Stokes System Research Articles

Optimal \(\boldsymbol{{L^2}}\) Error Estimates of Unconditionally Stable Finite Element Schemes for the Cahn–Hilliard–Navier–Stokes System

The paper is concerned with the analysis of a popular convex-splitting finite element method (FEM) for the Cahn–Hilliard–Navier–Stokes system, which has been widely used in practice. Since the method is based on a combined approximation to multiple variables involved in the system, the approximation to one of the variables may seriously affect the accuracy for others. Optimal-order error analysis for such combined approximations is challenging. The previous works failed to present optimal error analysis in -norm due to the weakness of the traditional approach. Here we first present an optimal error estimate in -norm for the convex-splitting FEMs. We also show that optimal error estimates in the traditional (interpolation) sense may not always hold for all components in the coupled system due to the nature of the pollution/influence from lower-order approximations. Our analysis is based on two newly introduced elliptic quasi-projections and the superconvergence of negative norm estimates for the corresponding projection errors. Numerical examples are also presented to illustrate our theoretical results. More important is that our approach can be extended to many other FEMs and other strongly coupled phase field models to obtain optimal error estimates.

A stable pressure-correction MAC scheme for the fluid-Poroelastic material interaction problem

In this paper, we propose and analyse pressure-correction schemes based on marker and cell (MAC) method for the linear Stokes–Biot system with a fixed interface. The implicit backward Euler scheme for the time discretization is used, whereas the coupling terms are treated explicitly. These schemes are computationally efficient in that we only solve two decoupled problems. And for Stokes equations, we solve one vector-valued elliptic equation and one scalar-value Poisson equation per time step. These methods have optimal order without the incompressibility constraint of the Stokes system. We prove rigorously that they are unconditionally stable and present the numerical experiments to show their performance.

Modified diffuse interface fluid model and its consistent energy-stable computation in arbitrary domains

Binary fluid flows in irregular domains are common phenomena in natural and industrial fields. To describe these physical processes, we herein present a novel fluid flow-coupled two-material model reflecting different wetting conditions on an arbitrary fluid-solid interface. A modified ternary Cahn–Hilliard (CH) diffuse interface model is adopted to capture the fluid-fluid interface. One of three components is fixed for all time to represent the profile of complex domains. By considering the contact angle condition on solid-fluid interface and equilibrium interface assumption, an extra wetting term is derived. To treat the fluid flow in domains with solid obstacles, we consider the entire system as a porous medium with variable permeability and add a large penalty term into the incompressible Navier–Stokes (NS) equations. It can be proven that the proposed modified Cahn–Hilliard–Navier–Stokes (CHNS) system satisfies the energy dissipation law (energy stability). Based on the scalar auxiliary variable (SAV) approach with appropriate correction techniques, the linear and consistent energy-stable scheme is developed. We introduce the numerical implementation and estimate the discrete version of energy stability in detail. The proposed method can be implemented on regular Cartesian grids with the absence of explicit boundary treatment. Moreover, the calculations are totally decoupled in each time step. Extensive numerical experiments not only indicate the expected accuracy and stability but also show the superior performance in arbitrary domains with different wetting conditions.

Low Mach number limit of the global solution to the compressible Navier–Stokes system for large data in the critical Besov space

In this paper, we consider the compressible Navier–Stokes system around the constant equilibrium states and prove the existence of a unique global solution for arbitrarily large initial data in the scaling critical Besov space provided that the Mach number is sufficiently small, and the incompressible part of the initial velocity generates the global solution of the incompressible Navier–Stokes equation. Moreover, we consider the low Mach number limit and show that the compressible solution converges to the solution of the incompressible Navier–Stokes equation in some space time norms.

