2023/01/24 by Matthew Rosenzweig · 1 citation
Engineering · Mathematics · Physics and Astronomy · #Advanced Thermodynamics and Statistical Mechanics #Fluid Dynamics and Turbulent Flows #Gas Dynamics and Kinetic Theory
paper · pdf · doi:10.1007/s11005-023-01630-w
openalex publication_date 2023/01/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Abstract A long-standing problem in mathematical physics is the rigorous derivation of the incompressible Euler equation from Newtonian mechanics. Recently, Han-Kwan and Iacobelli (Proc Am Math Soc 149:3045–3061, 2021) showed that in the monokinetic regime, one can directly obtain the Euler equation from a system of N particles interacting in \mathbb Td <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>T</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> </mml:math> , d≥ 2 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> , via Newton’s second law through a supercritical mean-field limit . Namely, the coupling constant λ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>λ</mml:mi> </mml:math> in front of the pair potential, which is Coulombic, scales like N-θ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>N</mml:mi> <mml:mrow> <mml:mo>-</mml:mo> <mml:mi>θ</mml:mi> </mml:mrow> </mml:msup> </mml:math> for some θ ∈ (0,1) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>θ</mml:mi> <mml:mo>∈</mml:mo> <mml:mo>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , in contrast to the usual mean-field scaling λ ∼ N-1 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>λ</mml:mi> <mml:mo>∼</mml:mo> <mml:msup> <mml:mi>N</mml:mi> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> . Assuming θ ∈ (1-(2)/(d(d+1)),1) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>θ</mml:mi> <mml:mo>∈</mml:mo> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>-</mml:mo> <mml:mfrac> <mml:mn>2</mml:mn> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>(</mml:mo> <mml:mi>d</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:mfrac> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , they showed that the empirical measure of the system is effectively described by the solution to the Euler equation as N→ ∞ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>→</mml:mo> <mml:mi>∞</mml:mi> </mml:mrow> </mml:math> . Han-Kwan and Iacobelli asked if their range for θ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>θ</mml:mi> </mml:math> was optimal. We answer this question in the negative by showing the validity of the incompressible Euler equation in the limit N→ ∞ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>→</mml:mo> <mml:mi>∞</mml:mi> </mml:mrow> </mml:math> for θ ∈ (1-(2)/(d),1) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>θ</mml:mi> <mml:mo>∈</mml:mo> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>-</mml:mo> <mml:mfrac> <mml:mn>2</mml:mn> <mml:mi>d</mml:mi> </mml:mfrac> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . Our proof is based on Serfaty’s modulated-energy method, but compared to that of Han-Kwan and Iacobelli, crucially uses an improved “renormalized commutator” estimate to obtain the larger range for θ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>θ</mml:mi> </mml:math> . Additionally, we show that for θ ≤ 1-(2)/(d) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>θ</mml:mi> <mml:mo>≤</mml:mo> <mml:mn>1</mml:mn> <mml:mo>-</mml:mo> <mml:mfrac> <mml:mn>2</mml:mn> <mml:mi>d</mml:mi> </mml:mfrac> </mml:mrow> </mml:math> , one cannot, in general, expect convergence in the modulated energy notion of distance.