A Two-Residual Compactness Framework for Suitable Navier–Stokes Limits
- 1School of Mathematical and Statistical Sciences, North-West University, Vanderbijlpark Campus, 1174 Hendrick Van Eck Boulevard, Vanderbijlpark 1900, South Africa
Abstract
We establish a compactness theorem for three-dimensional incompressible Navier–Stokes approximations with varying regularization mechanisms. Equation residuals are measured in a negative space compatible with the velocity, pressure, and convection, whereas errors in the local energy balance are represented by separate vanishing functionals. This separation covers residual stresses converging only in L3/2 and avoids an unjustified force–energy pairing. Uniform energy, normalized-pressure, and equation residual bounds yield a translation modulus whose constants are independent of the regulator and strong local convergence in every space-time Lebesgue class below 10/3, including the cubic class needed for the nonlinear energy flux. We prove stability of suitability and an exact identity relating the limiting local energy defect, the weak limit of approximate defects, and dissipation concentration. For a transport-mollified Friedrichs scheme on the three-torus, every admissibility condition is verified without invoking the abstract compactness theorem. Finally, classical one-scale epsilon regularity gives a fixed-scale lower bound for approximants converging to a singular suitable limit. This last conclusion is a consequence of the framework rather than a new epsilon regularity theorem. No interchange of regulator and shrinking-scale limits is asserted.
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BibTeX
@article{noutchie2026tworesidual,
title = {{A Two-Residual Compactness Framework for Suitable Navier–Stokes Limits}},
author = {Suares Clovis Oukouomi Noutchie},
journal = {Mathematics},
year = {2026},
month = oct,
doi = {10.3390/math14203787},
url = {https://www.mdpi.com/2227-7390/14/20/3787},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - A Two-Residual Compactness Framework for Suitable Navier–Stokes Limits
AU - Suares Clovis Oukouomi Noutchie
PY - 2026
DA - 2026/10/19
JO - Mathematics
DO - 10.3390/math14203787
UR - https://www.mdpi.com/2227-7390/14/20/3787
AB - We establish a compactness theorem for three-dimensional incompressible Navier–Stokes approximations with varying regularization mechanisms. Equation residuals are measured in a negative space compatible with the velocity, pressure, and convection, whereas errors in the local energy balance are represented by separate vanishing functionals. This separation covers residual stresses converging only in L3/2 and avoids an unjustified force–energy pairing. Uniform energy, normalized-pressure, and equation residual bounds yield a translation modulus whose constants are independent of the regulator and strong local convergence in every space-time Lebesgue class below 10/3, including the cubic class needed for the nonlinear energy flux. We prove stability of suitability and an exact identity relating the limiting local energy defect, the weak limit of approximate defects, and dissipation concentration. For a transport-mollified Friedrichs scheme on the three-torus, every admissibility condition is verified without invoking the abstract compactness theorem. Finally, classical one-scale epsilon regularity gives a fixed-scale lower bound for approximants converging to a singular suitable limit. This last conclusion is a consequence of the framework rather than a new epsilon regularity theorem. No interchange of regulator and shrinking-scale limits is asserted.
SN - 2227-7390
ER -