Two-dimensional Navier Stokes System Research Articles

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.

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

The Navier–Stokes system ut+(u·∇)u=Δu+∇P+f(x,t),∇·u=0,is considered along with homogeneous Dirichlet boundary conditions in a smoothly bounded planar domain Ω. It is firstly, inter alia, observed that if T>00$$\\end{document}]]> and ∫0T{∫Ω|f(x,t)|·ln12(|f(x,t)|+1)dx}2dt<∞,then for all divergence-free u0∈L2(Ω;R2), a corresponding initial-boundary value problem admits a weak solution u with u|t=0=u0. For any positive and nondecreasing L∈C0([0,∞)) such that L(ξ)ln12ξ→0asξ→∞,this is complemented by a statement on nonexistence of such a solution in the presence of smooth initial data and a suitably constructed f:Ω×(0,T)→R2 fulfilling ∫0T{∫Ω|f(x,t)|·L(|f(x,t)|)dx}2dt<∞.This resolves a fine structure in the borderline case p=1 and q=2 appearing in results on existence of weak solutions for sources in Lq((0,T);Lp(Ω;R2)) when p∈(1,∞] and q∈[1,∞] satisfy 1p+1q≤32, and on nonexistence if here p∈[1,∞) and q∈[1,∞) are such that 1p+1q>32\\frac{3}{2}$$\\end{document}]]>.

On the solvability of a boundary value problem for a two-dimensional system of Navier-Stokes equations in a cone

Due to the fact that the Navier-Stokes equations are involved in the formulation of a large number of interesting problems that are important from an applied point of view, these equations have been the object of attention of mechanics, mathematicians and other scientists for several decades in a row. But despite this, many problems for the Navier-Stokes equation remain unexplored to this day. In this work, we are exploring the solvability of a boundary value problem for a two-dimensional Navier-Stokes system in a non-cylindrical degenerating domain, namely, in a cone with its vertex at the origin. Previously, we studied cases of the linearized Navier-Stokes system or non-degenerating cylindrical domains, so this work is a logical continuation of our previous research in this direction. To the above-mentioned degenerate domain we associate a family of non-degenerate truncated cones, which, in turn, are formed by a oneto-one transformation into cylindrical domains, where for the problem under consideration we established uniform a priori estimates with respect to changes in the index of the domains. Further, using a priori estimates and the Faedo-Galerkin method, we established the existence, uniqueness of solution in Sobolev classes, and its regularity as the smoothness of the given functions increases.

Asymptotic Behavior of Attractors of the Two-Dimensional Navier–Stokes System in a Domain with Small Periodic Obstacles

Asymptotic Behavior of Attractors of the Two-Dimensional Navier–Stokes System in a Domain with Small Periodic Obstacles

Stability of planar exterior stationary flows with suction

We consider the two-dimensional Navier-Stokes system in a domain exterior to a disk. The system admits a stationary solution with critical decay O(|x|−1) written as a linear combination of the pure rotating flow and the flux carrier. We prove its nonlinear stability in large time for initial disturbances in L2 under smallness conditions, assuming that there is suction across the boundary, namely that the sign of coefficients of the flux carrier is negative. This result partially solves an open problem in the literature.

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.

Systematic search for extreme and singular behaviour in some fundamental models of fluid mechanics.

This review article offers a survey of the research program focused on a systematic computational search for extreme and potentially singular behaviour in hydrodynamic models motivated by open questions concerning the possibility of a finite-time blow-up in the solutions of the Navier-Stokes system. Inspired by the seminal work of Lu & Doering (2008 Ind. Univ. Math. 57, 2693-2727), we sought such extreme behaviour by solving PDE optimization problems with objective functionals chosen based on certain conditional regularity results and a priori estimates available for different models. No evidence for singularity formation was found in extreme Navier-Stokes flows constructed in this manner in three dimensions. We also discuss the results obtained for one-dimensional Burgers and two-dimensional Navier-Stokes systems, and while singularities are ruled out in these flows, the results presented provide interesting insights about sharpness of different energy-type estimates known for these systems. Connections to other bounding techniques are also briefly discussed. This article is part of the theme issue 'Mathematical problems in physical fluid dynamics (part 1)'.

