Stokes System Research Articles

Suppression of Blow-Up in Multispecies Patlak–Keller–Segel–Navier–Stokes System via the Poiseuille Flow in a Finite Channel

Suppression of Blow-Up in Multispecies Patlak–Keller–Segel–Navier–Stokes System via the Poiseuille Flow in a Finite Channel

A three-dimensional Keller-Segel-Navier–Stokes system involving subquadratic logistic degradation: global generalized solutions and eventual smoothness

In this paper, we consider a Keller-Segel-Navier–Stokes system involving subquadratic logistic degradation: nt+u·∇n=Δn-∇·(n∇c)+ρn-μnα,ct+u·∇c=Δc-c+n,ut+(u·∇)u=Δu+∇P+n∇ϕ,∇·u=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\left\\{ \\begin{array}{ccl} n_t + {\ extbf{u}}\\cdot \ abla n & =& \\Delta n-\ abla \\cdot (n\ abla c)+\\rho n- \\mu n^{\\alpha }, \\\\ \\, c_t + {\ extbf{u}}\\cdot \ abla c & =& \\Delta c -c+n, \\\\ {\ extbf{u}}_t+ ({\ extbf{u}}\\cdot \ abla ) {\ extbf{u}}& =& \\Delta {\ extbf{u}}+ \ abla P + n\ abla \\phi , \\\\ \ abla \\cdot {\ extbf{u}}& = & 0 \\end{array} \\right. \\end{aligned}$$\\end{document}in a three-dimensional smoothly bounded domain along with reasonably mild initial conditions and no-flux/no-flux/Dirichlet boundary conditions, where ρ∈R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\rho \\in {\\mathbb {R}}$$\\end{document} and μ>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu >0$$\\end{document}. The purpose of the present work is to firstly establish the generalized solvability for the model under the subquadratic exponent restriction α≥43\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha \\ge \\frac{4}{3}$$\\end{document}, which indicates that persistent Dirac-type singularities can be ruled out, and to secondly exhibit the eventual smoothness of these solutions under the stronger restriction α>53\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha > \\frac{5}{3}$$\\end{document} whenever ρ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\rho $$\\end{document} is not too large in the sense of (ρ++1)α-1ρ+≤δ0μα,(ρ++1)min{1,α-1}ρ+max{1,3-α}≤δ0μ2,ρ+≤δ0μ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\big (\\rho _++1\\big )^{\\alpha -1}{\\rho _+}\\le \\delta _0\\mu ^{\\alpha },\\qquad \\big (\\rho _++1\\big )^{\\min \\{1,\\,\\alpha -1\\}}{\\rho _+}^{\\max \\{1,\\,3-\\alpha \\}}\\le \\delta _0\\mu ^{2},\\qquad \\rho _+\\le \\delta _0\\mu \\end{aligned}$$\\end{document}for some δ0=δ0(α)>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\delta _0=\\delta _0(\\alpha )>0$$\\end{document}. These results especially extend the precedent works due to Winkler (J Functional Anal 276: 1339-1401, 2019; Comm Math Phys 367: 439–489, 2022), where, among other things, the corresponding studies focus on the case α=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha =2$$\\end{document} of quadratic degradation.

Well‐posedness and inviscid limits for the Keller–Segel–Navier–Stokes system of the parabolic–elliptic type

AbstractWe show the local well‐posedness of the Keller–Segel system of the parabolic–elliptic type coupled with the Navier–Stokes system for arbitrary initial data with Sobolev regularities, where the solution is uniformly bounded with respect to the viscosity. We also show the continuous dependence of the solutions with respect to the initial data. As a result of the uniform boundedness of the solutions, we obtain inviscid limits of the system. The proof is mainly based on a priori estimates in the Sobolev spaces.

The velocity tracking problem for Navier–Stokes equations with pointwise-integral control constraints in time-space

In this work, we study the velocity tracking control problem of the evolutionary two-dimensional Navier–Stokes system. The regularizing Tikhonov term is not involved in the cost functional and pointwise-integral control constraints in time-space are considered. First- and second-order optimality conditions are established using a suitable extended cone for the sufficient ones. We also address the numerical approximation of the control problem, proving the convergence of the discrete problems and establishing error estimates between the optimal discrete and continuous states.

A simple and efficient finite difference scheme to the Cahn–Hilliard–Navier–Stokes system equations

Central difference schemes are usually not suited to convective terms in a transport equation due to its oscillation nature when convection effect is pronounced. However, this work found that applying the standard central difference scheme to the convective term along with another central difference scheme to the fourth order diffusive term in the Cahn–Hilliard equation can realize nearly an order of magnitude speed rise, in the framework of a fully explicit finite difference scheme. The discretization was done on a semi-staggered grid where pressure was stored at the cell center and other variables were stored at the cell corners. To accelerate computation, a simple parallelism based on OpenMP was used. The scheme was tested in a number of cases and was compared with both analytical and experimental outcomes. The results showed that the scheme is efficient compared with a previous fully explicit finite difference scheme for the Cahn–Hilliard equation, and that a time step more than five times larger can be employed.

