  We study the curve diffusion flow for closed curves immersed in the Minkowski
plane $\mathcal{M}$, which is equivalent to the Euclidean plane endowed with a
closed, symmetric, convex curve called an indicatrix that scales the length of
a vector in $\mathcal{M}$ depending on its length. The indiactrix
$\partial\mathcal{U}$ (where $\mathcal{U}\subset\mathbb{R}^{2}$ is a convex,
centrally symmetric domain) induces a second convex body, the isoperimetrix
$\tilde{\mathcal{I}}$. This set is the unique convex set that miniminises the
isoperimetric ratio (modulo homothetic rescaling) in the Minkowski plane. We
prove that under the flow, closed curves that are initially close to a
homothetic rescaling of the isoperimetrix in an averaged $L^{2}$ sense exists
for all time and converge exponentially fast to a homothetic rescaling of the
isoperimetrix that has enclosed area equal to the enclosed area of the initial
immersion.
