## Advances in Differential Equations

- Adv. Differential Equations
- Volume 19, Number 1/2 (2014), 87-132.

### Large time decay properties of solutions to a viscous Boussinesq system in a half space

Pigong Han and Maria E. Schonbek

#### Abstract

We consider the long time behavior of weak and strong solutions of the $n$-dimensional viscous Boussinesq system in the half space, with $n\geq3$. The $L^r(\mathbb{R}^n_+)$-asymptotics of strong solutions and their first three derivatives, with $1\leq r\leq\infty$, are derived by combining $L^q-L^r$ estimates and properties of the fractional powers of the Stokes operator. For the $L^\infty-$asymptotics of the second order derivatives, the unboundedness of the projection operator $P: L^\infty(\mathbb{R}^n_+)\rightarrow L^\infty_\sigma(\mathbb{R}^n_+)$ is dealt with by an appropriate decomposition of the nonlinear term.

#### Article information

**Source**

Adv. Differential Equations Volume 19, Number 1/2 (2014), 87-132.

**Dates**

First available in Project Euclid: 12 November 2013

**Permanent link to this document**

https://projecteuclid.org/euclid.ade/1384278133

**Mathematical Reviews number (MathSciNet)**

MR3161657

**Zentralblatt MATH identifier**

1286.35206

**Subjects**

Primary: 35Q35: PDEs in connection with fluid mechanics 76D05: Navier-Stokes equations [See also 35Q30]

#### Citation

Han, Pigong; Schonbek, Maria E. Large time decay properties of solutions to a viscous Boussinesq system in a half space. Adv. Differential Equations 19 (2014), no. 1/2, 87--132. https://projecteuclid.org/euclid.ade/1384278133