Equations for Global Temperature Models
The Navier-Stokes equations are just a couple of the equations used in Climate Models, but they are the most complex set of equations used, and describes the movement of wind. Unfortunately there is no known solution to them yet (though if you come up with one, there is a million dollar prize waiting for you). Another equation used by complex models is the conservation of energy (which in layman's terms can be describes as):

How the temperature changes at a certain point = How fast wind is blowing warm/cold air to the point + How fast the air is cooling by expanding or warming by compression + Net heating/cooling (from the sun, clouds, greenhouse gases, etc.)

With more math, sophisticated techniques, and use of previous research, a prediction is made by the climate model, for just a couple seconds or minutes into the future. Then that prediction is used for the next start of the climate model. This is also how weather models work, but climate models repeat this process for years, decades, and centuries. Climate models also consider different factors from weather models, like what is in the air. Different models have different preferences and consider different scenarios.

You can find some of the results in the IPCC report.

But if you want a simple mathematical model, the simplest out there is the zero-dimensional energy balance model. It won't really predict the temperature to any sort of accurate degree, but it is a useful balance to know.