Boundedness and asymptotic behavior in a Keller-Segel(-Navier)-Stokes system with indirect signal production

This paper deals with the Keller-Segel(-Navier)-Stokes system with indirect signal production(⋆){nt+u⋅∇n=Δn−∇⋅(n∇v)+rn−μn2,vt+u⋅∇v=Δv−v+w,wt+u⋅∇w=Δw−w+n,ut+κ(u⋅∇)u=Δu−∇P+n∇Φ,∇⋅u=0 in a bounded and smooth domain Ω⊂RN(N=2,3) with no-flux boundary for n, v, w and no-slip boundary for u, where r∈R, μ≥0, κ∈{0,1} and Φ∈W2,∞(Ω). In the case without logistic source (r=μ=0), it is proved that for all suitably regular initial data, the associated initial-boundary value problem for the spatially two-dimensional Navier-Stokes system (⋆) admits a globally bounded classical solution. This result improves and extends the previously known ones. We point out that the same result to the corresponding two-dimensional Navier-Stokes system with direct signal production holds necessarily imposing some saturated chemotactic sensitivity, logistic damping or small total initial population mass. In the case coupled with logistic source (r∈R, μ>0), it is shown that for any reasonably regular initial data, the corresponding initial-boundary value problem for the spatially three-dimensional Stokes system (⋆) possesses a globally bounded classical solution, and that this solution stabilizes toward the corresponding spatially homogeneous equilibrium with the explicit convergence rates for the cases r<0, r=0 and r>0. We underline that the global boundedness of classical solution to the corresponding three-dimensional Stokes system with direct signal production was obtained only for μ≥23 (or sublinear signal production), and that the convergence result to the corresponding system with direct signal production was established only for r=0 and μ≥23. Our results rigorously confirm that the indirect signal production mechanism genuinely contributes to the global boundedness of classical solution to the Keller-Segel(-Navier)-Stokes system.

Large deviations and mixing for dissipative PDEs with unbounded random kicks

We study the problem of exponential mixing and large deviations for discrete-time Markov processes associated with a class of random dynamical systems. Under some dissipativity and regularisation hypotheses for the underlying deterministic dynamics and a non-degeneracy condition for the driving random force, we discuss the existence and uniqueness of a stationary measure and its exponential stability in the Kantorovich–Wasserstein metric. We next turn to the large deviations principle (LDP) and establish its validity for the occupation measures of the Markov processes in question. The proof is based on Kifer’s criterion for non-compact spaces, a result on large-time asymptotics for generalised Markov semigroup, and a coupling argument. These tools combined together constitute a new approach to LDP for infinite-dimensional processes without strong Feller property in a non-compact space. The results obtained can be applied to the two-dimensional Navier–Stokes system in a bounded domain and to the complex Ginzburg–Landau equation.

A global Stokes method of approximated particular solutions for unsteady two-dimensional Navier–Stokes system of equations

ABSTRACTThe unsteady two-dimensional Navier–Stokes system of equations, for viscous incompressible fluids are solved using a global method of approximated particular solutions (MAPS) in terms of a Stokes formulation, where the velocity and pressure fields are approximated from a linear superposition of particular solutions of a non-homogeneous Stokes system of equations, with a multiquadric (MQ) radial basis function (RBF) as non-homogeneous term. Steady-state solution of the flow problems considered in this work can be unstable at high Reynolds numbers (Re), corresponding to bifurcation of solutions that result in the appearance of new stable steady-state or periodic solutions. The main objective of this work is to present a global meshless numerical scheme able to predict these bifurcation points and concurrent new stable or periodic solutions. This is well known to be a very difficult task for any numerical scheme. An implicit first-order time-stepping scheme is used to approximate the transient term and the obtained nonlinear system of algebraic equations is solved by a Newton–Raphson method with variable step. Two steady-state and two transient problems are considered to validate the numerical scheme: the lid-driven cavity and backward-facing step (BFS) flows (steady-state problems) and the decaying Taylor–Green vortex and two-sided lid-driven cavity flows (transient problems). The first two problems are solved up to Re=10,000 and 2300, respectively. Results obtained are compared with corresponding benchmark numerical solutions, showing excellent agreement. Obtained numerical solutions for the decaying vortices at Re=100 shown excellent agreement with the corresponding analytical results. The transient problem of a rectangular two-sided lid-driven cavity flow is solved at Re=700. The influence of the cavity length, l, in determining the different structures of the flow pattern is studied for values of , showing that the scheme is able to reproduce the previously reported change in the flow pattern when l=2. Finally, the global Stokes MAPS are used to carry out nonlinear stability analyses of three steady-state problems: the sudden expansion, lid-driven cavity and BFS flows. Stable and unstable steady-state solutions at Re values greater than critical are predicted with the proposed numerical scheme. Our numerical results are consistent with previously stability analysis reported in the literature.

