Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.32296

Optimal time decay of the Boltzmann equation with cutoff soft potentials in the whole space

Yuanjie Lei·Xiongju Wei·Tong Yang·Huijiang Zhao
View original source ↗Read source document ↗Export metadata ↓

Abstract

Since Strain's work \cite{Strain-2012-KRM} on almost optimal time decay rates for the microscopic part of perturbative classical solutions to the Boltzmann system with soft potentials via the time-velocity splitting method, how to obtain the optimal decay of the microscopic part has remained unsolved. In this paper, we resolve this problem by developing a nonlinear energy estimate method different from the time-velocity splitting approach. The proof relies on a combination of a hierarchy of time-weighted energy estimates, the damping factor \(e^{-ν(v)(t-τ)}\), and a dyadic decomposition of velocity space.

Explore related subjects

math.AP

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Yuanjie Lei, Xiongju Wei, Tong Yang, Huijiang Zhao. 2026-10-23. Optimal time decay of the Boltzmann equation with cutoff soft potentials in the whole space. https://arxiv.org/abs/2610.32296

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

KEEP EXPLORING

Related papers

arXiv2026

Sharp Poincaré inequalities for the logarithmic smoothness of the Gauss map on hypersurfaces

We establish sharp lower bounds for two geometric functionals which evaluate the global variation of the Gauss map on the boundary of bounded open sets $Ω\subset\mathbb{R}^d$. These functionals arise as the endpoint case $s=0$ of the Gagliardo seminorm in $W^{s,p}(\partialΩ)$ of the normal vector field, for $p\in\{1,2\}$. We also show that balls are the unique minimizers. In addition, some applications to the fractional setting $s>0$ are given.

math.AP↗
arXiv2026

Upper estimates of the lifespan for the fractional wave equations with time-dependent damping and a power nonlinearity of subcritical and critical Fujita exponent

In this paper, we investigate the Cauchy problem of the fractional wave equation with time-dependent damping and the source nonlinearity $f(u)\approx |u|^p$: $$ \begin{cases} \partial_t^2u(t,x)+(-Δ)^{σ/2} u(t,x)+b(t) \partial_t u(t,x) =f(u(t,x)),\ &(t,x)\ \in [0,T)\times \mathbb{R}^N,\\ u(0,x)=u_0(x),\ \partial_tu(0,x)=u_1(x),\ &x\ \in\ \mathbb{R}^N, \end{cases} $$ where the damping coefficient behaves as $b(t)\approx (1+t)^{-β}$. We first establish local existence and uniqueness of mild and strong solutions in weighted Sobolev spaces by deriving weighted Sobolev estimates for the linear propagators and overcoming the spatial-weight coupling in the nonlinearity through the Bessel potential estimates. Then, in the subcritical and critical cases $1<p\leq p_c:=1+\frac σN$, we derive the upper estimates of the lifespan for the fractional Laplacian with $0<σ<2$ and time-dependent damping $β\in [-1, 1)$ via the framework of ordinary differential inequalities. The blow-up results, together with the global existence in the supercritical case obtained in \cite{LI}, show that the critical exponent for the fractional wave equation is $p_c=1+\fracσ{N}$ for $0<σ<2$. Moreover, combining with the lower estimate of lifespan derived in \cite{LI}, we conclude that the estimate in this paper is sharp. Note that our result for the critical case is completely new even in the classical case $b(t)=1$. We also consider the case of $β=1$, and obtain the upper estimate of the lifespan.

math.AP↗
arXiv2026

On the stationary Navier-Stokes equations in distorted pipes under energy-stable outflow boundary conditions

The steady motion of a viscous incompressible fluid in distorted pipes, of finite length, is modeled through the Navier-Stokes equations with mixed boundary conditions: the inflow is given by an arbitrary member of the Lions-Magenes class with positive influx, and the fluid motion is subject to a directional do-nothing boundary condition on the outlet, together with the standard no-slip assumption on the remaining walls of the domain. Existence of a weak solution to such Navier-Stokes system is proved without any restriction on the data (that is, inlet velocity and external force) by means of the Leray-Schauder Principle, in which the required a priori estimate is obtained by a contradiction argument that employs Bernoulli's law for solutions of the stationary Euler equations, as well as some properties of harmonic divergence-free vector fields. Under a suitable smallness assumption on the data, we also prove the unique solvability of the boundary-value problem.

math.AP↗