Isotropic realizability of a strain field for the incompressible two-dimensional Stokes equation

TL;DR

This paper investigates when a regular strain field in 2D can be realized by a positive viscosity in the incompressible Stokes equation, providing local, global, and counter-example results.

Contribution

It establishes conditions for local and global isotropic realizability of strain fields in 2D Stokes flow, including perturbation results and counter-examples.

Findings

Local realizability near non-vanishing points

Global realizability depends on solutions to a semilinear wave equation

Counter-example shows global realizability does not imply torus realizability

Abstract

In the paper we study the problem of the isotropic realizability in R^2 of a regular strain field e(U)=1/2(DU+DU^T) for the incompressible Stokes equation, namely the existence of a positive viscosity mu\textgreater{}0 solving the Stokes equation in R^2 with the prescribed field e(U). We show that if e(U) does not vanish at some point, then the isotropic realizability holds in the neighborhood of that point. The global realizability in R^2 or in the torus is much more delicate, since it involves the global existence of a regular solution to a semilinear wave equation the coefficients of which depend on the derivatives of U. Using the semilinear wave equation we prove a small perturbation result: If DU is periodic and close enough to its average for the C^4-norm, then the strain field is isotropically realizable in a given disk centered at the origin. On the other hand, a counter-example shows that the global realizability in R^2 may hold without the realizability in the torus, and it is discussed in connection with the associated semilinear wave equation. The case where the strain field vanishes is illustrated by an example. The singular case of a rank-one laminate field is also investigated.