Exact Solutions for Restricted Incompressible Navier--Stokes Equations with Dirichlet Boundary Conditions

TL;DR

This paper derives exact solutions for restricted incompressible Navier--Stokes equations with Dirichlet boundary conditions, revealing non-uniqueness of solutions and providing a method to compute kinetic energy explicitly over time.

Contribution

It introduces an additional ansatz to Navier--Stokes fields, enabling the derivation of exact solutions and analysis of solution uniqueness under Dirichlet boundary conditions.

Findings

Non-uniqueness of solutions with no-slip boundary conditions.

Finite L^2 norms of velocity fields can be achieved.

Kinetic energy can be computed explicitly as a function of time.

Abstract

This paper exposes how to obtain a relation that have to be hold for all free--divergence velocity fields that evolve according to Navier--Stokes equations. However, checking the violation of this relation requires a huge computational effort. To circumvent this problem it is proposed an additional antsatz to free-divergent Navier--Stokes fields. This makes available six degrees of freedom which can be tuned. When they are tuned adequately, it is possible to find finite $L^2$ norms of the velocity field for volumes of $\mathbb{R}^3$ and for $t\in[t_0,\infty)$. In particular, the kinetic energy of the system is bounded when the field components $u_i$ are class $C^3$ functions on $\mathbb{R}^3\times[t_0,\infty)$ that hold Dirichlet boundary conditions. This additional relation lets us conclude that Navier--Stokes equations with no-slip boundary conditions have not unique solution. Moreover, under a given external force the kinetic energy can be computed exactly as a funtion of time.