Math behind climate modeling. Let me start with a purely mathematical question. 
What are the partial differential equations used in modeling our climate? I do mean the exact mathematical description.
What are the initial conditions ? 
As far as I know, there should be at least 3 kind of equations. 


*

*The equation about Mass continuity (for mass).

*The equation for motion (of Gaz).

*Thermodynamics equation (for temperature). 


In order to study our climate I'm pretty sure that we need to take in account some "mathematized" chemical equations but I would not ask about them. 
Here is my main question: How precise is the prediction of these equations for our planet climate under the assumption that we know the initial conditions with exact precision ? (the last point is not realistic form physical viewpoint, I agree)
 A: Your question is a bit naive imho. There is not ONE big equation used to predict climate evolution and there is definitely no exact mathematical description of climate. 
There are numerous models though, each tailored to a specific aspect of the climate and with a specific domain of applicability. One of the first model that became famous in the sixties is the Lorenz attractor. Starting with a few general considerations on the climate, E. Lorenz devised an (infinite dimensional) equation and trimmed it down to just 3 unknowns, with the explicit goal to retain what he hoped could explain the difficulties of weather prediction. This showed that even the most simple ode related to weather can exhibit sensibility to initial conditions and chaotic behavior. That means that the slightest error in the initial conditions propagates very fast and lead to exponential divergence of the solutions, preventing long term predictions. That has been popularised under the term butterfly effect.
This does not mean however that some aspects of climate cannot be modelled in a rigorous and predictable way. Note also that probabilistic and statistical methods have since been proven to be pretty important in the study of chaotic dynamical systems, both from the theoretical and practical viewpoint. So you can't predict if the next holidays will be sunny, but you can go to the south pole and study ice cores in order to compute time series and get a glimpse at the general evolution of temperature on your planet.
Arguably, if you know the exact equations and the exact initial conditions of the universe (climate is affected by extraterrestrial factors, such as moon attraction or landing of meteorits), then you know the climate exactly for all time. And probably also the future and the past of every beings on earth. But you don't.
