You are subject to Australian consumer laws, in particular the Trade Practices Act which state:
# The goods must be of merchantable quality. That is, they must meet a basic level of quality and performance, taking into account their price and description. They also should be free from defects that were not obvious to you at the time of purchase.
# The goods must be fit for their purpose. That is, they should do what they are supposed to do and be suitable for any purpose that you might have made known to the supplier.
http://www.accc.gov.au/content/index.phtml/ite mId/815360
Now, I know Ugg boots are not cheap but even cheap boots should last more than 3 wears, so your boots are neither of merchantable quality nor fit for the purpose.
Knowing your rights and enforcing them are two different things. It will not be practicable for you to go to Australia and demand a refund but if they refuse to 'play ball' send an email of complaint to the official body:
http://www.accc.gov.au/content/index.phtml/ite mId/815327
Hopefully they will either help you resolve the problem or give you contact details of other organisations.