In theory the Mig-31 could take out a Blackbird. In practice it would be extremely difficult.
As it was pointed out, niether of these Migs could sustain the flight speeds of the Blackbird, so the Blackbird would simply out run it. But, let's
assume somehow that the Mig took a perfect course to intercept the Blackbird, we must then look at the weapons it would have to use against the
Blackbird.
The aircraft avionics enables the use of missiles with radar- and IR-homing heads. The FOXHOUND carries the long-range AA-9 air-to-air missiles, and
can engage 4 different targets simultaneouly with the M-9.The MiG-31 aircraft weapons mix comprises four R-33, four R-60M and two R-40T missiles and a
GSh-6-23 built-in gun.
GS.org
From what I have found, it seems the AA-9 is the only weapon able to engage the Blackbird.
The R-33 long-range missile was created for arming MiG-31 fighter-interceptors. It became operational in 1980 and is capable of engaging SR-71
strategic reconnaissance aircraft, B-52 and B-1 bombers, aircraft of front and transport aviation, and also helicopters and cruise missiles.
Global Security
However, one must keep in mind that the ability to engage a target does not equal the ability to have a probable or even moderate chance of actually
hitting the target. The reason for this is that when an enemy aircraft is flying at Mach 3, the missle must catch up to the aircraft and track it.
Everytime the blackbird would turn, it would send it hundreds of miles off course of the missle, and the missle would then have to compinsate. All the
while, the missle would be burning it's fuel. Even if it has a several hundred mile range, say 300 for arguments sake, the Blackbird would cover that
distance in just 9 minutes. So basically, the missle which probably travels in the low Mach 4 range, would have at best around 9 minutes to intercept
an aircraft that can put itself off a projected course by 10 miles in just 11 seconds!
So in this mans (admitedly amature) opinion, the Mig-31
could take down an SR-71 in theory, but the circumstance would have to be such that it
would be unrealistic in real world situations.
I would be intrested to see what some of the other aircraft buffs say about this one.