Advances in Differential Equations

On the $\delta$-primitive and Boussinesq type equations

M. Petcu and A. Rousseau

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In this article we consider the Primitive Equations without horizontal viscosity but with a mild vertical viscosity added in the hydrostatic equation, as in [13] and [16], which are the so-called $\delta-$Primitive Equations. We prove that the problem is well posed in the sense of Hadamard in certain types of spaces. This means that we prove the finite-in-time existence, uniqueness and continuous dependence on data for appropriate solutions. The results given in the 3D periodic space easily extend to dimension 2. We also consider a Boussinesq type of equation, meaning that the mild vertical viscosity present in the hydrostatic equation is replaced by the time derivative of the vertical velocity. We prove the same type of results as for the $\delta-$Primitive Equations; periodic boundary conditions are similarly considered.

Article information

Adv. Differential Equations, Volume 10, Number 5 (2005), 579-599.

First available in Project Euclid: 18 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 76U05: Rotating fluids
Secondary: 35B65: Smoothness and regularity of solutions 35Q35: PDEs in connection with fluid mechanics 76D99: None of the above, but in this section 86A05: Hydrology, hydrography, oceanography [See also 76Bxx, 76E20, 76Q05, 76Rxx, 76U05]


Petcu, M.; Rousseau, A. On the $\delta$-primitive and Boussinesq type equations. Adv. Differential Equations 10 (2005), no. 5, 579--599.

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