Fractional Higher Differentiability for Solutions of Stationary Stokes and Navier-Stokes Systems with Orlicz Growth

We consider weak solutions (u,π):Ω→ℝn×ℝ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(u,\\pi ):{\\Omega }\ o \\mathbb {R}^{n}\ imes \\mathbb {R}$\\end{document} to stationary ϕ-Navier-Stokes systems of the type −diva(x,Eu)+∇π+[Du]u=fdivu=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$ \\left \\{ \\begin {array}{ll} -\\mathrm {div~} a(x,\\mathcal {E} u)+\ abla \\pi +[Du]u=f \\\\ \\mathrm {div~} u=0 \\end {array} \\right . $\\end{document} in Ω⊂ℝn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}${\\Omega }\\subset \\mathbb {R}^{n}$\\end{document}, and to the corresponding ϕ-Stokes systems, in which the convective term [Du]u does not appear. In the above system, the function a(x,ξ) depends Hölder continuously on x and satisfies growth conditions with respect to the second variable expressed through a Young function ϕ. The notation Eu\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathcal {E} u$\\end{document} is used for the symmetric part of the gradient Du. We prove results on the fractional higher differentiability of both the symmetric part of the gradient Eu\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathcal {E} u$\\end{document} and of the pressure π.

Optimal well-posedness for the pressureless Euler–Navier–Stokes system

In this work, we investigate the Cauchy problem for the pressureless Euler–Navier–Stokes system in R3. We first establish the global small solutions of this system with critical regularity and then obtain the optimal time decay rate of the solutions by a suitable energy argument (independent of the spectral analysis). The proof crucially depends on non-standard product estimates and interpolations. In comparison with previous studies about time-decay by Choi and Jung [J. Math. Fluid Mech. 23, 99 (2021); arXiv:2112.14449], the smallness requirement of the low frequencies of initial data could be removed.

Fully decoupled energy-stable numerical schemes for two-phase coupled porous media and free flow with different densities and viscosities

In this article, we consider a phase field model with different densities and viscosities for the coupled two-phase porous media flow and two-phase free flow, as well as the corresponding numerical simulation. This model consists of three parts: a Cahn–Hilliard–Darcy system with different densities/viscosities describing the porous media flow in matrix, a Cahn–Hilliard–Navier–Stokes system with different densities/viscosities describing the free fluid in conduit, and seven interface conditions coupling the flows in the matrix and the conduit. Based on the separate Cahn–Hilliard equations in the porous media region and the free flow region, a weak formulation is proposed to incorporate the two-phase systems of the two regions and the seven interface conditions between them, and the corresponding energy law is proved for the model. A fully decoupled numerical scheme, including the novel decoupling of the Cahn–Hilliard equations through the four phase interface conditions, is developed to solve this coupled nonlinear phase field model. An energy-law preservation is analyzed for the temporal semi-discretization scheme. Furthermore, a fully discretized Galerkin finite element method is proposed. Six numerical examples are provided to demonstrate the accuracy, discrete energy law, and applicability of the proposed fully decoupled scheme.

The hydrodynamic limit for the inhomogeneous Vlasov–Navier–Stokes system

In this paper, we study the hydrodynamic limit for the fluid–particle model, which consists of the inhomogeneous incompressible Navier–Stokes equations coupled with the Vlasov equation through a drag force in a bounded domain of [Formula: see text] with a homogeneous Dirichlet boundary condition on the fluid velocity field and Maxwell boundary condition on the kinetic distribution function. The proof relies on the relative entropy argument, which extends the work of El Ghani [Asymptotic analysis for a Vlasov–Navier–Stokes system in a bounded domain, J. Hyperbolic Differ. Equ. 7 (2010) 191–210] to inhomogeneous incompressible Navier–Stokes–Vlasov equations.

Numerical analysis for backward Euler spectral discretization for Stokes equations with boundary conditions involving the pressure: part I

In this paper, we address the study of the time-dependent Stokes system with boundary conditions involving the pressure. We obtain existence and uniqueness for a class of Lipschitz-continuous domains. Next, a spectral discretizations of the problem is proposed combined with the backward Euler scheme. The discrete spaces are defined in a way to give exactly divergence-free discrete approximations for the velocity. Then, we prove the associated discrete inf–sup condition and derive a priori error estimates. Finally, some numerical experiments are presented.

