Comparison of decay of solutions to two compressible approximations to Navier-Stokes equations

Abstract

In this article, we use the decay character of initial data to compare the energy decay rates of solutions to different compressible approximations to the NavierStokes equations. We show that the system having a nonlinear damping term has slower decay than its counterpart with an advection-like term. Moreover, me characterize a set of initial data for which the decay of the first system is driven by the difference between the full solution and the solution to the linear part, while for the second system the linear part provides the decay rate.

Decay characterization of solutions to the viscous Camassa–Holm equations

“…In [18], Niche and Schonbek refined and extended Bjorland and Schonbek's work, more precisely, they defined the decay character of Λ s v 0 for v 0 ∈ H s (R n ), Λ = (−∆) 1/2 , s 0, and established its relation with the decay character of v 0 . Applications to upper and lower bounds for decay rates of solutions to the dissipative quasigeostrophic equations and to a compressible approximation to the 3D Navier-Stokes equations were also given there, see also the recent work [19]. Very recently in [7], Brandolese revisited and improved the theory of decay characters by Bjorland, Niche, and Schonbek, by taking advantage of the insight provided by the Littlewood-Paley analysis and the use of Besov spaces.…”

The Temporal decay rate of solutions to 2D dissipative quasi-geostrophic flows

“…We recall that there are some papers studied the decay estimate of solutions to dissipative equations by using the idea of decay character and Fourier splitting method (cf. [1,21,22,27]) and the reference cited therein). It is worth pointing out that, on the basis of Fourier splitting method and decay character r * , Niche and Schonbek [19], Ferreira, Niche and Planas [14] obtained some new results on the dissipative quasi-geostrophic equations and modified quasi-geostrophic equations.…”