Inertial manifolds for the hyperviscous Navier–Stokes equations

Gal, Ciprian G.; Guo, Yanqiu

doi:10.1016/j.jde.2018.06.011

Cited by 29 publications

(45 citation statements)

References 44 publications

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“…Finally, owing to another result of number theory (see Proposition 5.10), we can verify that the SAC (5.60) holds (see Theorem 5.11 below). Based on the above analysis, we can construct an N -dimensional IM for NSEs To our knowledge, the best known result for the 3D hyperviscous NSEs (1.1) was given in Gal and Guo [15] with the assumption θ 3 2 .…”

Section: Resultsmentioning

confidence: 99%

“…The idea was employed on a large class of dissipative equations [14] (see also [36,38]). A number of dynamical systems possess inertial manifolds, such as certain nonlinear reaction-diffusion equations in two [9,10,14] and three [27] dimensions, the Kuramoto-Sivashinsky equation [12], the Cahn-Hilliard equation [10,22], modified Navier-Stokes equations [15,18,21,25] and the von Kármán plate equations [7]. One may refer to [9,38] for the study of inertial manifolds for many dissipative PDEs.…”

Section: Introductionmentioning

confidence: 99%

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Inertial manifolds for the incompressible Navier-Stokes equations

Li,

Sun

2019

Preprint

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In this article, we devote to the existence of an N -dimensional inertial manifold for the incompressible Navier-Stokes equations in T d (d = 2, 3).Our results can be summarized as two aspects: Firstly, we construct an N -dimensional inertial manifold for the Navier-Stokes equations in T 2 ; Secondly, we extend slightly the spatial averaging method to the abstract case: ∂tuoperator with compact inverse and F is Lipschitz from a Hilbert space H to H), and then verify the existence of an N -dimensional inertial manifold for the hyperviscous Navier-Stokes equation with the hyperviscous index 5/4 in T 3 .

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Section: Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Inertial manifolds for the incompressible Navier-Stokes equations

Li,

Sun

2019

Preprint

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“…Dynamic changes in hydrostatic pressure and average flow rate in an elementary section of an anti-vibration segment of a horizontal pipeline were described by the Navier-Stokes fluid motion equation [11]. Many works [12][13][14][15][16][17][18][19][20] are devoted to the solution of various problems by this equation and the analysis of this equation itself. So, we take the Navier-Stokes equation in divergent form:…”

Section: Methodsmentioning

confidence: 99%

Simulation of fluid outflow from a channel with complex geometry

et al. 2020

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Topic of the paper is the simulation of fluid outflow from a channel with complex geometry. Subject of the study is the outflow of fluid from a channel with an arbitrary section, in particular, consisting of a parabolic inlet, cylindrical middle and hyperbolic outlet sections. The paper considers the integral form of determining the pressure and the average flow rate of the fluid in a channel with an arbitrary cross section. From the obtained expression for the pressure and from the equation of conservation of mass, given in integral form, it is necessary to determine the pressure and the average flow rate in successively located channels with a parabolic, cylindrical and hyperbolic section. Research methods are based on: Newton’s rheological law; the continuity equation and the Navier-Stokes equation, which are the basic equations of fluid motion; the method of mathematical modeling and the analytical method for their solution. The paper contains integral expressions for the hydrodynamic characteristics of a fluid flow in a channel with an arbitrary section. The pressure and average flow rate in the channel of successively located parabolic, cylindrical, hyperbolic segments are determined. Analytical expressions are obtained for the pressure and average flow rate of the liquid in a channel with an arbitrary cross section. As an example, the pressures and the average flow rate in the channel from the parabolic inlet, cylindrical middle and hyperbolic outlet sections are determined. In perspective, by substituting in them an arbitrary arithmetic expression of the channel cross-section, it is possible to determine the hydrodynamic characteristics of the flow with any geometry of the flow channel.

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“…in the case where A is a Laplacian in a bounded domain, it is satisfied in 1D case only) and is known to be sharp in the class of abstract semilinear parabolic equations (see [7,29,35,42] for more details), it can be relaxed for some concrete classes of PDEs. For instance, for scalar 3D reaction-diffusion equations (using the so-called spatial averaging principle, see [27]), for 1D reaction-diffusion-advection systems (using the proper integral transforms, see [23,24]), for 3D Cahn-Hilliard equations and various modifications of 3D Navier-Stokes equations (using various modifications of spatial-averaging, see [11,20,22,26]), for 3D complex Ginzburg-Landau equation (using the so-called spatio-temporal averaging, see [21]), etc. Note also that the global Lipschitz continuity assumption for the non-linearity F is not an essential extra restriction since usually one proves the well-posedness and dissipativity of the PDE under consideration before constructing the IM.…”

Section: Introductionmentioning

confidence: 99%

Smooth extensions for inertial manifolds of semilinear parabolic equations

Kostianko,

Zelik

2021

Preprint

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The paper is devoted to a comprehensive study of smoothness of inertial manifolds for abstract semilinear parabolic problems. It is well known that in general we cannot expect more than C 1,ε -regularity for such manifolds (for some positive, but small ε). Nevertheless, as shown in the paper, under the natural assumptions, the obstacles to the existence of a C n -smooth inertial manifold (where n ∈ N is any given number) can be removed by increasing the dimension and by modifying properly the nonlinearity outside of the global attractor (or even outside the C 1,ε -smooth IM of a minimal dimension). The proof is strongly based on the Whitney extension theorem. Contents 1. Introduction 1 2. Preliminaries I: Taylor expansions and Whitney Extension Theorem 6 3. Preliminaries II: Spectral gaps and the construction of an inertial manifold 10 4. Main result 18 5. Verifying the compatibility conditions 25 6. Examples and concluding remarks 30 References 34

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Inertial manifolds for the hyperviscous Navier–Stokes equations

Cited by 29 publications

References 44 publications

Inertial manifolds for the incompressible Navier-Stokes equations

Inertial manifolds for the incompressible Navier-Stokes equations

Simulation of fluid outflow from a channel with complex geometry

Smooth extensions for inertial manifolds of semilinear parabolic equations

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