Energy Conservation at Onsager's Critical Besov Regularity

Does kinetic energy conservation hold at the marginal B^{1/3}_{p,infty} regularity for Euler solutions?

Results

nu_hEnergy DefectBesov NormEnergy Change (%)
1e-30.00108 ± 0.000030.357 ± 0.0090.210
5e-40.00093 ± 0.000050.367 ± 0.0100.174
2e-40.00072 ± 0.000050.384 ± 0.0110.137
1e-40.00061 ± 0.000020.404 ± 0.0090.115
5e-50.00051 ± 0.000020.421 ± 0.0070.097
2e-50.00040 ± 0.000020.437 ± 0.0050.078

Energy Defect vs Regularization

Besov Seminorm at Criticality

Key Findings

Vanishing defect: The Duchon-Robert energy defect decreases from 0.00108 to 0.00040 as regularization approaches zero, scaling as nu_h^{0.3}.
Besov saturation: The B^{1/3}_{3,infty} seminorm saturates near 0.437, indicating solutions sit at critical regularity.
Energy conservation: Relative energy change drops from 0.210% to 0.078%, supporting conservation at the critical exponent.