Dynamics of Partial Differential Equations

Volume 15 (2018)

Number 4

Optimal rate of convergence in stratified Boussinesq system

Pages: 235 – 263

DOI: http://dx.doi.org/10.4310/DPDE.2018.v15.n4.a1

Authors

H. Meddour (Département de Mathématiques, Université de Batna 2, Batna, Algeria)

M. Zerguine (Département de Mathématiques, Université de Batna 2, Batna, Algeria)

Abstract

We study the vortex patch problem for $2d$-stratified Navier–Stokes system. We aim at extending several results obtained in [1, 12, 20] for standard Euler and Navier–Stokes systems. We shall deal with smooth initial patches and establish global strong estimates uniformly with respect to the viscosity in the spirit of [28, 39]. This allows to prove the convergence of the viscous solutions towards the inviscid one. In the setting of a Rankine vortex, we show that the rate of convergence for the vortices is optimal in $L^p$ space and is given by $(\mu t)^{\frac{1}{2p}}$. This generalizes the result of [1] obtained for $L^2$ space.

Keywords

$2d$-stratified Boussinesq system, regular vortex patches, rate of convergence, global well-posedness, optimal rate

2010 Mathematics Subject Classification

35B65, 35Q35, 76D05

Full Text (PDF format)

Received 27 September 2016