[Paper Review] Some infinite matrix analysis, a Trotter product formula for dissipative operators, and an algorithm for the incompressible Navier-Stokes equation

Joerg Kampen|arXiv (Cornell University)|Dec 11, 2012

Advanced Mathematical Physics Problems11 references3 citations

TL;DR

This paper presents a novel global numerical scheme for the incompressible Navier-Stokes equation on the n-torus using an infinite-dimensional ODE system in Fourier space, leveraging a Trotter product formula for dissipative operators and infinite matrix analysis. It establishes global existence and uniform boundedness of solutions for smooth (polynomially decaying) initial data via a time-dilatation-based damping mechanism, ensuring convergence in strong dual Sobolev norms for any viscosity $ u > 0$. The method provides a priori estimates and an algorithmic framework with rigorous convergence guarantees.

ABSTRACT

We introduce a global scheme on the n-torus of a controlled incompressible Navier-Stokes equation in terms of a coupled controlled infinite ODE-system of Fourier-modes with smooth data. We construct a scheme of global approximations related to linear partial integrodifferential equations in dual space which are uniformly bounded in dual Sobolev spaces with polynomially decaying modes. The scheme is based on some infinite matrix algebra related to weakly singular integrals, and a Trotter-product formula for dissipative operators which leads to rigorous existence results and a uniform bound for the solutions of the successive approximating linearized equations in dual space which may be otherwise represented as formal solutions in the sense of an iterated Dyson formalism.

Motivation & Objective

• To develop a globally convergent numerical scheme for the incompressible Navier-Stokes equation on the n-torus with smooth initial data.
• To establish rigorous a priori bounds on solutions of iterated linear partial integro-differential equations derived from the Navier-Stokes system.
• To construct a time-dilatation-based damping mechanism that preserves global upper bounds in strong dual Sobolev norms.
• To provide a convergence proof for an Euler-type Trotter product scheme in the context of dissipative operators and infinite-dimensional systems.
• To define an algorithmic framework using real and complex Fourier bases that ensures convergence in strong norms for any positive viscosity $ u > 0$.

Proposed method

• The Navier-Stokes equation is transformed into an infinite system of ODEs in Fourier space, representing velocity modes as an infinite vector in a sequence space with polynomial decay.
• A controlled, auto-controlled scheme is introduced using a time-dilatation transformation to induce damping via dissipative modes, ensuring global boundedness.
• The method relies on infinite matrix algebra involving weakly singular integrals and applies a Trotter product formula for dissipative operators to approximate the solution evolution.
• The scheme is formulated in both complex and real Fourier bases to support numerical implementation and error control on real spaces.
• A priori estimates are derived for the iterated linear equations, with bounds preserved at each time step using a compactness argument in strong dual Sobolev spaces.
• Convergence of the Euler-type Trotter scheme is rigorously analyzed in the appendix, with conditions on parameters $r, u, u r^2 = u_0$ ensuring stability and bounded growth.

Experimental results

Research questions

• RQ1 Can a global numerical scheme be constructed for the incompressible Navier-Stokes equation on the torus that ensures uniform boundedness of solutions for all time with smooth initial data?
• RQ2 How can a Trotter product formula for dissipative operators be rigorously applied to infinite-dimensional systems to yield existence and convergence results?
• RQ3 What role does time-dilatation-based damping play in preserving upper bounds on solution norms in strong dual Sobolev spaces?
• RQ4 Can a priori estimates be established for the linearized, iterated equations in the scheme, enabling global existence proofs without relying on external control functions?
• RQ5 How does the algorithmic framework based on Fourier modes ensure convergence in strong norms for arbitrary viscosity $ u > 0$?

Key findings

• Global existence of solutions to the incompressible Navier-Stokes equation on the n-torus is proven for smooth initial data with polynomially decaying Fourier modes.
• The scheme ensures uniform upper bounds in $H^p$-type dual norms for $p > rac{n}{2}+1$, with the bound preserved at each time step via a damping mechanism.
• A priori estimates are established for the linearized, iterated equations in the scheme, enabling a compactness argument in strong dual Sobolev spaces.
• The Euler-type Trotter product scheme converges under the condition $ u r^2 = u_0$, with growth of the original velocity components bounded by $ u_0 u^2$ on time intervals of length $a$, which can be offset by small viscosity.
• The algorithm is numerically implementable in both real and complex Fourier bases, with convergence guaranteed in strong norms for any $ u > 0$, and the external control function is optional and only affects zero modes.
• The analysis shows that turbulence may be understood as a dynamical property of the system, consistent with long-standing suggestions in the literature.