Boundedness in a two-dimensional attraction–repulsion Navier–Stokes system with consumption of chemoattractant and sub-quadratic degradation

In this paper, we study an attraction–repulsion Navier–Stokes system with consumption of chemoattractant and sub-quadratic degradation: nt+u⋅∇n=Δn−χ∇⋅(n∇c)+ξ∇⋅(n∇v)+λn−μnθ; ct+u⋅∇c=Δc−nc; τvt+u⋅∇v=Δv−v+n; ut+κ(u⋅∇)u=Δu+∇P+n∇ϕ; ∇⋅u=0, x∈Ω, t>0 in a bounded and smooth domain Ω⊂R2 with no-flux/Dirichlet boundary conditions, where χ,ξ are positive constants, τ∈{0,1} and θ∈(1,2). We present the global boundedness of attraction–repulsion Navier–Stokes system with consumption of chemoattractant and sub-quadratic degradation.

Enhancing the viscosity-splitting method to solve the time-dependent Navier–Stokes equations

The viscosity-splitting (VS) scheme for solving the incompressible time-dependent Navier–Stokes equations consists to decouple the full problem into two easy subproblems to resolve. This method splits the nonlinearity (convection term) and the incompressibility into two different steps. As a result, every subproblem is easier to solve than the Navier–Stokes system. The semi-implicit approach is mostly used in the first step for the nonlinearity term and the intermediate velocity is calculated by dropping the pressure gradient. We then resolve a linear Stokes problem in the next step.We present a modified viscosity-splitting scheme (Incremental viscosity-splitting) (IVS) to improve this method. This scheme adds the gradient of the pressure (calculated at the previous time step) in the first step and modifies the Stokes equations in the second step to predict a better value of the intermediate velocity. A manufactured solution and two well-known benchmarks will be used to demonstrate the effectiveness of the suggested strategy.

Boundedness of solutions to a 2D chemotaxis-Navier–Stokes system with general sensitivity and nonlinear diffusion

This paper deals with the chemotaxis-Navier–Stokes system with the density equation nt+u⋅∇n=Δnm−∇⋅(nS(x,n,c)∇c)in a bounded domain Ω⊂R2. This paper gives results for global existence, boundedness and stabilization of solutions to the system provided that m>1+4α3,|S(x,n,c)|≤S0nα,S0>0,α≥0,without smallness conditions on initial data.

Differential-difference systems in the analysis of weak solvability of initial-boundary value problems with a spatial variable in a network-like domain

In the work, the approach and the corresponding methods, which make it possible to construct a priori estimates of weak solutions of a differential-difference system with a spatial variable varying in a multidimensional network-like domain are indicated. Such estimates in spaces of summable functions are used to find solvability conditions for boundary value problems of various types for differential-difference systems. In addition, a priori estimates are used to justify the application of the method of discretization with respect to the time variable (semi-discretization) to the analysis of the weak solvability of initial-boundary value problems and the subsequent construction of approximations of weak solutions. The rationale for the approach used is the fact that in a fairly wide class applied analysis of the problems of transporting continuous media networks-like carriers, the representation of mathematical models of the process using the formalisms of differential-difference systems is the only tool for effectively solving these problems. For example, the reduction of a differential system (initial-boundary value problem) to the corresponding differential-difference system makes it possible not only to significantly simplify the analysis of problems of optimal control of a differential system (since this analysis reduces to studying the problem of optimal control of a system of elliptic equations), but also, using classical methods of control theory for elliptic systems, algorithmize the original problem. The reduction used often facilitates establishing the conditions for the existence and uniqueness of optimal control of a differential system. These problems also include a fairly large range of studies of non-stationary network-like hydrodynamic processes and flow phenomena. As an illustration of the approach used and the results obtained, the analysis of the solvability of the linearized Navier–Stokes system is given and the ways of studying the nonlinear Navier–Stokes system are indicated.

Global weak solution for a chemotaxis Navier–Stokes system with [formula omitted]-Laplacian diffusion and singular sensitivity

In this paper, we consider a chemotaxis-Navier–Stokes system with p-Laplacian diffusion and singular sensitivity in a bounded convex domain Ω⊂R3 with smooth boundary. It is shown that under an appropriate hypothesis for p and δ, there exists a locally bounded global weak solution of the corresponding initial–boundary problem.

