Existence ‘in the large’ of a solution to primitive equations in a domain with uneven bottom

Abstract

The existence and uniqueness theorems ‘in the large’ are proved for a system of primitive equations in the Cartesian coordinates in a domain with an uneven bottom. The original equations are slightly modified: some terms containing mixed derivatives are omitted because they are small. Namely, it is proved that for arbitrary time period [0,T] in a spatial domain Ω = {(x,y,z) | (x,y) ∈ Ω′, z ∈ [0,H(x,y)]}, for an arbitrary viscosity coefficients ν,ν1 > 0, any depth H ∈ C2(Ω′), H ⩾ H0 > 0, and any initial conditions there exists a unique weak solution and the norms are continuous with respect to t, where s is the vertical variable.