Local in time solution to ES-BGK model with correct Prandtl number

In this paper, we study the existence of a unique mild solution for the ellipsoidal BGK model with the critical Prandtl parameter $ \nu = -1/2 $. The key difficulty is the breakdown of the equivalence between the local temperature and the temperature tensor in this critical case. To overcome this, we use the fact that the quadratic polynomial of the temperature tensor can be written as the difference of the directional temperatures, which enables us to control the temperature tensor from below for a finite time.