Uniform momentum-derivative control for nonlinear characteristics

Establish uniform boundedness in the spatial and momentum variables of $t^{-1}\nabla_v\mathcal{X}(0,t,x,v)$ for the nonlinear characteristics of the Vlasov–wave system, so that the change of variables $y(v)=\mathcal{X}(0,t,x,v)$ yields the required $t^{-3}$ decay of velocity averages.

Background

To derive optimal decay for momentum averages, the paper considers using the initial-position map of the nonlinear characteristics as a change of variables. This approach would require uniform control of the momentum derivative of the characteristic map after division by time. The authors explicitly state that they are unable to prove this bound, and consequently use a dynamical logarithmic correction instead.

References

However, consistently with the discussion in Sections \ref{subsubsec0}--\ref{subsubsec1}, we are unable to prove uniform boundedness in $(x,v)$ of $t{-1}\nabla_v \mathcal{X} (0,t,x,v)$. This prevents us from deriving the $t{-3}$ decay of the integral in dernierlabel through the change of variables $y(v)=\mathcal{X} (0,t,x,v)$.

dernierlabel:

∫Rv3∣h(t,x,v)∣dv≲∫Rv3dvω(t,x,v)sup⁡w∈Rv3ω(t,x,w)∣h(t,x,w)∣≲1⟨t+∣x∣⟩3sup⁡w∈Rv3ω(t,x,w)∣h(t,x,w)∣,\int_{R^3_v} \big|h(t,x,v) \big| d v \lesssim \int_{R^3_v} \frac{d v}{\pmb{\omega}(t,x,v)} \sup_{w \in R^3_v}\pmb{\omega}(t,x,w) \big|h(t,x,w) \big| \lesssim \frac{1}{\langle t+|x| \rangle^3} \sup_{w \in R^3_v} \pmb{\omega}(t,x,w) \big|h(t,x,w) \big|,

— Global dynamics of Vlasov--wave systems without the null condition under exponential momentum decay  (2608.27053 - Bigorgne, 27 Aug 2026) in Section 1.2, “Deriving optimal decay estimates,” discussion preceding the construction of the dynamical correction