By Larin A. A.

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**Additional resources for A Boundary Value Problem for a Second-Order Singular Elliptic Equation in a Sector on the Plane**

**Example text**

140) wa = ω s = 0, ∂v s gpa ∂v a ρs0 νvs = −¯ ρa νva ( ¯ )2 a = αv (v a − v s ), ∂z RT ∂p s a a s s S ∂T a a a gp 2 ∂T = −cp ρ¯ νT ( ¯ ) = αT (T a − T s ), cp ρ0 νT ∂z RT ∂z ∂q ∂S = = 0. ∂pa ∂z Weak formulation of the PEs For the sake of simplicity we restrict ourselves to a regional problem using the beta-plane approximation. 6The same equation appears in [24] with ρ¯a replaced by ρa . Replacing ρa by ρ¯a is a necessary simplification for the developments below. SOME MATHEMATICAL PROBLEMS IN GFD 14-1-04 43 The function spaces that we introduce are similar to those used for the ocean and the atmosphere; hence V = V1 × V2 × V3 , H = H1 × H2 × H3 , where Vi = Via × Vis , Hi = Hia × His , the spaces Via , Hia , Vis , His being exactly like those of the atmosphere and the ocean respectively.

The governing equations In Ma , the variable is U a = (v a , T a , q) and in Ms the variable is U s = (v s , T s , S); we set also U = {U a , U s } , or alternatively v = {v a , v s } , T = {T a , T s } . 29), introducing only a superscript s for w, p, ρ0 , Fv , FT , FS , Tr , Sr , as well as the eddy viscosity coefficients µv , νv , etc. 136), and the differential operators are changed accordingly. Boundary Conditions Except for Γi , the boundary conditions are the same as for the ocean and the atmosphere taken separately.

Proof. 1 for dimension three. 1 and prove by contradiction that t∗ = t1 . 57) lim sup ||U (t)|| = +∞. 57). The bounds for ||U (t)|| will be derived sequentially: we will show successively that uz , ux , are in L∞ (0, t0 ; L2 (M)) and L2 (0, t0 ; H 1 (M)) where ϕx = ∂ϕ/∂x and ϕz = ∂ϕ/∂z; then we will prove at once that v, T and S are in L∞ (0, t0 ; H 1 ) and L2 (0, t0 ; H 2 ). In fact we will give the proofs for uz , ux , T the other quantities being estimated in exactly the same way. For the sake of simplicity, we assume hereafter that g = (gv , gT ) = 0.