Global well‐posedness of the 3D primitive equations with only horizontal eddy diffusivity and delays in both convective and heat source terms

Abstract

In ocean and atmospheric dynamics, the strong horizontal turbulent mixing tends to produce horizontal eddy viscosity, so the horizontal scale is much larger than the vertical scale. At the same time, as a fluid, the motion of the atmosphere or the ocean has a delayed property to some extent. In this paper, we consider 3D viscous primitive equations with only horizontal eddy diffusivity and delays in both convective and heat source terms. With some suitable assumptions on delay term, we prove the existence and uniqueness of strong solution with initial data and obtain the existence and uniqueness of the stationary solution. Moreover, we prove that the strong solution converges exponentially to the stationary solution under some suitable assumptions.