Sony Ericsson entered into the world of Android with the Xperia X10 Mini. The phone is unique as it was the smallest Android smartphone of its time. Earlier back, when it was released, it lacked flash player as Flash was not supported in Android 1.6. Even back, when it received Android 2.2 Froyo update from Sony Ericsson, it lacked flash plugin. Fortunately, we have an excellent community called xda which has released Flash player for Xperia X10 Mini and it handles youtube videos like a charm.
SE might have underestimated the capabilities of ARM v7 processor that is used inside the X10 Mini. That is why we always suggest you to compare phones before you shell out that hard earned money.
Why? Because, other sony ericsson phones like Xperia Play and Sony Xperia Arco play flash just fine out of the box. You can download the player from here.