If you check on Google Books Ngram Viewer, you will see that the expressions war on and war against are just about equally popular.
In my experience, war on is used typically for popular campaigns such as war on waste, war on plastic and war on drugs. In each of these war is being used as a metaphor for a campaign linked to ecology, pollution or health concerns.
War against is used mainly for actual wars in which one nation, tribe or group fights against another.
Sometimes the distinction is blurred. For example, one can choose whether a campaign to eradicate mosquitoes is a war on or a war against the pests.
As far as smog is concerned, you can wage a war on the causes but hardly on smog itself (except as a popular catch-phrase) and certainly not against it.