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### Open Access Journals

DCDS-B

We study the two dimensional primitive equations in the presence of multiplicative stochastic forcing. We prove the existence and uniqueness of solutions in a fixed probability space. The proof is based on finite dimensional approximations, anisotropic Sobolev estimates, and weak convergence methods.

DCDS

Let $\Omega=[0,L_1]\times[0,L_2]\times[0,\epsilon]$ where
$L_1,L_2>0$ and $\epsilon\in(0,1)$.
We consider the Navier-Stokes equations
with periodic boundary conditions
and
prove that if

$ \|\| \nabla u_0\|\|_{L^2(\Omega)} \le \frac{1}{C(L_1,L_2)\epsilon^{1/6}} $

then there exists a unique global smooth solution with the initial datum $u_0$.

DCDS

This article examines a class of singular perturbation systems in
the presence of a small white noise. Modifying the renormalization
group procedure developed by Chen, Goldenfeld and Oono
[6], we derive an
associated reduced system which we use to construct an approximate
solution that separates scales. Rigorous results demonstrating that
these approximate solutions remain valid with high probability on
large time scales are established. As a special case we infer new
small noise asymptotic results for a class of processes exhibiting a
physically motivated cancellation property in the nonlinear term.
These results are applied to some concrete perturbation systems arising in
geophysical fluid dynamics and in the study of turbulence. For each
system we exhibit the associated renormalization group equation which
helps decouple the interactions between the different scales inherent
in the original system.

DCDS

In this article we study the global existence
of strong solutions of the Primitive Equations (PEs)
for the large scale ocean under the small depth hypothesis.
The small depth hypothesis implies that the domain $M_\varepsilon$ occupied
by the ocean is a thin domain,
its thickness parameter $\varepsilon$
is the aspect ratio between its vertical and horizontal scales.
Using and generalizing the methods developed in [23], [24],
we establish
the global existence of strong solutions for initial data and volume and
boundary 'forces', which belong to large
sets in their respective phase spaces, provided $\varepsilon$ is sufficiently
small. Our proof of the existence
results for the PEs is based on precise estimates of the dependence of a
number of
classical constants on the thickness $\varepsilon$ of the domain.
The extension of the results to the atmosphere or the
coupled ocean and atmosphere or to other relevant boundary conditions will
appear elsewhere.

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