## Differential and Integral Equations

- Differential Integral Equations
- Volume 21, Number 9-10 (2008), 837-849.

### Uniform gradient bounds for the primitive equations of the ocean

Igor Kukavica and Mohammed Ziane

#### Abstract

In this paper, we consider the 3D primitive equations of the ocean in the case of the Dirichlet boundary conditions on the side and bottom boundaries. We provide an explicit upper bound for the $H^{1}$ norm of the solution. We prove that, after a finite time, this norm is less than a constant which depends only on the viscosity $\nu$, the force $f$, and the domain $\Omega$. This improves our previous result from [7] where we established the global existence of strong solutions with an argument which does not give such explicit rates.

#### Article information

**Source**

Differential Integral Equations, Volume 21, Number 9-10 (2008), 837-849.

**Dates**

First available in Project Euclid: 20 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1356038588

**Mathematical Reviews number (MathSciNet)**

MR2483337

**Zentralblatt MATH identifier**

1224.35346

**Subjects**

Primary: 35Q35: PDEs in connection with fluid mechanics

Secondary: 35B45: A priori estimates 35B65: Smoothness and regularity of solutions 76D03: Existence, uniqueness, and regularity theory [See also 35Q30] 76U05: Rotating fluids 86A05: Hydrology, hydrography, oceanography [See also 76Bxx, 76E20, 76Q05, 76Rxx, 76U05]

#### Citation

Kukavica, Igor; Ziane, Mohammed. Uniform gradient bounds for the primitive equations of the ocean. Differential Integral Equations 21 (2008), no. 9-10, 837--849. https://projecteuclid.org/euclid.die/1356038588