Dynamics of Partial Differential Equations

Volume 15 (2018)

Number 4

Optimal rate of convergence in stratified Boussinesq system

Pages: 235 – 263

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


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

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


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.


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

2010 Mathematics Subject Classification

35B65, 35Q35, 76D05

Received 27 September 2016

Published 5 December 2018