From nonisothermal BGK to Euler Maxwellians via relative entropy

We study the hydrodynamic limit of the nonisothermal BGK model toward Euler Maxwellians. For a prescribed sufficiently smooth Euler solution, we use relative entropy to compare a BGK solution with the corresponding Euler Maxwellian. The result is conditional: for BGK solutions satisfying a uniform sixth velocity-moment bound and uniform $L^{\mathrm{\infty }}$ bounds on their macroscopic velocity, temperature, and inverse temperature, we obtain a stability estimate uniform in time. The key new ingredient is the control of an additional velocity-cubic term in the relative entropy identity. For well-prepared initial data, this yields strong $L^1$ convergence of the BGK solution and its local Maxwellians to the target Euler Maxwellian, together with convergence of the associated macroscopic quantities.