Global bounded solution in an attraction repulsion Chemotaxis-Navier-Stokes system with Neumann and Dirichlet boundary conditions

This paper deals with an attraction–repulsion Chemotaxis-Navier–Stokes system with Dirichlet boundary for the attraction signal and Neumann boundary for the repulsion signal. Based on the work of Winkler (2020) and Wang et al. (2022), by using a series estimates, it is shown that in two dimension the classical solution of the system is globally bounded, under the condition of small initial values ‖n0‖L1(Ω) in the explicit expressions for ‖c0‖L∞(Ω) and attraction–repulsion coefficients.

Global Classical Solution to the Strip Problem of 2D Compressible Navier–Stokes System with Vacuum and Large Initial Data

Global Classical Solution to the Strip Problem of 2D Compressible Navier–Stokes System with Vacuum and Large Initial Data

Quasi-neutral Limit of Nernst–Planck–Navier–Stokes System

Quasi-neutral Limit of Nernst–Planck–Navier–Stokes System

Global existence of weak solutions for 2D chemotaxis‐Navier–Stokes system with fractional diffusion

In this paper, we consider the following 2D incompressible chemotaxis‐Navier–Stokes equations with the fractional diffusion where and . The gravitational potential is generally supposed to be sufficiently smooth given functions. By establishing the new priori estimates, we get the global well‐posedness for the above system with large initial data.

Dynamically regularized Lagrange multiplier schemes with energy dissipation for the incompressible Navier-Stokes equations

In this paper, we present efficient numerical schemes based on the Lagrange multiplier approach for the Navier-Stokes equations. By introducing a dynamic equation (involving the kinetic energy, the Lagrange multiplier, and a regularization parameter), we form a new system which incorporates the energy evolution process but is still equivalent to the original equations. Such nonlinear system is then discretized in time based on the backward differentiation formulas, resulting in a dynamically regularized Lagrange multiplier (DRLM) method. First- and second-order DRLM schemes are derived and shown to be unconditionally energy stable with respect to the original variables. The proposed schemes require only the solutions of two linear Stokes systems and a scalar quadratic equation at each time step. Moreover, with the introduction of the regularization parameter, the Lagrange multiplier can be uniquely determined from the quadratic equation, even with large time step sizes, without affecting accuracy and stability of the numerical solutions. Fully discrete energy stability is also proved with the Marker-and-Cell (MAC) discretization in space. Various numerical experiments in two and three dimensions verify the convergence and energy dissipation as well as demonstrate the accuracy and robustness of the proposed DRLM schemes.

Stability of the composite wave of two planar viscous shock waves for the compressible Navier–Stokes system

We are concerned with the nonlinear stability of the composite wave consisting of two planar viscous shock waves to the three-dimensional compressible Navier–Stokes system. It is shown that if the shock strengths are suitably small but not necessarily of the same order of magnitude, and the initial perturbations are suitably small without the zero-mass conditions, then there exists a unique, globally strong solution in time to the compressible Navier–Stokes system, which asymptotically approaches the corresponding composite wave up to time-dependent shifts in L^{\infty} norm. The proof employs the weighted relative entropy method, an L^{2} -contraction technique with time-dependent shifts to the shocks developed by Kang and Vasseur [J. Eur. Math. Soc. (JEMS) 23 (2021), 585–638; Invent. Math. 224 (2021), 55–146]. We perform the stability analysis within the original H^{2} -perturbation framework instead of using the anti-derivative technique. Compared with the previous work of the author and Wang [J. Eur. Math. Soc. (JEMS) (2023), DOI 10.4171/JEMS/1486] for the single shock case, a major difficulty is the construction of shifts to ensure that the two shock waves are well separated.

Air Leakages at Microvalves: Pressure Decay Measurements and Extended Continuum Modelling of Knudsen Flows

To improve the performance of valves in relation to the leakage rate, a comprehensive evaluation of the valve characteristics and behavior during pressure exposure is important. Often, these low gas flow rates below 0.1 cm3/min cannot be accurately measured with conventional flow sensors. This paper presents a small and low-cost test rig for measuring gas leakage rates accurately, even far below 0.1 cm3/min, with the pressure decay method. These leakage flows are substantiated with a flow model, where we demonstrate the feasibility of modeling those gas flows with an extended Navier–Stokes framework to obtain more accurate theoretical predictions. As expected, the comparison to the experimental results proves that the classical Navier–Stokes system is unsuitable for modeling Knudsen flows. Hence, self-diffusion of gas, a wall-slip boundary condition, and an effective mean free path model were introduced in a physically evident manner. In terms of the calculated mass flow, while self-diffusion and slip boundary conditions explain deviations from the classical Navier–Stokes equation for Knudsen numbers already smaller than 1, the effective mean free path model has an effect, especially when Kn > 1. For simplified conditions, an analytical solution was presented and compared to the results of an OpenFOAM CFD-solver for flow rates through more complex gap-flow geometries of the flap valve. Hereby, acceptable deviations between 10% and 20% were observed. A comparison with measurement results was carried out. The reproducibility of the measurement method was verified by comparing multiple measurements of one silicon microvalve sample to a state-of-the-art flow sensor. Three geometrically similar passive silicon microvalves were measured with air overpressure decreasing from 15 kPa relative to atmospheric pressure. Maximum gas volume flowing in a blocking direction of 1–26 µL/min with high reproducibility and marginal noise were observed.

