On the Energy Scaling Behaviour of a Singularly Perturbed Tartar Square

TL;DR

This paper establishes near-optimal energy scaling laws for a singular perturbation problem related to the Tartar square, revealing the complex interplay of nonlinearity and negative Sobolev spaces.

Contribution

It introduces a novel lower bound matching the upper bound up to logarithmic factors, using a bootstrap argument in Fourier space for the Tartar square problem.

Findings

Derived an almost optimal scaling law for the Tartar square problem.

Established a new lower bound using a bootstrap argument in Fourier space.

Provided evidence of the infinite order of lamination in the problem.

Abstract

In this article we derive an (almost) optimal scaling law for a singular perturbation problem associated with the Tartar square. As in \cite{W97, C99}, our upper bound quantifies the well-known construction which is used in the literature to prove flexibility of the Tartar square in the sense of flexibility of approximate solutions to the differential inclusion. The main novelty of our article is the derivation of an (up to logarithmic powers matching) ansatz free lower bound which relies on a bootstrap argument in Fourier space and is related to a quantification of the interaction of a nonlinearity and a negative Sobolev space in the form of "a chain rule in a negative Sobolev space". Both the lower and the upper bound arguments give evidence of the involved "infinite order of lamination".