Dynamics of Partial Differential Equations

Volume 17 (2020)

Number 2

Almost sure existence of global weak solutions to the Boussinesq equations

Pages: 165 – 183

DOI: https://dx.doi.org/10.4310/DPDE.2020.v17.n2.a4

Authors

Weinan Wang (Department of Mathematics, University of Southern California, Los Angeles, Cal., U.S.A.)

Haitian Yue (Department of Mathematics, University of Southern California, Los Angeles, Cal., U.S.A.)

Abstract

In this paper, we show that after a suitable randomization of the initial data in the negative order Sobolev spaces $H^{-\alpha}$ with $0 \lt \alpha \lt 1 / 2$, there exist almost sure global weak solutions to the Boussinesq equations in $\mathbb{R}^d$ and $\mathbb{T}^d$, when $d = {2, 3}$. Furthermore, we prove that the global weak solutions are unique in dimension two.

Keywords

Boussinesq equations, almost sure well-posedness, random data, negative order Sobolev spaces

2010 Mathematics Subject Classification

35Q30, 76D05

Received 4 September 2019

Published 24 February 2020