Global existence of solutions for hyperbolic Navier–Stokes equations in three space dimensions

Abdelhedi, Bouthaina

doi:10.3233/asy-181503

Cited by 11 publications

(8 citation statements)

References 14 publications

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“…This completes the proof of part (1) of Theorem 1.1. Part (2). We also deduce from (2.25) and (2.27) that aK e…”

Section: ) and Definition A3 Below)mentioning

confidence: 58%

“…It is therefore natural to consider a hyperbolic Navier-Stokes system by adding the term η∂ 2 t u to the classical Navier-Stokes system (N S), where η = 1 c 2 is a small parameter. The hyperbolic Navier-Stokes system (1.5) has been extensively studied in the literature and one may check [2,5,13,18,22,23,24] and the references therein. The justification of this system by using the Cattaneo's law follows by considering that the stress tensor is given by the solution of the retarded equation (1.3) τ (t + η) = −P Id + ν ∇U + (∇U ) ⊤ (t).…”

Section: Introductionmentioning

confidence: 99%

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Global hydrostatic approximation of hyperbolic Navier-Stokes system with small Gevrey class data

Paicu¹,

Zhang²

2021

Preprint

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We investigate the hydrostatic approximation of a hyperbolic version of Navier-Stokes equations, which is obtained by using Cattaneo type law instead of Fourier law, evolving in a thin strip R × (0, ε). The formal limit of these equations is a hyperbolic Prandtl type equation. We first prove the global existence of solutions to these equations under a uniform smallness assumption on the data in Gevrey 2 class. Then we justify the limit globally-in-time from the anisotropic hyperbolic Navier-Stokes system to the hyperbolic Prandtl system with such Gevrey 2 class data. Compared with [27] for the hydrostatic approximation of 2-D classical Navier-Stokes system with analytic data, here the initial data belong to the Gevrey 2 class, which is very sophisticated even for the well-posedness of the classical Prandtl system (see [14,32]), furthermore, the estimate of the pressure term in the hyperbolic Prandtl system arises additional difficulties.

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“…This completes the proof of part (1) of Theorem 1.1. Part (2). We also deduce from (2.25) and (2.27) that aK e…”

Section: ) and Definition A3 Below)mentioning

confidence: 58%

Section: Introductionmentioning

confidence: 99%

Global hydrostatic approximation of hyperbolic Navier-Stokes system with small Gevrey class data

Paicu¹,

Zhang²

2021

Preprint

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“…The classical Navier-Stokes equation has infinite propagation speed which is non-physical, and in order to avoid the non-physical feature Cattaneo [6] proposed the Cattaneo law instead of the classical Fourier law. Up to now, there have been extensive works on the hyperbolic Navier-Stokes equation; see for instance [2,5,8,34,37,38] and the references therein. Similar to the parabolic Naver-Stokes equation, a boundary layer will appear when investigating the hyperbolic counterpart in a bounded domain complemented with the no-slip boundary conditions, and its governing equation can be derived as below (see Appendix A below), just following the classical Prandtl's ansatz,…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Gevrey well-posedness of the hyperbolic Prandtl equations

Li¹,

Xu²

2021

Preprint

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We study the 2D and 3D Prandtl equations of degenerate hyperbolic type, and establish without any structural assumption the Gevrey well-posedness with Gevrey index ≤ 2. Compared with the classical parabolic Prandtl equations, the loss of the derivatives, caused by the hyperbolic feature coupled with the degeneracy, can't be overcame by virtue of the classical cancellation mechanism that developed for the parabolic counterpart. Inspired by the abstract Cauchy-Kowalewski theorem and by virtue of the hyperbolic feature, we give in this text a straightforward proof, basing on an elementary L 2 energy estimate. In particular our argument does not involve the cancellation mechanism used efficiently for the classical Prandtl equations.

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“…There have been many results on the hyperbolic Navier-Stokes equations, cf. [2,4,8,40,44,45]. By performing proper change of scales as in [42] we have the following hyperbolic version of the anisotropic Navier-Stokes equations (1.1):…”

Section: Introduction and The Main Resultsmentioning

confidence: 99%

3D Hyperbolic Navier-Stokes Equations in a Thin Strip: Global Well-Posedness and Hydrostatic Limit in Gevrey Space

Li¹,

Yang²

2022

CMAA

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We consider a hyperbolic version of three-dimensional anisotropic Navier-Stokes equations in a thin strip and its hydrostatic limit that is a hyperbolic Prandtl type equations. We prove the global-in-time existence and uniqueness for the two systems and the hydrostatic limit when the initial data belong to the Gevrey function space with index 2. The proof is based on a direct energy method by observing the damping effect in the systems.

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Global existence of solutions for hyperbolic Navier–Stokes equations in three space dimensions

Cited by 11 publications

References 14 publications

Global hydrostatic approximation of hyperbolic Navier-Stokes system with small Gevrey class data

Global hydrostatic approximation of hyperbolic Navier-Stokes system with small Gevrey class data

Gevrey well-posedness of the hyperbolic Prandtl equations

3D Hyperbolic Navier-Stokes Equations in a Thin Strip: Global Well-Posedness and Hydrostatic Limit in Gevrey Space

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