L^p theory for Boussinesq system with Dirichlet boundary conditions

ABSTRACT

We consider the stationary Boussinesq system with non-homogeneous Dirichlet boundary conditions in a bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^3$ of class ${\mathcal{C}}^{1,1}$ with a possibly disconnected boundary. We prove the existence of weak solutions in $W^{1,p}\left(\mathrm{\Omega }\right)$, strong solutions in $W^{2,p}\left(\mathrm{\Omega }\right)$ and very weak solutions in $L^p\left(\mathrm{\Omega }\right)$ of the stationary Boussinesq system by assuming that the fluxes of the velocity are sufficiently small. Finally, as it is expected, we obtain the uniqueness of the solution by considering small data.

COMMUNICATED BY:

• Grigory Panasenko

KEYWORDS:

• Boussinesq system
• non-homogeneous Dirichlet boundary conditions
• weak solution
• strong solution
• very weak solution

AMS SUBJECT CLASSIFICATIONS:

• 35Q30
• 49N60
• 76D03

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 This process of applying successively the existence of generalized solutions and the uniqueness of very weak solutions for the Oseen problem in order to conclude that $\mathit{u}$ and π are more regular, we use it several times. So, we will refer to this process as the Oseen argument.

Additional information

Funding

The third author is partially supported by PFBasal-01 (CeBiB) and AFB170001 (CMM) projects, Fondecyt [grant number 1140773], and from the Regional Program STIC-AmSud Project MOSTICOW.