Jiahong Wu
Inviscid Limits and Regularity Estimates for the Solutions
of 2-D Dissipative Quasi-geostrophic Equations
(24K, latex)
ABSTRACT. We discuss two important topics of turbulence theory: inviscid limit
and decay of Fourier spectrum for the 2-D dissipative quasi-geostrophic
(QGS) equations. In the first part
we consider inviscid limits for both smooth and weak
solutions of the 2-D dissipative QGS equations and
prove that the classical solutions with smooth initial data tend to
the solutions of the corresponding non-dissipative equations as the
dissipative coefficient tends to zero. Here the convergence is in the
strong $L^2$ sense and we give the optimal
convergence rate. For the weak solutions of the dissipative QGS equations
with $L^2$ initial data, we obtain weak $L^2$ inviscid limit result.
In the second part we
use the methods of Foias-Temam \cite{FT} and Doering-Titi \cite{DT}
developed for the Navier-Stokes equations
to establish exponential decay of spatial Fourier
spectrum for the solutions of the dissipative QGS equations,
but we treat general norms and our method of
estimating the nonlinear terms are different.
\end{abstract}