Decay and stability in L p for strong solutions of the Cauchyproblem for the Navier-Stokes equations

dedicated to professor kyu ya masuda on the occasion of his sixtieth birthdayWe consider the Navier Stokes system with slowly decaying external forcesWe show that the energy norm of a weak solution has non-uniform decay,under suitable conditions on the data f and a which make the energy of solution bounded in time. Also, we show the exact rate of the decay (uniform decay) of the energy,for external forces with a given explicit decay rate.

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“…On the other hand, if we choose a more rapid decay for f the decay for the solution is determined by the nonlinear term (cf. [19,20,27]). …”

Section: Then the Decay Results Readsmentioning

confidence: 97%

Energy Decay for a Weak Solution of the Navier–Stokes Equation with Slowly Varying External Forces

Ogawa

Rajopadhye

Schonbek

1997

Journal of Functional Analysis

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“…We are going to explore necessary and sufficient conditions so that the above limit exists but is not zero. For more references, see Oliver and Titi [26], Maria E. Schonbek [27,28], Michael Wiegner [39,40].…”

Section: Previous Resultsmentioning

confidence: 98%

“…Suppose also that the global solutions of problems (1)-(2) satisfy u ∈ L q (R + , L p (R n )). See [2,3,39] and [43] for the same assumption.…”

Section: Hypothesesmentioning

confidence: 94%

New results of general n-dimensional incompressible Navier–Stokes equations

Zhang

2008

Journal of Differential Equations

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Let u = u(x, t, u 0 ) represent the global strong/weak solutions of the Cauchy problems for the general n-dimensional incompressible Navier-Stokes equationswhere the spatial dimension n 2, 0 ε 1 is a constant and β = (β 1 , β 2 , . . . , β n ) T ∈ R n is a constant vector. Note that if ε = 0 and β = 0, then the problem reduces to the traditional Navier-Stokes equations. Let the scalar functions φ ij ∈ C 2 (R n ) ∩ L 1 (R n ), ∂φ ij ∂x j

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“…Recent papers that consider closely related topics are [Se,VS,W2]. Secchi [Se] utilized energy methods to prove his instability result for the 3-dimensional case.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Many mathematicians have studied the asymptotic behavior of solutions to (1) and have made significant progress in this area [H,K,KM,Wl,W2,W3,S,V,Z3]. They verified that solutions to the n(> 2)-dimensional problem have the same rates of decay as those of the heat equations, provided the initial velocities are in the same class.…”

mentioning

confidence: 99%

Long time uniform stability for solutions of $n$-dimensional Navier-Stokes equations

Zhang¹

1999

Quart. Appl. Math.

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Decay and stability in L p for strong solutions of the Cauchyproblem for the Navier-Stokes equations

Cited by 22 publications

References 6 publications

Energy Decay for a Weak Solution of the Navier–Stokes Equation with Slowly Varying External Forces

Energy Decay for a Weak Solution of the Navier–Stokes Equation with Slowly Varying External Forces

New results of general n-dimensional incompressible Navier–Stokes equations

Long time uniform stability for solutions of $n$-dimensional Navier-Stokes equations

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