## Nagoya Mathematical Journal

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

#### 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

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

Mathematical Reviews number (MathSciNet)
MR1711383

Zentralblatt MATH identifier
0939.35150

#### 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

#### References

• A. Majda, Vorticity and the mathematical theory of incompressible fluid flow , Princeton University graduate course lecture note (1986–1987).