The stochastic primitive equations in two space dimensions with multiplicative noise
Nathan Glatt-Holtz Mohammed Ziane
Discrete & Continuous Dynamical Systems - B 2008, 10(4): 801-822 doi: 10.3934/dcdsb.2008.10.801
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.
keywords: Stochastic Partial Differential Equations Primitive Equations Existence and Uniqueness. Galerkin Approximation Geophysical Fluid Dynamics
Regularity of the Navier-Stokes equation in a thin periodic domain with large data
Igor Kukavica Mohammed Ziane
Discrete & Continuous Dynamical Systems - A 2006, 16(1): 67-86 doi: 10.3934/dcds.2006.16.67
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 $u_0$.

keywords: thin domains weak solutions strong solutions Navier-Stokes equations regularity.
Singular perturbation systems with stochastic forcing and the renormalization group method
Nathan Glatt-Holtz Mohammed Ziane
Discrete & Continuous Dynamical Systems - A 2010, 26(4): 1241-1268 doi: 10.3934/dcds.2010.26.1241
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.
keywords: Stochastic Analysis Renormalization Group Method White Noise Singular Perturbation.
The primitive equations on the large scale ocean under the small depth hypothesis
Changbing Hu Roger Temam Mohammed Ziane
Discrete & Continuous Dynamical Systems - A 2003, 9(1): 97-131 doi: 10.3934/dcds.2003.9.97
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.
keywords: primitive equations atmosphere sciences. oceanography Nonlinear partial differential equations

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