Quasigeostrophic Equations for Fractional Powers of Infinitesimal Generators. (7th February 2019) - Abadias, Luciano; Miana, Pedro J.
Abstract : 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 convolutionC 0 -semigroup of positive kernel onL p ( R n ), with1 ≤ p < ∞ . Firstly, we give remarkable pointwise and integral inequalities involving the fractional powers( - A ) α for0 ≤ α ≤ 1 . We use these estimates to obtainL p -decayment of solutions of the above quasigeostrophic equation. These results extend the case of fractional derivatives (takingA = Δ, the Laplacian), which has been studied in the literature.