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Decay estimates for large velocities in the Boltzmann equation without cutoff
Analysis of PDEs · Submitted 17 Apr 2018
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We establish quantitative estimates of \emph{propagation} in time of "pointwise polynomial moments", i.e. $\sup\_{x,v} f(t,x,v) (1+|v|)^q$, $q \ge 0$.
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We consider solutions $f=f(t,x,v)$ to the full (spatially inhomogeneous) Boltzmann equation with periodic spatial conditions $x \in \mathbb T^d$, for hard and moderately soft potentials \emph{without the angular cutoff assumption}, and under the \emph{a priori} assumption that the main hydrodynamic fields, namely the local mass $\int\_v f(t,x,v)$ and local energy $\int\_v f(t,x,v)|v|^2$ and local entropy $\int\_v f(t,x,v) \ln f(t,x,v)$, are controlled along time. We establish quantitative estimates of \emph{propagation} in time of "pointwise polynomial moments", i.e. $\sup\_{x,v} f(t,x,v) (1+|v|)^q$, $q \ge 0$. In the case of hard potentials, we also prove \emph{appearance} of these moments for all $q \ge 0$. In the case of moderately soft potentials we prove the \emph{appearance} of low-order pointwise moments.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We consider solutions $f=f(t,x,v)$ to the full (spatially inhomogeneous) Boltzmann equation with periodic spatial conditions $x \in \mathbb T^d$, for hard and moderately soft potentials \emph{without the angular cutoff assumption}, and under the \emph{a priori} assumption that the main hydrodynamic fields, namely the local mass $\int\_v f(t,x,v)$ and local energy $\int\_v f(t,x,v)|v|^2$ and local entropy $\int\_v f(t,x,v) \ln f(t,x,v)$, are controlled along time. | background 0.71 | 0.41 | 0.05 | asserted 0.54 |
| s1 | We establish quantitative estimates of \emph{propagation} in time of "pointwise polynomial moments", i.e. $\sup\_{x,v} f(t,x,v) (1+|v|)^q$, $q \ge 0$. | result 0.78 | 0.96 | 0.57 | asserted 1.00 |
| s2 | In the case of hard potentials, we also prove \emph{appearance} of these moments for all $q \ge 0$. | result 0.98 | 0.95 | 0.21 | asserted 1.00 |
| s3 | In the case of moderately soft potentials we prove the \emph{appearance} of low-order pointwise moments. | result 0.97 | 0.96 | 0.13 | asserted 1.00 |
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The paper
Original abstract
We consider solutions $f=f(t,x,v)$ to the full (spatially inhomogeneous) Boltzmann equation with periodic spatial conditions $x \in \mathbb T^d$, for hard and moderately soft potentials \emph{without the angular cutoff assumption}, and under the \emph{a priori} assumption that the main hydrodynamic fields, namely the local mass $\int\_v f(t,x,v)$ and local energy $\int\_v f(t,x,v)|v|^2$ and local entropy $\int\_v f(t,x,v) \ln f(t,x,v)$, are controlled along time. We establish quantitative estimates of \emph{propagation} in time of "pointwise polynomial moments", i.e. $\sup\_{x,v} f(t,x,v) (1+|v|)^q$, $q \ge 0$. In the case of hard potentials, we also prove \emph{appearance} of these moments for all $q \ge 0$. In the case of moderately soft potentials we prove the \emph{appearance} of low-order pointwise moments.
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References
How the literature uses it
9 arXiv papers cite this in 19 sentences
Papers citing it · 6 sentences without a confident label
Builds on it 1 paper
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The present article extends the result of [ref] , specifically the self-generating upper bounds, to the case $γ+ 2s > 2$.
Velocity Decay Estimates for Boltzmann equation with hard potentials · Introduction · 1.00
Uses its method or tool 3 papers
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The proof of Lemma [ref] combines, at its core, the spreading argument of the cut-off case that goes back to Carleman [ref] , used here in the form developed in [ref] , coercivity estimates on the collision operator at grazing collisions developed in [ref] , and finally a barrier argument similar to [ref] .
Gaussian lower bounds for the Boltzmann equation without cut-off · Spreading lemma around zero · 0.99
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On a technical level, we make heavy use of the decomposition of $Q(f,f)$ into a fractional diffusion operator $Q_s(f,f)$ and a lower-order term $Q_ns(f,f)$ (see Section [ref]), taking inspiration from [ref] and using some of their estimates out of the box.
Local well-posedness of the Boltzmann equation with polynomially decaying initial data · Proof ideas · 1.00
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(This is Theorem [ref].) The proof of this estimate is based on the method of [ref] .
Velocity Decay Estimates for Boltzmann equation with hard potentials · Proof strategy · 0.98
Cites it as support 1 paper
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The fact that our initial data is allowed to decay merely polynomially is crucial in establishing <Ref> because polynomial decay in $v$ in the initial data is propagated forward in time (see [ref] ) but it is not currently known if the same is true for Gaussian decay.
Local well-posedness of the Boltzmann equation with polynomially decaying initial data · Introduction · 0.98
Cites it as background 7 papers
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* In [ref] , working in the case of moderately soft potentials ($γ∈(-2,2)$ and $γ+2s∈[0,2]$), Imbert-Mouhot-Silvestre showed that pointwise polynomial decay in $v$ is propagated forward (i.e. if $f_in(x,v) ≲|v|^-q$, then $f(t,x,v) ≲|v|^-q$) if $γ∈(-2,0]$, and in the case $γ∈(0,2)$, this decay is self-generating (i.e. $f(t,x,v) ≲t^-p|v|^-q$ some $p,q>0$, for arbitrary initial data).
Velocity Decay Estimates for Boltzmann equation with hard potentials · Introduction · 0.85
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Previous results of conditional regularity include, for the Boltzmann equation and under the conditions above, the proof of $L^∞$ bounds in [ref] , the proof of a weak Harnack inequality and Hölder continuity in [ref] , the proof of polynomially decaying upper bounds in [ref] , and the proof of Schauder estimates to bootstrap higher regularity estimates in [ref] .
Gaussian lower bounds for the Boltzmann equation without cut-off · The program of conditional regularity · 0.80
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In the next subsections, we present the progresses made so far in the case of moderately soft potentials: the estimate in $L^∞$ for $t>0$ was obtained in [ref] , the local Hölder regularity in [ref] , and finally the polynomial pointwise decay estimates in [ref] .
De Giorgi-Nash-Moser and H{\"o}rmander theories: new interplay · Previous works and a conjecture · 0.93
What it cites · 5 sentences without a confident label
Builds on 2 papers
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The method was extended in [ref] , and in [ref] (using estimates from [ref] ) where the authors prove exponential (but not exactly Gaussian) upper bound for the space homogeneous non-cutoff Boltzmann equation.
On pointwise exponentially weighted estimates for the Boltzmann equation · Pointwise decay (weighted $L^∞$ estimates) · 0.82
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The method was extended in [ref] , and in [ref] (using estimates from [ref] ) where the authors prove exponential (but not exactly Gaussian) upper bound for the space homogeneous non-cutoff Boltzmann equation.
Exponentially-tailed regularity and time asymptotic for the homogeneous Boltzmann equation · Pointwise decay (weighted $L^∞$ estimates) · 0.82
Uses the method or tool of 1 paper
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In future work, we intend to address the forth step using the Schauder estimates from [ref] .
The Schauder estimate for kinetic integral equations · The Boltzmann equation with long-range interactions · 0.91
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