Attractors of equations of non-Newtonian fluid dynamics

This survey describes a version of the trajectory-attractor[4] method, which is applied to study the limit asymptotic behaviour of solutions of equations of non-Newtonian fluid dynamics. The trajectory-[4]attractor method emerged in papers of the Russian mathematicians Vishik and Chepyzhov and the American mathematician Sell under the condition that the corresponding trajectory spaces be invariant under the translation semigroup. The need for such an approach was caused by the fact that for many equations of mathematical physics for which the Cauchy initial-value problem has a global (weak) solution with respect to the time, the uniqueness of such a solution has either not been established or does not hold. In particular, this is the case for equations of fluid dynamics. At the same time, trajectory spaces invariant under the translation semigroup could not be constructed for many equations of non-Newtonian fluid dynamics. In this connection, a different approach to the construction of trajectory attractors for dissipative systems was proposed in papers of Zvyagin and Vorotnikov without using invariance of trajectory spaces under the translation semigroup and is based on the topological lemma of Shura– Bura. This paper presents examples of equations of non-Newtonian fluid dynamics (the Jeffreys system describing movement of the Earth's crust, the model of motion of weak aqueous solutions of polymers, a system with memory) for which the aforementioned construction is used to prove the existence of attractors in both the autonomous and the non-autonomous cases. At the beginning of the paper there is also a brief exposition of the results of Ladyzhenskaya on the existence of attractors of the two-dimensional Navier–Stokes system and the result of Vishik and Chepyzhov for the case of attractors of the three-dimensional Navier–Stokes system. Bibliography: 34 titles.

Recovering a Constant in the Two-Dimensional Navier–Stokes System with No Initial Condition

We deal with the $$2D$$ 2 D -Navier---Stokes system endowed with Cauchy boundary conditions, but with no initial condition. We assume that the right-hand side is of the form $$\beta f_0+f_1$$ β f 0...

A global meshless collocation particular solution method for solving the two-dimensional Navier–Stokes system of equations

The two-dimensional Navier–Stokes system of equations for incompressible fluids is solved by the method of approximate particular solutions (MAPS) in its global formulation. The fluid velocity and pressure fields are approximated by a linear superposition of particular solutions of a Stokes non-homogeneous system of equations with multiquadric (MQ) radial basis function as the source term. The nonlinear convective terms of the momentum equations are linearly approximated by using a guess value of the velocity field, and the resulting linear system of equations is solved by a simple direct iterative scheme (Picard iteration), with the velocity guess given by the solution at the previous iteration. Although the continuity equation is not explicitly imposed in the resulting formulation, the scheme is mass conservative because the particular solutions exactly satisfy the mass conservation equation. The proposed numerical scheme is validated by comparison of the obtained numerical results with the corresponding analytical solution of the Kovasznay flow problem at different Reynolds numbers, Re. From this analysis, it is observed that the MAPS results are stable and accurate for a wide range of shape parameter values. In addition, lid-driven cavity flow problems in rectangular and triangular domains up to Re=3200 and Re=1000, respectively, and the backward-facing step at Re=800 are solved, and the results obtained are compared with corresponding benchmark numerical solutions, showing excellent agreement.

Averaging of the planetary 3D geostrophic equations with oscillating external forces

In this article, we consider a non-autonomous three-dimensional planetary geostrophic model of the ocean with a singularly oscillating external force depending on a small parameter ϵ. We prove the existence of the uniform global attractor A ϵ . Furthermore, using the method of [11] in the case of the two-dimensional Navier–Stokes systems, we study the convergence of A ϵ as ϵ goes to zero.

