On the Characterization of the Navier–Stokes Flows with the Power-Like Energy Decay

Skalák, Zdenĕk

doi:10.1007/s00021-014-0164-7

Cited by 6 publications

(8 citation statements)

References 11 publications

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“…It completes well known Wiegner's Theorem by giving several necessary and sufficient conditions for the validity of bounds from below for the decay of the energy. It improves [2,Theorem 6.5], and completes an earlier study by Skalák [9] where only the upper bounds were discussed.…”

Section: Application To Nonlinear Dissipative Systems: the Navier-stosupporting

confidence: 76%

“…Therefore, under the same restrictions on σ and the conditions of item (iii), it easily follows that u = e t∆ u 0 + w satisfies the estimates as in (5.1). The proof of the implication (iv) ⇒ (iii) relies on the so-called inverse Wiegner's theorem recently established by Z. Skalák [9]: his results asserts (among other things) that if u is a weak solution of the Navier-Stokes equation as above, satisfying u(t) 2 (1 + t) −σ (with 0 ≤ σ ≤ 5/4) then e t∆ u 0 2…”

Section: Application To Nonlinear Dissipative Systems: the Navier-stomentioning

confidence: 98%

“…The proof of the implication (iv) ⇒ (iii) relies on the so-called inverse Wiegner's theorem recently established by Z. Skalák [9]: his results asserts (among other things) that if u is a weak solution of the Navier-Stokes equation as above, satisfying u(t) 2…”

Section: Application To Nonlinear Dissipative Systems: the Navier-sto...mentioning

confidence: 98%

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Characterization of Solutions to Dissipative Systems with Sharp Algebraic Decay

Brandolese¹

2016

SIAM J. Math. Anal.

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We characterize the set of functions u0 ∈ L 2 (R n ) such that the solution of the problem ut = Lu in R n × (0, ∞) starting from u0 satisfy upper and lower bounds of the formHere L is in a large class of linear pseudo-differential operator with homogeneous symbol (including the Laplacian, the fractional Laplacian, etc.). Applications to nonlinear PDEs will be discussed: in particular our characterization provides necessary and sufficient conditions on u0 for a solution of the Navier-Stokes system to satisfy sharp upper-lower decay estimates as above.In doing so, we will revisit and improve the theory of decay characters by C. Bjorland, C. Niche, and M.E. Schonbek, by getting advantage of the insight provided by the Littlewood-Paley analysis and the use of Besov spaces.

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Section: Application To Nonlinear Dissipative Systems: the Navier-stosupporting

confidence: 76%

Section: Application To Nonlinear Dissipative Systems: the Navier-stomentioning

confidence: 98%

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Characterization of Solutions to Dissipative Systems with Sharp Algebraic Decay

Brandolese¹

2016

SIAM J. Math. Anal.

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“…The derivation in [11,26] is actually much simpler than Wiegner's original proof. The inverse Wiegner's theorem above was first obtained by Z. Skalák in dimension n = 3 (see [20], Theorem 3.1) using a very elaborated argument. The proof of Theorem 7.1 given here is much simpler and is based on monotonicity ideas if n = 2.…”

Section: Proof Of Lemma 22mentioning

confidence: 98%

Upper and Lower H<sup>m</sup> Estimates for Solutions to Dissipative Systems

Guterres,

Niche,

Perusato

et al. 2022

Preprint

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In this work we introduce a novel approach to generate lower and upper L2 estimates for solution derivatives of arbitrary order to a general class of dissipative systems in the case that such estimates are available for the solutions themselves. Our method also works in reverse order: from the L2 estimates of solution derivatives of some (arbitrary) order we can derive lower and upper L2 estimates for the solutions and then to their derivatives of any order. This procedure is based on very simple monotonicity properties combined with standard energy estimates in physical space, following previous ideas of Kreiss, Hagstrom, Lorenz and Zingano. For simplicity, it is applied here in the context of algebraic rates, but the method can be used in other contexts as well (exponential, logarithm, and so forth).

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“…A natural question is whether the converse property holds, that is, if the knowledge that ‖𝒖(⋅, 𝑡)‖ 𝐿 2 (ℝ 𝑛 ) = 𝑂(1 + 𝑡) −𝛼 for some 0 < 𝛼 ⩽ (𝑛 + 2)∕4 will entail that ‖𝒗(⋅, 𝑡)‖ 𝐿 2 (ℝ 𝑛 ) = 𝑂(1 + 𝑡) −𝛼 as well. A positive answer was given by Skalák's "inverse Wiegner's theorem" [15], using an elaborated argument.…”

Section: Introductionmentioning

confidence: 99%

On the topological size of the class of Leray solutions with algebraic decay

Brandolese,

Perusato,

Zingano

2023

Bulletin of London Math Soc

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In 1987, Michael Wiegner in his seminal paper (J. Lond. Math. Soc. (2), 35 (1987) 303–313) provided an important result regarding the energy decay of Leray solutions to the incompressible Navier–Stokes in : if the associated Stokes flows had their norms bounded by for some , then the same would be true of . The converse also holds, as shown by Skalák (J. Math. Fluid Mech. 16 (2014) 431–446) and by our analysis below, which uses a more straightforward argument. As an application of these results, we discuss the genericity problem of algebraic decay estimates for Leray solutions of the unforced Navier–Stokes equations. In particular, we prove that Leray solutions with algebraic decay generically satisfy two‐sided bounds of the form .

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On the Characterization of the Navier–Stokes Flows with the Power-Like Energy Decay

Cited by 6 publications

References 11 publications

Characterization of Solutions to Dissipative Systems with Sharp Algebraic Decay

Characterization of Solutions to Dissipative Systems with Sharp Algebraic Decay

Upper and Lower H<sup>m</sup> Estimates for Solutions to Dissipative Systems

On the topological size of the class of Leray solutions with algebraic decay

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