From nonisothermal BGK to Euler Maxwellians via relative entropy

We study the hydrodynamic limit of the nonisothermal BGK model toward smooth Euler Maxwellians. For a prescribed smooth Euler solution, we derive a relative entropy stability estimate between a BGK solution and the associated Maxwellian. The main new ingredient is the control of an additional velocity-cubic term in the relative entropy identity. Under a uniform sixth velocity-moment bound and suitable bounds on the BGK macroscopic quantities, we obtain a uniform-in-time relative entropy estimate. For well-prepared initial data, this yields strong $L^1$ convergence of the BGK solution and the local Maxwellians to the target Euler Maxwellian, together with convergence of the associated macroscopic quantities.