Energy equality and renormalization of Navier-Stokes-Vlasov equations
Abstract
We study weak solutions of the incompressible and the compressible Navier--Stokes--Vlasov systems on a bounded domain, where a viscous fluid is coupled to a dispersed particle phase through a drag force; the fluid obeys the no-slip boundary condition and the particles are subject to specular reflection. For solutions with Besov regularity, we show that the Vlasov equation is renormalizable in the sense of DiPerna--Lions, and we give regularity thresholds for the validity of the global energy equality. If the distribution function f belongs to Bpα,∞ in space and velocity with time integrability p, and the fluid velocity u belongs to Bqβ,∞ jointly in space and time with integrability q, then the energy equality holds provided α+β>1 and β exceeds an explicit bound depending on p, q and the dimension, under a natural integrability assumption on the pressure; an analogous criterion holds for the compressible system when the density is bounded from above and below. The condition αp>1 imposed on f is not merely technical: it is exactly the range in which the macroscopic momentum flux possesses a boundary trace, so that the specular reflection condition passes to the macroscopic energy balance. When the particle phase is absent, our thresholds coincide with the criteria β>1/2+1/q and β>3/q obtained by Yu for the incompressible Navier--Stokes equations alone. The proofs combine a boundary cutoff argument with commutator estimates based on a scale-localized Besov modulus, together with a boundary-strip analysis.
Related
- On energy equality of the 3D anisotropic Navier-Stokes equations
- Existence of a weak solution to the Navier-Stokes equations enjoying the energy equality
- Global existence of smooth solutions to the Navier-Stokes equations for initial data with infinite energy
- Energy Decay Induced by Localized Coercivity in the 3D Navier--Stokes Equations
- A second-order energy-stable low-regularity integrator for the Navier--Stokes equations
BibTeX
@misc{teng2026energy,
title = {{Energy equality and renormalization of Navier-Stokes-Vlasov equations}},
author = {Hanyu Teng},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.36579},
eprint = {2610.36579},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.36579},
}
RIS
TY - UNPB
TI - Energy equality and renormalization of Navier-Stokes-Vlasov equations
AU - Hanyu Teng
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.36579
UR - https://arxiv.org/abs/2610.36579
AB - We study weak solutions of the incompressible and the compressible Navier--Stokes--Vlasov systems on a bounded domain, where a viscous fluid is coupled to a dispersed particle phase through a drag force; the fluid obeys the no-slip boundary condition and the particles are subject to specular reflection. For solutions with Besov regularity, we show that the Vlasov equation is renormalizable in the sense of DiPerna--Lions, and we give regularity thresholds for the validity of the global energy equality. If the distribution function f belongs to B_p^α,∞ in space and velocity with time integrability p, and the fluid velocity u belongs to B_q^β,∞ jointly in space and time with integrability q, then the energy equality holds provided α+β>1 and β exceeds an explicit bound depending on p, q and the dimension, under a natural integrability assumption on the pressure; an analogous criterion holds for the compressible system when the density is bounded from above and below. The condition αp>1 imposed on f is not merely technical: it is exactly the range in which the macroscopic momentum flux possesses a boundary trace, so that the specular reflection condition passes to the macroscopic energy balance. When the particle phase is absent, our thresholds coincide with the criteria β>1/2+1/q and β>3/q obtained by Yu for the incompressible Navier--Stokes equations alone. The proofs combine a boundary cutoff argument with commutator estimates based on a scale-localized Besov modulus, together with a boundary-strip analysis.
ER -