Remark on the Local Well-Posedness of Compressible Non-Newtonian Fluids with Initial Vacuum

We discuss in this short note the local-in-time strong well-posedness of the compressible Navier–Stokes system for non-Newtonian fluids on the three dimensional torus. We show that the result established recently by Kalousek, Mácha, and Nečasova in https://doi.org/10.1007/s00208-021-02301-8 can be extended to the case where vanishing density is allowed initially. Our proof builds on the framework developed by Cho, Choe, and Kim in https://doi.org/10.1016/j.matpur.2003.11.004 for compressible Navier–Stokes equations in the case of Newtonian fluids. To adapt their method, special attention is given to the elliptic regularity of a challenging nonlinear elliptic system. We show particular results in this direction, however, the main result of this paper is proven in the general case when elliptic W2,p\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$W^{2,p}$$\\end{document}-regularity is imposed as an assumption. Also, we give a finite time blow-up criterion.

On Isolated Singularities for the Stationary Navier–Stokes System

The classical problem of removable singularities is considered for solutions to the stationary Navier–Stokes system in dimension n≥3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n\\ge 3$$\\end{document} and an old theorem of Shapiro (TAMS 187:335–363, 1974) is recovered and extended to solutions in a half ball vanishing on the flat boundary. Moreover, for n=4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n=4$$\\end{document} it is proved that there are not distributional solutions, smooth away from the singularity and such that u(x)=O(|x|-1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$u(x)=O(|x|^{-1})$$\\end{document}.

Multi-vortices and lower bounds for the attractor dimension of 2d Navier-Stokes equations

A new method for obtaining lower bounds for the dimension of attractors for the Navier–Stokes equations, which does not use Kolmogorov flows, is presented. Using this method, exact estimates of the dimension are obtained for the case of equations on a plane with Ekman damping. Similar estimates were previously known only for the case of periodic boundary conditions. In addition, similar lower bounds are obtained for the classical Navier–Stokes system in a two-dimensional bounded domain with Dirichlet boundary conditions.

Global existence in a two-dimensional chemotaxis-(Navier)-Stokes system with sub-logarithmic sensitivity

This paper considers the following chemotaxis-(Navier)-Stokes system{nt+u⋅∇n=Δn−∇⋅(nχ(c)∇c),ct+u⋅∇c=Δc−f(n)c,ut+κ(u⋅∇)u=Δu+∇P+n∇Φ in a smooth bounded domain Ω⊂R2 with no-flux/no-flux/Dirichlet boundary conditions, whereχ(c)=1cαandf(n)=nγ with α∈(0,1) and γ∈(0,1). We proved that if ‖c0‖L∞(Ω)<1, then for any κ∈R the problem possesses a global classical solution.

Existence of Solutions to the Patlak–Keller–Segel–Navier–Stokes System

Existence of Solutions to the Patlak–Keller–Segel–Navier–Stokes System

Capturing non-equilibrium in hypersonic flows: Insights from a two-temperature model in polyatomic rarefied gases

The study utilizes a two-temperature model to analyze non-equilibrium in normal shocks within hypersonic flows in polyatomic rarefied gases. Derived from the extended second law of thermodynamics, this model separates translational and internal temperatures in polyatomic gases, providing a more accurate depiction of non-equilibrium gas flow compared to classical theories like the Navier–Stokes and Fourier (NSF) system. Notably, the analysis reveals that the two-temperature model incorporates an additional contribution to the heat flux due to the gradient of the dynamic temperature, resulting in improved accuracy, especially for high Mach numbers. Results show that the model gives satisfactory shock density and temperature profiles up to Mach 10, with very good agreement observed up to Mach 6.1 compared to the classical NSF model. We conduct an order of magnitude analysis on the dynamic temperature and heat flux gradients appearing in the new constitutive equation using the Mott-Smith method. This analysis highlights the impact of these terms on accurately modeling polyatomic gas behavior in high-speed flows. The effects of bulk viscosity and incoming temperature on shock profiles are also investigated, contributing to a better understanding of shock wave structures in polyatomic gases and their implications for hypersonic flow dynamics.

On Hydrodynamic Limits of the Vlasov–Navier–Stokes System

We introduce a framework to justify hydrodynamic limits of the Vlasov–Navier–Stokes system. We specifically study high friction regimes, which take into account the fact that particles of the dispersed phase are light (resp. small) compared to the fluid part, and lead to the derivation of Transport–Navier–Stokes (resp. Inhomogeneous Navier–Stokes) systems.

Externally forced blow-up and optimal spaces for source regularity in the two-dimensional Navier–Stokes system

Externally forced blow-up and optimal spaces for source regularity in the two-dimensional Navier–Stokes system