Decay Results for Weak Solutions of the Navier-Stokes Equations on R<sup>n</sup>

Wiegner, Michael

doi:10.1112/jlms/s2-35.2.303

Cited by 303 publications

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“…In this section, we adapt the results in [13] to the particular case where N = 3 and L ≡ Λ + 1 2 acts on the space of divergence free vector fields L 2 (m) defined in (24). Remark that div(Λf) = L div(f), so that Λ preserves the divergence free condition.…”

Section: A Spectrum Of the Operator λmentioning

confidence: 99%

“…As no external force is applied, it is intuitively clear that all finite-energy solutions of (2) should converge, as time goes to infinity, to the rest state u ≡ 0, p ≡ const. As a matter of fact, if u(x, t) is any global weak solution in L 2 (R 3 ) satisfying the energy inequality, it is known that u(·, t) L 2 → 0 as t → ∞ (see Wiegner [24]). Moreover, if…”

Section: Introductionmentioning

confidence: 99%

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Long-time asymptotics of the Navier-Stokes and vorticity equations on ℝ³

Gallay

Wayne

2002

Philosophical Transactions of the Royal Society of London. Seri

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We use the vorticity formulation to study the long-time behavior of solutions to the Navier-Stokes equation on R 3 . We assume that the initial vorticity is small and decays algebraically at infinity. After introducing self-similar variables, we compute the long-time asymptotics of the rescaled vorticity equation up to second order. Each term in the asymptotics is a self-similar divergence-free vector field with Gaussian decay at infinity, and the coefficients in the expansion can be determined by solving a finite system of ordinary differential equations. As a consequence of our results, we are able to characterize the set of solutions for which the velocity field satisfies u(·, t) L 2 = o(t −5/4 ) as t → +∞. In particular, we show that these solutions lie on a smooth invariant submanifold of codimension 11 in our function space.

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Section: A Spectrum Of the Operator λmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Long-time asymptotics of the Navier-Stokes and vorticity equations on ℝ³

Gallay

Wayne

2002

Philosophical Transactions of the Royal Society of London. Seri

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“…Note in particular that kuðtÞk 2 ¼ Oðt À1 Þ as t ! y; so the result of [20] suggests that we would need some symmetry condition to find solutions which decay more rapidly than stated in Theorem 1.1. In fact, [16] proves We know (see [2]…”

Section: à1=2àð1à1=rþmentioning

confidence: 99%

On Two-Dimensional Navier-Stokes Flows with Rotational Symmetries

He¹,

Miyakawa

2006

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Abstract. Navier-Stokes flows are found on R 2 that decay in time more rapidly than observed in general. The decay rate is determined in accordance with the order of symmetry with respect to the action of dihedral groups of orthogonal transformations. Contrary to the previous work [13], the basic existence result is proved with no restriction on the size of initial data. Our result extends that of [3] under di¤erent assumptions on the initial data. Unlike [3], the proofs are all carried out without using estimates for the associated vorticity transport equations. Key Words and Phrases. Navier-Stokes system, Cauchy problem, Group symmetry, Asymptotic behavior, Weighted estimate.2000 Mathematics Subject Classification Numbers. 35Q30, 76D05. Statement of the resultsThis paper studies the Cauchy problem for the Navier-Stokes system in R 2 :Here, x ¼ ðx 1 ; x 2 Þ A R 2 , t b 0, q t ¼ q=qt; u ¼ uðx; tÞ ¼ ðu k ðx; tÞÞ 2 k¼1 and p ¼ pðx; tÞ denote unknown velocity and pressure, respectively; and a ¼ aðxÞ is a given initial velocity. The kinematic viscosity is normalized. The standard notation is used for di¤erential operators and function spaces and the summation convention is employed for repeated indices.We are interested in finding a (unique) solution u to (1.1) that satisfies kuðtÞk r a cð1 þ tÞ Àðmþ1Þ=2Àð1À1=rÞ ð1 a r a yÞ ð1:2Þ for some m A N U f0g. (Here and in what follows k Á k r denotes the L r -norm.) In view of the structure of solutions to parabolic equations, it is reasonable to guess that such solutions would decay su‰ciently rapidly as t þ jxj ! y; and our purpose is to specify a class of initial data that provide us with solutions satisfying (1.2).

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“…Esta questão somente foi resolvida (positivamente) 50 anos mais tarde por Kato [6] e subsequentemente também por outros autores [5,9,17]. Vários desenvolvimentos e extensões importantes de (3) vem sendo estabelecidos (ver e.g.…”

Section: Introductionunclassified

Taxas de Decaimento para Fluidos Magneto-Micropolares

Perusato¹,

Guterres²

2018

Proceeding Series of the Brazilian Society of Computational and Applied Mathematics

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A partir daí foi possível obter algumas interessantes consequências, como por exemplo:Palavras-chave. Decaimento assintótico, MHD Micropolar, Estimativas L p , EstimativasḢ s . IntroduçãoEm seu célebre artigo [8], de 1934, Leray construiu soluções (fracas) globais de energia finita) para as equações de Navier-Stokes em R 3 . Tais soluções (fracas) de Leray podem ser construídas de maneira análoga para os seguinte sistema MHD Micropolar:obtendo assim, as mesmas importantes propriedades existentes para a equação de Navier-Stokes. Como por exemplo, a existência de t * > 0 suficientemente grande, tal queNo sistema (1)

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Decay Results for Weak Solutions of the Navier-Stokes Equations on Rⁿ

Cited by 303 publications

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Long-time asymptotics of the Navier-Stokes and vorticity equations on ℝ³

Long-time asymptotics of the Navier-Stokes and vorticity equations on ℝ³

On Two-Dimensional Navier-Stokes Flows with Rotational Symmetries

Taxas de Decaimento para Fluidos Magneto-Micropolares

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Decay Results for Weak Solutions of the Navier-Stokes Equations on Rn

Cited by 303 publications

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Long-time asymptotics of the Navier-Stokes and vorticity equations on ℝ3

Long-time asymptotics of the Navier-Stokes and vorticity equations on ℝ3

On Two-Dimensional Navier-Stokes Flows with Rotational Symmetries

Taxas de Decaimento para Fluidos Magneto-Micropolares

Contact Info

Product

Resources

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Decay Results for Weak Solutions of the Navier-Stokes Equations on Rⁿ

Long-time asymptotics of the Navier-Stokes and vorticity equations on ℝ³

Long-time asymptotics of the Navier-Stokes and vorticity equations on ℝ³