Large Time Behavior of the Navier-Stokes Flow

Brandolese, Lorenzo; Schonbek, María E.

doi:10.1007/978-3-319-13344-7_11

Cited by 13 publications

(18 citation statements)

References 155 publications

(207 reference statements)

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“…This constrasts with the case

ε = 0

of the Navier–Stokes equations: indeed, solutions of the Navier–Stokes equations are known to decay as

{∥ u false(t false) ∥}_{2} \sim t^{- (n + 2) / 4}

as soon as

u_{0}

is well localized, see , and sometimes even at faster rates (for example, under appropriate symmetries). See contribution for an up‐to‐date review of decay issues for the Navier–Stokes flows. Remark Theorem corrects one of the results of the paper .…”

Section: Statement Of the Main Resultsmentioning

confidence: 99%

On a non-solenoidal approximation to the incompressible Navier-Stokes equations

Brandolese

2017

J. London Math. Soc.

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We establish an asymptotic profile that sharply describes the behavior as t→∞ for solutions to a non‐solenoidal approximation of the incompressible Navier–Stokes equations introduced by Temam. The solutions of Temam's model are known to converge to the corresponding solutions of the classical Navier–Stokes, for example, in L loc 3false(double-struckR+×double-struckR3false), provided ε→0, where ε>0 is the physical parameter related to the artificial compressibility term. However, we show that such model is no longer a good approximation of Navier–Stokes for large times: indeed, its solutions can decay much slower as t→∞ than the corresponding solutions of Navier–Stokes.

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“…This constrasts with the case

ε = 0

of the Navier–Stokes equations: indeed, solutions of the Navier–Stokes equations are known to decay as

{∥ u false(t false) ∥}_{2} \sim t^{- (n + 2) / 4}

as soon as

u_{0}

Section: Statement Of the Main Resultsmentioning

confidence: 99%

On a non-solenoidal approximation to the incompressible Navier-Stokes equations

Brandolese

2017

J. London Math. Soc.

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“…(see [7],THEOREM A), with t * = 0 if n = 2. For more on solution properties, see e.g., [1][2][3][4][5][6][8][9][10][11].…”

Section: Introductionmentioning

confidence: 99%

“…as t → ∞, with the generic limitation α ≤ (n + 2)/4, see [8,22,23]. (For the exceptional case of faster decaying solutions, see [8,22,24,25].)…”

Section: Introductionmentioning

confidence: 99%

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A Note on Stokes Approximations to Leray Solutions of the Incompressible Navier–Stokes Equations in ℝn

Rigelo¹,

Zingano

2021

Fluids

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In the early 1980s it was well established that Leray solutions of the unforced Navier–Stokes equations in Rn decay in energy norm for large t. With the works of T. Miyakawa, M. Schonbek and others it is now known that the energy decay rate cannot in general be any faster than t−(n+2)/4 and is typically much slower. In contrast, we show in this note that, given an arbitrary Leray solution u(·,t), the difference of any two Stokes approximations to the Navier–Stokes flow u(·,t) will always decay at least as fast as t−(n+2)/4, no matter how slow the decay of ∥u(·,t)∥L2(Rn) might be.

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“…For the incompressible Navier-Stokes equation, the existence of global weak solutions to the initial value problem has been established by Leray [58] and Hopf [48]. For comprehensive results regarding the Navier-Stokes equation, we refer the interested readers to [20,35,57,59,75,76,78,80], the references therein, and [1, 7, 21, 36-38, 40-42, 52, 53, 71] for results in two and general dimensions.…”

Section: Introductionmentioning

confidence: 99%

Global existence of free-energy solutions to the 2D Patlak--Keller--Segel--Navier--Stokes system with critical and subcritical mass

Lai¹,

Wei²,

Zhou³

2021

Preprint

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We consider a coupled Patlak-Keller-Segel-Navier-Stokes system in R 2 that describes the collective motion of cells and fluid flow, where the cells are attracted by a chemical substance and transported by ambient fluid velocity, and the fluid flow is forced by the friction induced by the cells. The main result of the paper is to show the global existence of free-energy solutions to the 2D Patlak-Keller-Segel-Navier-Stokes system with critical and subcritical mass.

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Large Time Behavior of the Navier-Stokes Flow

Cited by 13 publications

References 155 publications

On a non-solenoidal approximation to the incompressible Navier-Stokes equations

On a non-solenoidal approximation to the incompressible Navier-Stokes equations

A Note on Stokes Approximations to Leray Solutions of the Incompressible Navier–Stokes Equations in ℝn

Global existence of free-energy solutions to the 2D Patlak--Keller--Segel--Navier--Stokes system with critical and subcritical mass

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