Air strikes COULD do it, but only if they pick the right targets. It wouldn't eradicate them (you need boots on the ground for that)...but, it can certainly halt expansion and shrink the area of containment by destroying armor, weapon emplacements, income generators, command and control centers, airfields, etc. (this would effectively limit their supply lines, so shrinking their area of influence)
I get that politically, we NEED ISIS right now....
1) Keeps the situation in Syria hostile enough to prevent the Syrian President's agreement for the Russian Natural Gas pipeline to actually happen.
2) Keeps Arab coalition partners awake at night, and more willing to cooperate rather than see ISIS enter THEIR country.
3) Is building a basis for eventually establishing permanent US bases in Iraq and other neighboring countries (which is the ONLY way to really keep the peace there after this is all over).
Simply put, the Arab coalition partners' troops have always been poorly trained, poorly equipped, and quite honestly, pure cowards...even when they had numbers and equipment advantages. They simply don't want to fight fanatics. They drop their guns and run...and there really isn't anything that will change this.
Mark my words, if we ever truly do want to finish it, we will eventually NEED to put American and European boots on the ground. However, there is a LOT of enemy softening that can be done well in advance of this need, to truly minimize risk and exposure here.
Then, to prevent ISIS from reforming, there will have to be a US/European ongoing and permanent military presence (i.e. bases), or the second we leave, we just open up the same can of worms.
To that end though, we need to be COMPENSATED for doing so. Not just the goodness of our hearts here. No reason to expect us to spend billions so they get all the benefit. No frickin' way.......
That all said, allowing ISIS to continue to expand territory seems counter-productive to our goals. We certainly have the power in the region to prevent it, so why aren't we? The ONLY reason I can fathom is that some players aren't playing ball, so we're letting them see what happens. Only thing I can think of that makes sense.