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  1. Consider a one-dimensional simple small-amplitude solution (ϱ(bkg), v1(bkg)) to the isentropic compressible Euler equations which has smooth initial data, coincides with a constant state outside a compact set, and forms a shock in finite time. Viewing (ϱ(bkg), v1(bkg)) as a plane-symmetric solution to the full compressible Euler equations in three dimensions, we prove that the shock-formation mechanism for the solution (ϱ(bkg), v1(bkg)) is stable against all sufficiently small and compactly supported perturbations. In particular, these perturbations are allowed to break the symmetry and have nontrivial vorticity and variable entropy. Our approach reveals the full structure of the set of blowup-points at the first singular time: within the constant-time hypersurface of first blowup, the solution’s first-order Cartesian coordinate partial derivatives blow up precisely on the zero level set of a function that measures the inverse foliation density of a family of characteristic hypersurfaces. Moreover, relative to a set of geometric coordinates constructed out of an acoustic eikonal function, the fluid solution and the inverse foliation density function remain smooth up to the shock; the blowup of the solution’s Cartesian coordinate partial derivatives is caused by a degeneracy between the geometric and Cartesian coordinates, signified by the vanishing of the inverse foliation density (i.e., the intersection of the characteristics). 
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  2. Consider the Vlasov–Poisson–Landau system with Coulomb potential in the weakly collisional regime on a33-torus, i.e.∂<#comment/>tF(t,x,v)+vi∂<#comment/>xiF(t,x,v)+Ei(t,x)∂<#comment/>viF(t,x,v)=ν<#comment/>Q(F,F)(t,x,v),E(t,x)=∇<#comment/>Δ<#comment/>−<#comment/>1(∫<#comment/>R3F(t,x,v)dv−<#comment/>∫<#comment/>−<#comment/>T3∫<#comment/>R3F(t,x,v)dvdx),\begin{align*} \partial _t F(t,x,v) + v_i \partial _{x_i} F(t,x,v) + E_i(t,x) \partial _{v_i} F(t,x,v) = \nu Q(F,F)(t,x,v),\\ E(t,x) = \nabla \Delta ^{-1} (\int _{\mathbb R^3} F(t,x,v)\, \mathrm {d} v - {{\int }\llap {-}}_{\mathbb T^3} \int _{\mathbb R^3} F(t,x,v)\, \mathrm {d} v \, \mathrm {d} x), \end{align*}withν<#comment/>≪<#comment/>1\nu \ll 1. We prove that forϵ<#comment/>>0\epsilon >0sufficiently small (but independent ofν<#comment/>\nu), initial data which areO(ϵ<#comment/>ν<#comment/>1/3)O(\epsilon \nu ^{1/3})-Sobolev space perturbations from the global Maxwellians lead to global-in-time solutions which converge to the global Maxwellians ast→<#comment/>∞<#comment/>t\to \infty. The solutions exhibit uniform-in-ν<#comment/>\nuLandau damping and enhanced dissipation.

    Our main result is analogous to an earlier result of Bedrossian for the Vlasov–Poisson–Fokker–Planck equation with the same threshold. However, unlike in the Fokker–Planck case, the linear operator cannot be inverted explicitly due to the complexity of the Landau collision operator. For this reason, we develop an energy-based framework, which combines Guo’s weighted energy method with the hypocoercive energy method and the commuting vector field method. The proof also relies on pointwise resolvent estimates for the linearized density equation.

     
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  3. Consider the linear transport equation in 1D under an external confining potential \begin{document}$ \Phi $\end{document}:

    For \begin{document}$ \Phi = \frac {x^2}2 + \frac { \varepsilon x^4}2 $\end{document} (with \begin{document}$ \varepsilon >0 $\end{document} small), we prove phase mixing and quantitative decay estimates for \begin{document}$ {\partial}_t \varphi : = - \Delta^{-1} \int_{ \mathbb{R}} {\partial}_t f \, \mathrm{d} v $\end{document}, with an inverse polynomial decay rate \begin{document}$ O({\langle} t{\rangle}^{-2}) $\end{document}. In the proof, we develop a commuting vector field approach, suitably adapted to this setting. We will explain why we hope this is relevant for the nonlinear stability of the zero solution for the Vlasov–Poisson system in \begin{document}$ 1 $\end{document}D under the external potential \begin{document}$ \Phi $\end{document}.

     
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