed method. To facilitate further study, we provide the C codes for the typical numerical simulations at http://github.com/yang521.

Local Strong Solutions to the Full Compressible Navier--Stokes System with Temperature-Dependent Viscosity and Heat Conductivity

Local Strong Solutions to the Full Compressible Navier--Stokes System with Temperature-Dependent Viscosity and Heat Conductivity

Quotient and Solutions of the One-Dimensional Navier–Stokes System

Quotient and Solutions of the One-Dimensional Navier–Stokes System

Boundedness and asymptotic stabilization in a two-dimensional Keller–Segel–Navier–Stokes system with sub-logistic source

This paper mainly deals with a Keller–Segel–Navier–Stokes model with sub-logistic source in a two-dimensional bounded and smooth domain. For a large class of cell kinetics including sub-logistic sources, it is shown that under an explicit condition involving the chemotactic strength, asymptotic “damping” rate and initial mass of cells, the associated no-flux/no-flux/Dirichlet problem possesses a global and bounded classical solution. Moreover, a systematical treatment has been conducted on convergence of bounded solutions toward constant equilibrium in [Formula: see text] for sub- and standard logistic sources. In such chemotaxis-fluid setting, our boundedness improves known blow-up prevention by logistic source to blow-up prevention by sub-logistic source, indicating standard logistic source is not the weakest damping source to prevent blow-up, and our stability improves known algebraic convergence under quadratic degradation to exponential convergence under log-correction of quadratic degradation, implying log-correction of quadratic degradation quickens the decay of bounded solutions. These findings significantly improve and extend previously known ones.

An error analysis of the finite element method of the Stokes equations with parameter based on the singular function expansion

AbstractIn this paper, we introduce a mixed finite element method of the Stokes system with parameter on a non‐convex polygon. This model problem is considered in the time‐discretization of the unstationary Stokes equations, and it is mentioned in Kweon's study that the velocity field and the pressure function are composed of singular parts and smoother remainders due to the corner singularities of the solution near each re‐entrant corner. We propose a stable numerical scheme to approximate the coefficients in singular part and the regular part of solution, based on the extraction formulae of the coefficients in singular part. Furthermore, to show the reliability of the proposed method, we derive the absolute error estimates of the approximations of coefficients, and the and error estimates of the finite element solution corresponding to the regular part.

Attractors of 2D Navier–Stokes system of equations in a locally periodic porous medium

This article deals with two-dimensional Navier–Stokes system of equations with rapidly oscillating term in the equations and boundary conditions. Studying the problem in a perforated domain, the authors set homogeneous Dirichlet condition on the outer boundary and the Fourier (Robin) condition on the boundary of the cavities. Under such assumptions it is proved that the trajectory attractors of this system converge in some weak topology to trajectory attractors of the homogenized Navier–Stokes system of equations with an additional potential and nontrivial right hand side in the domain without pores. For this aim, the approaches from the works of A.V. Babin, V.V. Chepyzhov, J.-L. Lions, R. Temam, M.I. Vishik concerning trajectory attractors of evolution equations and homogenization methods appeared at the end of the XX-th century are used. First, we apply the asymptotic methods for formal construction of asymptotics, then, we verify the leading terms of asymptotic series by means of the methods of functional analysis and integral estimates. Defining the appropriate axillary functional spaces with weak topology, we derive the limit (homogenized) system of equations and prove the existence of trajectory attractors for this system. Lastly, we formulate the main theorem and prove it through axillary lemmas.

Variational resolution of outflow boundary conditions for incompressible Navier–Stokes

This paper focuses on the so-called weighted inertia-dissipation-energy variational approach for the approximation of unsteady Leray–Hopf solutions of the incompressible Navier–Stokes system. Initiated in (Ortiz et al 2018 Nonlinearity 31 5664–82), this variational method is here extended to the case of non-Newtonian fluids with power-law index r ⩾ 11/5 in three space dimension and large nonhomogeneous data. Moreover, boundary conditions are not imposed on some parts of boundaries, representing, e.g., outflows. Correspondingly, natural boundary conditions arise from the minimisation. In particular, at walls we recover boundary conditions of Navier-slip type. At outflows and inflows, we obtain the condition . This provides the first theoretical explanation for the onset of such boundary conditions.

