[Paper Review] Random attractor of the 3D viscous primitive equations driven by fractional noises

Guoli Zhou|arXiv (Cornell University)|Apr 18, 2016

Stability and Controllability of Differential Equations32 references3 citations

TL;DR

This paper establishes the existence of a random attractor (strong attractor) for the 3D viscous primitive equations under non-periodic boundary conditions driven by infinite-dimensional fractional Brownian motion. The authors introduce a novel method based on the compactness of the solution operator and the existence of a bounded absorbing set, overcoming the limitations of traditional Sobolev embedding techniques in high-regularity spaces, and provide a general framework applicable to dissipative stochastic PDEs.

ABSTRACT

We develop a new and general method to prove the the existence of the random attractor (strong attractor) for the primitive equations (PEs) of large-scale ocean and atmosphere dynamics under $non$-$periodic$ boundary conditions and driven by infinite-dimensional additive fractional Wiener processes. In contrast to our new method, the common method, compact Sobolev embedding theorem, is to obtain the uniform $a$ $priori$ estimates in some Sobolev space whose regularity is high enough. But this is very complicated for the 3D stochastic PEs with the $non$-$periodic$ boundary conditions. Therefore, the existence of universal attractor ( weak attractor) was established in previous work. The main idea of our method is that we first derive that $\mathbb{P}$-almost surely the solution operator of stochastic PEs is compact. Then we construct a compact absorbing set by virtue of the compact property of the the solution operator and the existence of a absorbing set. We should point out that our method has some advantages over the common method of using compact Sobolev embedding theorem, i.e., if the random attractor in some Sobolev space do exist in view of the common method, our method would then further implies the existence of random attractor in this space. The present work provides a general way for proving the existence of random attractor for common classes of dissipative stochastic partial differential equations and improves the existing results concerning random attractor of stochastic PEs. In a forth coming paper, we use this new method to prove the existence of strong attractor for the stochastic moist primitive equations, improving the results, the existence of weak (universal) attractor of the deterministic model.

Motivation & Objective

• To establish the existence of a random attractor (strong attractor) for the 3D viscous primitive equations under non-periodic boundary conditions.
• To overcome the technical challenges of applying the compact Sobolev embedding theorem in high-regularity spaces for 3D stochastic PEs with non-periodic boundaries.
• To develop a general method applicable to a broad class of dissipative stochastic partial differential equations.
• To improve upon prior results that only established the existence of a weak (universal) attractor in the deterministic setting.
• To lay the foundation for proving strong attractors in more complex models, such as stochastic moist primitive equations.

Proposed method

• Introduce a new method to prove the existence of a random attractor by first establishing that the solution operator is compact P-almost surely.
• Use the compactness of the solution operator in conjunction with the existence of a bounded absorbing set to construct a compact absorbing set.
• Replace the standard approach relying on compact Sobolev embeddings with a dynamical systems argument based on the intrinsic compactness of the solution operator.
• Apply this method to the 3D stochastic primitive equations driven by infinite-dimensional fractional Wiener processes with Hurst parameter H > 1/2.
• Use energy estimates and Gronwall-type inequalities to derive uniform a priori bounds in H^1 and H^2 norms for velocity and temperature fields.
• Employ Minkowski and Hölder inequalities, interpolation inequalities, and Sobolev embeddings to control nonlinear terms in the energy estimates.

Experimental results

Research questions

• RQ1 Can a strong (random) attractor exist for the 3D viscous primitive equations under non-periodic boundary conditions driven by fractional noise?
• RQ2 Is it possible to prove the existence of a random attractor without relying on compact Sobolev embeddings in high-regularity spaces?
• RQ3 Does the proposed method yield stronger results than the classical approach when both are applicable?
• RQ4 Can this method be generalized to other classes of dissipative stochastic PDEs?
• RQ5 What is the regularity and compactness structure of the long-time dynamics of the stochastic primitive equations with fractional noise?

Key findings

• The solution operator of the 3D stochastic primitive equations is compact P-almost surely, enabling the construction of a compact absorbing set.
• A random attractor (strong attractor) exists in the phase space H^1 × H^1 for the 3D viscous primitive equations under non-periodic boundary conditions.
• The method ensures the existence of a random attractor in Sobolev spaces where the classical method based on compact Sobolev embedding would also apply, but with stronger dynamical implications.
• Uniform a priori estimates in H^1 and H^2 norms are established for the velocity and temperature fields, uniformly in time and for all ω in a full measure set.
• The existence of a compact absorbing set is proven via the interplay between the compactness of the solution operator and the existence of a bounded absorbing set.
• The results are robust to the choice of parameters, and the method applies even when ν_i = μ_i = 1, with generalizations to other values straightforward.