Quasigeostrophic Equations for Fractional Powers of Infinitesimal Generators

Abstract EN

In this paper we treat the following partial differential equation, the quasigeostrophic equation: ∂ / ∂ t + u · ∇ f = - σ - A α f , 0 ≤ α ≤ 1 , where ( A , D ( A ) ) is the infinitesimal generator of a convolution C 0 -semigroup of positive kernel on L p ( R n ) , with 1 ≤ p < ∞ .

Firstly, we give remarkable pointwise and integral inequalities involving the fractional powers ( - A ) α for 0 ≤ α ≤ 1 .

We use these estimates to obtain L p -decayment of solutions of the above quasigeostrophic equation.

These results extend the case of fractional derivatives (taking A = Δ , the Laplacian), which has been studied in the literature.