A generalized Stokes system with a non-smooth slip boundary condition.

A class of quasi variational-hemivariational inequalities in reflexive Banach spaces is studied. The inequalities contain a convex potential, a locally Lipschitz superpotential and an implicit obstacle set of constraints. Results on the well-posedness including existence, uniqueness, dependence of solution on the data and the compactness of the solution set in the strong topology are established. The applicability of the results is illustrated by the steady-state generalized Stokes model of a generalized Newtonian incompressible fluid with a non-monotone slip boundary condition. This article is part of the theme issue 'Non-smooth variational problems and applications'.

Inviscid limit for the full viscous MHD system with critical axisymmetric initial data

This paper establishes the global well-posedness issue for the full viscous MHD equations in the axisymmetric setting. Global solutions are obtained in critical Besov spaces uniformly to the viscosity when the resistivity is fixed in the spirit of [Abidi H, Hmidi T, Keraani S. On the global well-posedness for the axisymmetric Euler equations. Math. Ann. 2010;347:15–41.], [Hassainia Z. On the global well-posedness of the 3D axisymmetric resistive MHD equations. Ann. Henri Poincaré. 2022;23:2877-2917], [Hmidi T, Zerguine M. Inviscid limit axisymmetric Navier–Stokes system. Differential and Integral Equations. 2009;22(11–12):1223–1246.]. Furthermore, strong convergence in the resolution spaces with a rate of convergence is also studied.

Higher Order Fractional Differentiability for the Stationary Stokes System

Higher Order Fractional Differentiability for the Stationary Stokes System

Stabilization of 3D Navier–Stokes Equations to Trajectories by Finite-Dimensional RHC

Local exponential stabilization of the three-dimensional Navier–Stokes system to a given reference trajectory via receding horizon control (RHC) is investigated. The RHC enters as the linear combinations of a finite number of actuators. The actuators are spatial functions and can be chosen in particular as indicator functions whose supports cover only a part of the spatial domain.

Global large solutions and incompressible limit for the compressible Navier–Stokes system with capillarity

Consider the Cauchy problem for the barotropic compressible Navier–Stokes–Korteweg equations in the whole space Rd (d≥2), supplemented with large initial velocity v0 and almost constant initial density ϱ0. In the two-dimensional case, the global solutions are shown in the critical Besov spaces framework without any restrictions on the size of the initial velocity, provided that the pressure admits a stability condition and the volume viscosity is sufficiently large. The result still holds for the higher dimensional case d≥3 under the additional assumption that the classical incompressible Navier-Stokes equations, supplemented with the initial velocity as the Helmholtz projection of v0, admits a global strong solution.

Model for Aqueous Polymer Solutions with Damping Term: Solvability and Vanishing Relaxation Limit.

The main aim of this paper is to investigate the solvability of the steady-state flow model for low-concentrated aqueous polymer solutions with a damping term in a bounded domain under the no-slip boundary condition. Mathematically, the model under consideration is a boundary value problem for the system of strongly nonlinear partial differential equations of third order with the zero Dirichlet boundary condition. We propose the concept of a full weak solution (velocity–pressure pair) in the distributions sense. Using the method of introduction of auxiliary viscosity, the acute angle theorem for generalized monotone nonlinear operators, and the Krasnoselskii theorem on the continuity of the superposition operator in Lebesgue spaces, we obtain sufficient conditions for the existence of a full weak solution satisfying some energy inequality. Moreover, it is shown that the obtained solutions of the original problem converge to a solution of the steady-state damped Navier–Stokes system as the relaxation viscosity tends to zero.

