Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

From nonisothermal BGK to Euler Maxwellians via relative entropy

Nuno J. Alves

Abstract

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.

From nonisothermal BGK to Euler Maxwellians via relative entropy

Abstract

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 convergence of the BGK solution and the local Maxwellians to the target Euler Maxwellian, together with convergence of the associated macroscopic quantities.

Paper Structure

This paper contains 7 sections, 11 theorems, 135 equations.

Table of Contents

  1. Introduction
  2. Assumptions and main theorem
  3. Convergence to Maxwellians
  4. Relative entropy identity
  5. Auxiliary lemmas
  6. Proof of Theorem \ref{['thm_BGK-E']}
  7. Proof of Corollary \ref{['cor:strongconv']}

Key Result

Theorem 2.1

Let $f^\varepsilon$ be a smooth solution of the BGK equation eq:BGK on $[0,T)\times\mathbb{R}^d_x\times\mathbb{R}^d_v$ with $f^\varepsilon\ge0$ and unit total mass Assume that (A) and (B) hold. Then there exists a positive constant independent of $\varepsilon$, such that Consequently, if at $t=0$, then

Theorems & Definitions (21)

  • Theorem 2.1
  • Corollary 2.2
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Proposition 3.3
  • proof
  • Proposition 3.4
  • proof
  • ...and 11 more