Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.36579

Energy equality and renormalization of Navier-Stokes-Vlasov equations

Hanyu Teng
View original source ↗Read source document ↗Export metadata ↓

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 $B_p^{α,\infty}$ in space and velocity with time integrability $p$, and the fluid velocity $\mathbf{u}$ belongs to $B_q^{β,\infty}$ 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 $β>\frac12+\frac1q$ and $β>\frac3q$ 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.

Explore related subjects

math.AP

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Hanyu Teng. 2026-10-27. Energy equality and renormalization of Navier-Stokes-Vlasov equations. https://arxiv.org/abs/2610.36579

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Local rigidity of conformally Euclidean metrics for the anisotropic Helmholtz equation at fixed energy

We prove local rigidity of smooth conformally Euclidean metrics $g_0$ on smooth compact domains $M\subset\mathbb{R}^n$, $n\ge3$, for the anisotropic Helmholtz equation $(-Δ_g -k^2)u =0$ at a fixed energy $k^2>0$ which is not a Dirichlet eigenvalue of $-Δ_{g_0}$ in $M$. Any smooth metric sufficiently close to $g_0$ in $H^s(M)$, with integer $s>n/2+1$, and having the same Dirichlet-to-Neumann map, is its pullback by a smooth diffeomorphism fixing the boundary pointwise. The proof uses a Helmholtz embedding to fix the gauge and a compactness argument together with a characterization of the kernel of the linearization.

math.AP↗
arXiv2026

A self-contained proof of the variational characterization of the principal eigenvalue of $-Δ+q$ under general Robin boundary conditions of non-classical type

In this paper, we are going to deliver a completely self-contained proof of the variational characterization of the principal eigenvalue of $-Δ+q$ in $Ω$ under Robin boundary conditions of non-classical type, $\frac{\partial u}{\partial n}+β(x)u=0$ on $\partialΩ$, where $β(x)$ is allowed to change sign on $\partialΩ$. The classical case occurs in the very special case when $β\geq 0$ on $\partialΩ$.

math.AP↗
arXiv2026

Large data global strong solutions to the quantum Navier-Stokes equations with degenerate viscosity

This work is devoted to study the Cauchy problem for the quantum Navier-Stokes equations in $\mathbb R^N$, $N=2,3$, with density dependent viscosities $μ(ρ)=ρ,λ(ρ)=0$, quantum capillarity $κ(ρ)=1/ρ$, which was derived by Dunn and Serrin (see Arch. Ration. Mech. Anal. 88(2):95-133, 1985) as a phase transition model. For arbitrarily large initial data satisfying suitable regularity and decay assumptions, with density positive at every finite point and vanishing at infinity, we prove the existence and uniqueness of global strong solutions without symmetry assumptions. The admissible pressure range is $γ\ge 1$ in two dimensions and $1\leγ< 8/3$ in three dimensions. On every finite time interval, the density remains pointwise comparable to its initial profile, and both the effective velocity and the logarithmic density gradient remain bounded. The central estimates require no uniform positive lower bound for the density. When $γ>1$, a suitable combination of the density and effective velocity equations allows the maximum principle to yield an $L^\infty$ bound for the effective velocity using only an upper bound for the density. We then introduce the relative density with respect to a fixed profile determined by the initial data. In the isothermal case $γ=1$, we combine estimates for this relative density with a scalar inequality involving the effective velocity and the logarithmic density to close the coupled estimates and obtain the effective velocity bound. Combining the relative-density bounds with weighted gradient estimates and localized De Giorgi iteration, we control the logarithmic density gradient and close the higher-order estimates.

math.AP↗