Nonlinear Potential Estimates for Generalized Stokes System

In this paper, we consider the generalized stationary Stokes system with p -growth and Dini-\({\text {BMO}}\) regular coefficients. The main purpose is to establish pointwise estimates for the shear rate and the associated pressure to such Stokes system in terms of an unconventional nonlinear Havin–Maz’ya–Wolff type potential of the nonhomogeneous term in the plane. As a consequence, a symmetric gradient \(L^{\infty }\) estimate is obtained. Moreover, we derive potential estimates for the weak solution to the Stokes system without additional regularity assumptions on the coefficients in higher dimensional space.

Homogenization of MHD flows in porous media

The paper is concerned with the homogenization of a nonlinear differential system describing the flow of an electrically conducting, incompressible and viscous Newtonian fluid through a periodic porous medium, in the presence of a magnetic field. We introduce a variational formulation of the differential system equipped with boundary conditions. We show the existence of a solution of the variational problem, and derive uniform estimates of the solutions depending on the characteristic parameters of the flow. Using the two-scale convergence method, we rigorously derive a two-scale equation for the two-scale current density, and a two-pressure Stokes system. We derive, in the case of constant magnetic permeability, an explicit relation expressing the macroscopic velocity as a function of the macroscopic Lorentz force, the pressure gradient, the external body force, and the macroscopic current density, via two permeability filtration tensors. When the magnetic field is absent, this relation reduces to the Darcy law.

Global weak solutions to the Vlasov–Poisson–Fokker–Planck–Navier–Stokes system

We consider the compressible Vlasov–Poisson–Fokker–Planck–Navier–Stokes system in a three‐dimensional bounded domain with nonhomogeneous Dirichlet boundary conditions. The system describes the evolution of charged particles ensemble dispersed in an isentropic fluid. For the adiabatic coefficient , we establish the global existence of weak solutions to this system with arbitrary large initial and boundary data.

Non-isothermal non-Newtonian flow problem with heat convection and Tresca's friction law

We consider an incompressible non-isothermal fluid flow with non-linear slip boundary conditions governed by Tresca's friction law. We assume that the stress tensor is given as where θ is the temperature, π is the pressure, u is the velocity and is the strain rate tensor of the fluid while p is a real parameter. The problem is thus given by the p-Laplacian Stokes system with subdifferential type boundary conditions coupled to a elliptic equation describing the heat conduction in the fluid. We establish first an existence result for a family of approximate coupled problems where the coupling term in the heat equation is replaced by a bounded one depending on a parameter , by using a fixed point technique. Then we pass to the limit as δ tends to zero and we prove the existence of a solution to our original coupled problem in Banach spaces depending on p for any .

Non-homogeneous Dirichlet-transmission problems for the anisotropic Stokes and Navier-Stokes systems in Lipschitz domains with transversal interfaces

This paper is build around the stationary anisotropic Stokes and Navier-Stokes systems with an L^infty -tensor coefficient satisfying an ellipticity condition in terms of symmetric matrices in {mathbb {R}}^{ntimes n} with zero matrix traces. We analyze, in L^2-based Sobolev spaces, the non-homogeneous boundary value problems of Dirichlet-transmission type for the anisotropic Stokes and Navier-Stokes systems in a compressible framework in a bounded Lipschitz domain with a transversal Lipschitz interface in {mathbb {R}}^n, nge 2 (n=2,3 for the nonlinear problems). Thus, the interface intersects transversally the boundary of the Lipschitz domain and divides the domain into two Lipschitz sub-domains. First, we use a mixed variational approach to prove the well-posedness of linear problems related to the anisotropic Stokes system. Then we show the existence of a weak solution to the Dirichlet and Dirichlet-transmission problems for the nonlinear anisotropic Navier-Stokes system. This is done by implementing the Leray-Schauder fixed point theorem and using various results and estimates from the linear case, as well as the Leray-Hopf and some other norm inequalities. Explicit conditions for uniqueness of solutions to the nonlinear problems are also provided.