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

Abstract

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.

Existence and stability of steady-state solutions with finite energy for the Navier–Stokes equation in the whole space

“…A particularly useful technique for estimating energy decay is the Fourier Splitting Method which was used in [31] to establish energy decay for initial data u 0 ∈ L 2 ∩ L 1 and later for initial data u 0 ∈ L 2 in [25]. Other works in this area include [2], [17], [18], [22], [23], [24], [27], [28], [29], [30], [35], and [36].…”

“…We give a formal proof here which can be made precise by considering an approximating sequence, see [25] for details. To see the first inequality multiply the PDE (4.1) by e 2ν△(t+s)φ * φ * w and integrate from s to t. The assumptions are enough to ensure all integrals are finite and this multiplication makes sense.…”

“…Proof. Following [25] we bound first the low frequencies using (4.6) and then the high frequencies using (4.7) and the Fourier Splitting Method. To show the low frequencies tend to zero we start by estimating the integrals on the RHS of (4.6):…”

Long time behavior for a dissipative shallow water model

“…The proof is based on ideas of [15], [24]: We decompose the L 2 -norm of the Fourier transform of the weak solution u as follows , the fundamental solution of the heat equation at t = 1. We estimate separately the low frequencies and the high energy frequencies terms in (3.13).…”

“…Under suitable conditions on the forcing term and of the initial datum, we show that the energy norm of weak solution has non-uniform decay . A weak solution which satisfies a generalized energy inequality is constructed following [22,24,15]. Then using the Fourier splitting method [31,32,37] non uniform L 2 decay is obtained.…”