Global well-posedness of 3D inhomogeneous Navier–Stokes system with small unidirectional derivative

In Liu and Zhang (2020 Arch. Ration. Mech. Anal. 235 1405–44); Liu et al (2020 Arch. Ration. Mech. Anal. 238 805–43), the authors proved that as long as the one-directional derivative of the initial velocity is sufficiently small in some scaling invariant spaces, then the (anisotropic) Navier–Stokes (NS) system has a unique global solution. The goal of this paper is to extend this type of result to the 3D inhomogeneous (density-dependent) NS system. More precisely, given initial density such that and the initial velocity with belonging to , then the inhomogeneous NS system has a unique global solution provided that being sufficiently small for some bounded function f depending on and . This provide a more general result that of Chemin et al (2014 Commun. Math. Phys. 272 529–66); Chemin and Zhang (2015 Commun. PDE 40 878–96).

Global solutions to the free boundary value problem of a chemotaxis-Navier–Stokes system

In this paper, we investigate the global solvability of the chemotaxis-Navier–Stokes system on a three-dimensional moving domain of finite depth, bounded below by a rigid flat bottom and bounded above by the free surface. Completing the system with physical boundary conditions on the free surface that match the boundary descriptions in the experiments and numerical simulations, we establish the global existence and uniqueness of solutions near a constant state , where is the saturation value of the oxygen on the free surface. To the best of our knowledge, this is the first analytical work for the well-posedness of chemotaxis-Navier–Stokes system on a time-dependent domain.

Continuum Percolation in Stochastic Homogenization and the Effective Viscosity Problem

This contribution is concerned with the effective viscosity problem, that is, the homogenization of the steady Stokes system with a random array of rigid particles, for which the main difficulty is the treatment of close particles. Standard approaches in the literature have addressed this issue by making moment assumptions on interparticle distances. Such assumptions, however, prevent clustering of particles, which is not compatible with physically-relevant particle distributions. In this contribution, we take a different perspective and consider moment bounds on the size of clusters of close particles. On the one hand, assuming such bounds, we construct correctors and prove homogenization. On the other hand, based on subcritical percolation techniques, these bounds are shown to hold for various mixing particle distributions with nontrivial clustering. As a by-product of the analysis, we also obtain similar homogenization results for compressible and incompressible linear elasticity with unbounded random stiffness.

How far do indirect signal production mechanisms regularize the three-dimensional Keller–Segel–Stokes system?

How far do indirect signal production mechanisms regularize the three-dimensional Keller–Segel–Stokes system?

Large deviations principle via Malliavin calculus for the Navier–Stokes system driven by a degenerate white-in-time noise

The purpose of this paper is to establish the Donsker–Varadhan type large deviations principle (LDP) for the two-dimensional stochastic Navier–Stokes system. The main novelty is that the noise is assumed to be highly degenerate in the Fourier space. The proof is carried out by using a criterion for the LDP developed in [17] in a discrete-time setting and extended in [26] to the continuous-time. One of the main conditions of that criterion is the uniform Feller property for the Feynman–Kac semigroup, which we verify by using Malliavin calculus.

Well-posedness of a Navier–Stokes–Cahn–Hilliard system for incompressible two-phase flows with surfactant

We investigate a diffuse-interface model that describes the dynamics of viscous incompressible two-phase flows with surfactant. The resulting system of partial differential equations consists of a sixth-order Cahn–Hilliard equation for the difference of local concentrations of the binary fluid mixture coupled with a fourth-order Cahn–Hilliard equation for the local concentration of the surfactant. The former has a smooth potential, while the latter has a singular potential. Both equations are coupled with a Navier–Stokes system for the (volume averaged) fluid velocity. The evolution system is endowed with suitable initial conditions, a no-slip boundary condition for the velocity field and homogeneous Neumann boundary conditions for the phase functions as well as for the chemical potentials. We first prove the existence of a global weak solution, which turns out to be unique in two dimensions. Stronger regularity assumptions on the initial data allow us to prove the existence of a unique global (respectively, local) strong solution in two (respectively, three) dimensions. In the two-dimensional case, we derive a continuous dependence estimate with respect to the norms controlled by the total energy. Moreover, we establish instantaneous regularization properties of global weak solutions for [Formula: see text]. In particular, we show that the surfactant concentration stays uniformly away from the pure states 0 and 1 after some positive time.