[Paper Review] Constructive analysis of the Navier-Stokes equation

Joerg Kampen|arXiv (Cornell University)|Apr 26, 2010

Stochastic processes and financial applications6 references3 citations

TL;DR

This paper presents a globally convergent, time-discretized numerical scheme for the incompressible Navier-Stokes equations in Leray projection form, using dynamic control of velocity and its gradients via a feedback mechanism. It proves convergence to a bounded global classical solution for smooth initial data with polynomial decay at infinity, extending to initial-boundary value problems and compact manifolds.

ABSTRACT

A global time-discretized scheme for the Navier-Stokes equation system in its Leray projection form is defined. It is shown that the scheme converges to a bounded global classical solution for smooth data which have polynomial decay at infinity. Furthermore, the algorithm proposed is extended to the situation of initial-boundary value problems. Algorithms constructed in a different context (cf. [4, 10, 5, 9]) may be used within the proposed scheme in order to compute the solution of Leray's form of the Navier-Stokes system. The main idea for global existence is to define a control function dynamically and show explicitly that the scheme which solves a controlled Navier-Stokes type equation can control the modulus of velocity and the first derivatives of velocity to be bounded. The method described here can be extended to Navier-Stokes equations on compact manifolds which is done in a subsequent paper.

Motivation & Objective

• To develop a constructive, globally convergent time-discretized scheme for the incompressible Navier-Stokes equations in Leray projection form.
• To establish global existence of classical solutions for smooth initial data with polynomial decay at spatial infinity.
• To extend the scheme to initial-boundary value problems and compact Riemannian manifolds.
• To control the growth of velocity and its first derivatives through a dynamically defined control function.
• To provide a computationally implementable framework using existing algorithms for solving the Leray-form Navier-Stokes system.

Proposed method

• A time-discretized scheme is constructed for the Leray projection form of the Navier-Stokes system, treating the pressure term via a singular integral operator involving the Poisson kernel.
• The method employs a dynamically defined control function to bound the velocity and its first derivatives, ensuring global existence.
• The scheme is extended to initial-boundary value problems using a Robin-type boundary condition with a feedback term involving the normal derivative and a coefficient function αi.
• Solutions are represented via fundamental solutions Γkl of the linearized equation, with corrections δvρ,k,l,i expressed through integral equations involving kernels KΓ and forcing terms.
• The boundary correction terms are solved via a Fredholm-type integral equation with a kernel KΓ that combines the normal derivative and the feedback coefficient.
• A Levy-type series expansion is used to represent the boundary correction functions φi and φik, enabling iterative computation of the solution.

Experimental results

Research questions

• RQ1 Can a time-discretized scheme be constructed that ensures global convergence to a classical solution of the incompressible Navier-Stokes equations?
• RQ2 How can the growth of velocity and its derivatives be controlled in the absence of external forces, given the nonlinear quadratic integral term in the pressure?
• RQ3 Is it possible to extend the scheme to initial-boundary value problems with non-trivial boundary conditions?
• RQ4 Can the method be adapted to the Navier-Stokes equations on compact Riemannian manifolds?
• RQ5 What is the role of the Leray projection form in enabling constructive analysis and numerical implementation?

Key findings

• The proposed time-discretized scheme converges to a bounded global classical solution for smooth initial data with polynomial decay at infinity.
• The control function is dynamically defined to ensure uniform boundedness of velocity and its first derivatives throughout time.
• The solution representation via fundamental solutions and integral equations allows for constructive computation using existing solvers from related contexts.
• The method is extendable to initial-boundary value problems by incorporating Robin-type boundary conditions with a feedback mechanism.
• The scheme is formally extendable to Navier-Stokes equations on compact manifolds, as demonstrated in a subsequent paper.
• The boundary correction terms are computed via a convergent Levy-type series expansion of the integral equation solution.