[Paper Review] Time analyticity with higher norm estimates for the 2D Navier-Stokes equations

Ciprian Foiaş, Michael S. Jolly+4 more|arXiv (Cornell University)|Dec 3, 2013

Navier-Stokes equation solutions17 references3 citations

TL;DR

This paper establishes uniform, explicit bounds on higher-order Sobolev norms $|A^eta u|$ for solutions of the 2D Navier-Stokes equations on the global attractor, complexified in time, under periodic boundary conditions. It proves that if any solution in the attractor lies in $\mathcal{D}(A^\alpha)$, then all do—establishing a universal 'all for one, one for all' principle—while also showing that the zero solution can only belong to the attractor if the forcing $g$ lies in a specific Gevrey-type function class $\mathcal{C}(\sigma^*)$. The results rely on refined inductive estimates in complex time strips and analytic semigroup theory.

ABSTRACT

This paper establishes bounds on norms of all orders for solutions on the global attractor of the 2D Navier-Stokes equations, complexified in time. Specifically, for periodic boundary conditions on $[0,L]^2$, and a force $g\in\calD(A^{\frac{α-1}{2}})$, we show there is a fixed strip about the real time axis on which a uniform bound $|A^αu|< m_ανκ_0^α$ holds for each $α\in \bN$. Here $ν$ is viscosity, $\k0=2π/L$, and $m_α$ is explicitly given in terms of $g$ and $α$. We show that if any element in $\calA$ is in $\D(A^α)$, then all of $\calA$ is in $\D(A^α)$, and likewise with $\D(A^α)$ replaced by $C^\infty(Ω)$. We demonstrate the universality of this "all for one, one for all" law on the union of a hierarchal set of function classes. Finally, we treat the question of whether the zero solution can be in the global attractor for a nonzero force by showing that if this is so, the force must be in a particular function class.

Motivation & Objective

• To establish uniform, explicit bounds on $|A^\alpha u|$ for all $\alpha \in \mathbb{N}$ in a complex time strip for 2D Navier-Stokes solutions on the global attractor.
• To prove a universal 'all for one, one for all' regularity principle: if one solution in the attractor is in $\mathcal{D}(A^\alpha)$, then all are, and likewise for $C^\infty(\Omega)$.
• To characterize the class of forcing terms $g$ for which the zero solution can belong to the global attractor, showing it must lie in a specific Gevrey-type class $\mathcal{C}(\sigma^*)$.
• To refine estimates on the width of the time-analyticity strip by reducing it by a factor of 2 at each order $\alpha$.
• To demonstrate that the attractor's regularity is fully determined by the regularity of any single element, via inductive estimates in complex time.

Proposed method

• Complexify time and analyze solutions as $H_{\mathbb{C}}$-valued analytic functions in a strip $\mathcal{N}_\alpha$ around the real axis.
• Use inductive estimates on $|A^{\alpha/2} u(\zeta)|$ for $\zeta \in \mathcal{N}_\alpha$, leveraging the Stokes operator $A$ and the Helmholtz-Leray projection.
• Apply the Riesz-Fréchet theorem and weak analyticity to lift bounds from $|A^{\alpha/2} u|$ to strong analyticity of $A^{\alpha/2} u(\zeta)$ in $H_{\mathbb{C}}$.
• Use Vitali's theorem and convergence of Galerkin approximations $u_\kappa$ to pass limits and establish existence and bounds for the full solution $u(\zeta)$.
• Derive differential inequalities for $\frac{d}{d\rho}|A^{\alpha/2} u(t_0 + \rho e^{i\theta})|^2$ to control growth and ensure uniform bounds.
• Introduce the function class $\mathcal{C}(\sigma) = \{ u \in C^\infty(\Omega) : \sup_\alpha |A^{\alpha/2} u| \exp(-\sigma \alpha^2 / 2) < \infty \}$ to characterize the regularity hierarchy of the attractor.

Experimental results

Research questions

• RQ1 Under what conditions is the global attractor of the 2D Navier-Stokes equations uniformly analytic in complex time, and what is the width of the strip of analyticity for each order $\alpha$?
• RQ2 Can the regularity of a single solution in the global attractor imply the regularity of all solutions? Specifically, does $u \in \mathcal{D}(A^\alpha)$ for some $u \in \mathcal{A}$ imply $\mathcal{A} \subset \mathcal{D}(A^\alpha)$?
• RQ3 What is the precise function class that the forcing $g$ must belong to if the zero solution is in the global attractor?
• RQ4 How do the bounds on $|A^\alpha u|$ scale with $\alpha$, and can the width of the analyticity strip be reduced in a controlled way as $\alpha$ increases?
• RQ5 Is there a universal regularity hierarchy—such as $\bigcup_{\sigma>0} \mathcal{C}(\sigma)$—that characterizes the attractor’s smoothness, and does the 'all for one, one for all' law hold on such classes?

Key findings

• For each $\alpha \in \mathbb{N}$, there exists a uniform strip $\mathcal{N}_\alpha$ of width $\delta_\alpha$ such that $|A^\alpha u(\zeta)| < m_\alpha \nu \kappa_0^\alpha$ for all $\zeta \in \mathcal{N}_\alpha$, with $m_\alpha$ explicitly depending on $g$ and $\alpha$.
• The width of the analyticity strip satisfies $\delta_{\alpha+1} = \delta_\alpha / 2$, allowing for sharper bounds at higher orders.
• If any solution in the global attractor $\mathcal{A}$ lies in $\mathcal{D}(A^\alpha)$, then all solutions in $\mathcal{A}$ lie in $\mathcal{D}(A^\alpha)$, establishing a universal 'all for one, one for all' regularity law.
• The same law holds for $C^\infty(\Omega)$: if one solution is smooth, all are.
• The attractor lies in $\bigcup_{\sigma>0} \mathcal{C}(\sigma)$, a hierarchal family of Gevrey-type classes with $\mathcal{C}(\sigma_1) \subsetneq \mathcal{C}(\sigma_2)$ for $\sigma_1 < \sigma_2$, and $\bigcup_{\sigma>0} \mathcal{C}(\sigma) \subsetneq C^\infty(\Omega)$.
• If the zero solution is in $\mathcal{A}$, then the forcing $g$ must lie in $\mathcal{C}(\sigma^*)$ for a specific $\sigma^*$, narrowing the search for such forces.