A non-autonomous two-phase flow model with oscillating external force and its global attractor

In this article, we consider a non-autonomous diffuse interface model for an isothermal incompressible two-phase flow in a two-dimensional bounded domain. Assuming that the external force is singularly oscillating and depends on a small parameter ϵ , we prove the existence of the uniform global attractor A ϵ . Furthermore, using the method similar to that of Chepyzhov and Vishik (2007) [22] in the case of the two-dimensional Navier–Stokes systems, we study the convergence of A ϵ as ϵ goes to zero. Let us mention that the nonlinearity involved in the model considered in this article is slightly stronger than the one in the two-dimensional Navier–Stokes system studied in Chepyzhov and Vishik (2007) [22].

Analysis of semidiscretization of the compressible Navier–Stokes equations

The objective of this work is to investigate the time discretization of two-dimensional Navier–Stokes system with the slip boundary conditions. First, the existence of weak solutions for fixed time step △ t > 0 is presented and then the limit passage as △ t → 0 + is carried out. The proof is based on a new technique established for the steady Navier–Stokes equations by P.B. Mucha and M. Pokorný [P.B. Mucha, M. Pokorný, On a new approach to the issue of existence and regularity for the steady compressible Navier–Stokes equations, Nonlinearity 19 (8) (2006) 1747–1768] which enables to estimate the growth of L ∞ norm of the density when △ t goes to 0.

Averaging of a 3D Lagrangian averaged Navier-Stokes-$\alpha$ model with oscillating external forces

In this article, we consider a non-autonomous three-dimensional Lagrangian averaged Navier-Stokes-$\alpha$ equations with a singularly oscillating external force depending on a small parameter $\epsilon.$ We prove the existence of the uniform global attractor $A^\epsilon.$ Furthermore, using the method of [18] in the case of the two-dimensional Navier-Stokes systems, we study the convergence of $A^\epsilon$ as $\epsilon$ goes to zero.

Global regularity for generalized anisotropic Navier–Stokes equations

We consider the global Cauchy problem for the three-dimensional generalized anisotropic Navier–Stokes system with the viscosity term \documentclass[12pt]{minimal}\begin{document}$\Delta _h u-M^2u$\end{document}Δhu−M2u, where M is a Fourier multiplier whose symbol \documentclass[12pt]{minimal}\begin{document}$m:\mathbb {R}\rightarrow \mathbb {R}^+$\end{document}m:R→R+ is non-negative. The case \documentclass[12pt]{minimal}\begin{document}$m(\xi _3)=|\xi _3|$\end{document}m(ξ3)=|ξ3| is essentially the Navier–Stokes system, while the case m = 0 is the anisotropic Navier–Stokes system. We obtain the global regularity in the case that \documentclass[12pt]{minimal}\begin{document}$m(\xi _3)\ge {|\xi _3|^{\frac{3}{2}}}/{g(|\xi _3|)}$\end{document}m(ξ3)≥|ξ3|32/g(|ξ3|) for all sufficiently large \documentclass[12pt]{minimal}\begin{document}$|\xi _3|$\end{document}|ξ3| where \documentclass[12pt]{minimal}\begin{document}$g:=\mathbb {R}^+\rightarrow \mathbb {R}^+$\end{document}g:=R+→R+ is a nondecreasing function such that \documentclass[12pt]{minimal}\begin{document}$\int ^\infty _1\frac{ds}{ g^2(s) s\ln s}=+\infty$\end{document}∫1∞dsg2(s)slns=+∞. Then, using the method of descent, we can obtain the classical global regularity result of the two-dimensional Navier–Stokes system.

Non-autonomous planetary 3D geostrophic equations with oscillating external force and its global attractor

In this article, we consider a non-autonomous three-dimensional planetary geostrophic model of the ocean with a singulary oscillating external force depending on a small parameter ϵ . We prove the existence of the uniform global attractor A ϵ . Furthermore, using the method of Chepyzhov and Vishik (2007) [3] in the case of the two-dimensional Navier–Stokes systems, we study the convergence of A ϵ as ϵ goes to zero.

Navier-Stokes System on the Unit Square with no Slip Boundary Condition

We consider the two-dimensional Navier-Stokes system on the unit square with no-slip boundary condition. The nonlinear evolution equation for the stream function is studied. Under some hypothesis, we show that the decay of Fourier modes of solutions is power-like.