## Nagoya Mathematical Journal

- Nagoya Math. J.
- Volume 155 (1999), 55-80.

### Local existence and blow-up criterion of Hölder continuous solutions of the Boussinesq equations

Dongho Chae, Sung-Ki Kim, and Hee-Seok Nam

#### Abstract

In this paper we prove the local existence and uniqueness of $C^{1+\gamma}$ solutions of the Boussinesq equations with initial data $v_0$, $\theta_0 \in C^{1+\gamma}$, $\omega_0, \Delta\theta_0\in L^q$ for $0 < \gamma < 1$ and $1 < q < 2$. We also obtain a blow-up criterion for this local solutions. More precisely we show that the gradient of the passive scalar $\theta$ controls the breakdown of $C^{1+\gamma}$ solutions of the Boussinesq equations.

#### Article information

**Source**

Nagoya Math. J., Volume 155 (1999), 55-80.

**Dates**

First available in Project Euclid: 27 April 2005

**Permanent link to this document**

https://projecteuclid.org/euclid.nmj/1114631256

**Mathematical Reviews number (MathSciNet)**

MR1711383

**Zentralblatt MATH identifier**

0939.35150

**Subjects**

Primary: 35Q35: PDEs in connection with fluid mechanics

Secondary: 35B05: Oscillation, zeros of solutions, mean value theorems, etc. 76B03: Existence, uniqueness, and regularity theory [See also 35Q35]

#### Citation

Chae, Dongho; Kim, Sung-Ki; Nam, Hee-Seok. Local existence and blow-up criterion of Hölder continuous solutions of the Boussinesq equations. Nagoya Math. J. 155 (1999), 55--80. https://projecteuclid.org/euclid.nmj/1114631256