Taylor dispersion and phase mixing in the non-cutoff Boltzmann equation on the whole space (Q6581866)

The authors consider the Boltzmann equation for a distribution function \( F=F(t,x,v)\): \(\partial _{t}F+v\cdot \nabla F=\nu \mathcal{Q}(F,F)\), \(t\geq 0\), \(x,v\in \mathbb{R}^{d}\) (or \(\mathbb{T}^{d}\)), \(d\geq 2\), where \(1/\nu \geq 0\) is the Knudsen number or inverse collision frequency. The authors describe the long-time dynamics of this equation in the limit \(\nu \rightarrow 0\), the weakly collisional limit. The collision operator \( \mathcal{Q}\) accounting for elastic and binary collisions between the gas molecules is defined as \(\mathcal{Q}(f,g)(v)=\frac{1}{2}\int_{\mathbb{R} ^{d}}\int_{\mathbb{S}^{d-1}}B(v-v_{\ast },\sigma )(f^{\prime }g_{\ast }^{\prime }-fg_{\ast })dv_{\ast }d\sigma \), with \(v^{\prime }=\frac{ v+v_{\ast }}{2}+\frac{\left\vert v-v_{\ast }\right\vert }{2}\sigma \), \( v_{\ast }^{\prime }=\frac{v+v_{\ast }}{2}-\frac{\left\vert v-v_{\ast }\right\vert }{2}\sigma \), \(f^{\prime }(v)=f(v^{\prime })\), \(g_{\ast }^{\prime }=g(v_{\ast }^{\prime })\), \(f=f(v)\), \(g_{\ast }=g(v_{\ast })\). The non-negative kernel \(B=B(z,\sigma ):\mathbb{R}^{d}\times \mathbb{S} ^{d-1}\rightarrow \mathbb{R}\) depends only on \(\left\vert z\right\vert \) and the angle between the vectors \(z\) and \(\sigma \) and the authors take \( B(z,\sigma )=\Phi (\left\vert z\right\vert )b(\cos(\theta ))\), \(\cos(\theta )= \frac{z}{\left\vert z\right\vert }\), \(0\leq \theta \leq \frac{\pi }{2}\), where the kinetic factor \(\Phi \) is given by \(\Phi (\left\vert z\right\vert )=\left\vert z\right\vert ^{\gamma }\), and the collision angle contains a singularity \(b(\cos(\theta ))\approx K\theta ^{1-(d+2s)}\), as \(\theta \rightarrow 0^{+}\), \(0<s<1\). The parameter \(\gamma \) leads to three classes: soft potentials if \(-d\leq \gamma +2s\leq 0\) and \(\gamma >\max\{-d,-d/2-2s\}\), moderately soft potentials if \(-2s<\gamma <0\), and hard potentials if \( \gamma \geq 0\). \N\NIn the case \(\mathbb{R}^{d}\), the main result proves that for \(\gamma \) either soft, moderately soft, or hard, there exists \( \varepsilon _{0}>0\) (independent of \(\nu \)) such that if the initial datum \(f \) satisfies \(\sum_{\left\vert \alpha \right\vert +\beta \leq s\sigma }\left\Vert \left\langle v\right\rangle ^{M+M^{\prime }}\partial _{\alpha }^{\beta }f\right\Vert _{L_{x,v}^{2}}+\left\Vert \left\langle v\right\rangle ^{M+M^{\prime }}\partial _{\alpha }^{\beta }f_{in}\right\Vert _{L_{v}^{2}L_{x}^{1}}=\varepsilon \leq \varepsilon _{0}\), for integers \( M,M^{\prime }>d\), then a Taylor dispersion/enhanced dissipation estimate, a low-frequency decay estimate if \(\frac{2s}{1+2s}J>d\), for some integer \(J\), and an almost uniform Landau damping for the density hold. \N\NIn the case \( \mathbb{T}^{d}\), the main result proves that if the initial datum \(f_{in}\) satisfies \(\sum_{\left\vert \alpha \right\vert +\beta \leq s\sigma }\left\Vert \left\langle v\right\rangle ^{M+M^{\prime }}\partial _{\alpha }^{\beta }f_{in}\right\Vert _{L_{x,v}^{2}}=\varepsilon \leq \varepsilon _{0}\), then for all \(\nu \in (0,1)\) an enhanced dissipation estimate and for all \(\beta <\sigma -d\), a uniform Landau damping estimate hold. For the proofs, the authors introduce the linearized problem \(\partial _{t}f+v\cdot \nabla _{x}f+\nu \mathcal{L}f=0\), where \((\mathcal{L}f)(v)=-\frac{1}{2\sqrt{\mu }} \iint\limits_{\mathbb{R}^{d}\times \mathbb{S}^{d-1}}-B(v-v_{\ast },\sigma )\mu \mu _{\ast }(\frac{f^{\prime \prime }}{\sqrt{\mu ^{\prime }}}+\frac{ f_{\ast }^{\prime \prime }}{\sqrt{\mu _{\ast }^{\prime }}}-\frac{f}{\sqrt{ \mu }}+\frac{f_{\ast }^{\prime }}{\sqrt{\mu _{\ast }}})dv_{\ast }d\sigma \). \N\NThey apply the Fourier transform: \(\partial _{t}\widehat{f}+ikv\widehat{f} +\nu \mathcal{L}\widehat{f}=0\), and they distinguish between two regimes: \( \left\vert k\right\vert \gg \nu \) the enhanced dissipation regime, and \( \left\vert k\right\vert \ll \nu \) the Taylor dispersion regime. \NIn the enhanced dissipation regime, the authors introduce an energy functional which involves a natural dissipation term associated with the negative-definite contributions that come from the time derivative. They prove a monotonicity estimate for a weighted sum between the energy and the dissipation term, from which they derive a suitable polynomial decay estimate for the energy. \NIn the Taylor dispersion regime, the authors introduce the (non-closed) hydrodynamic system: \(\partial _{t}\rho +\nabla _{x}\cdot m=0\), \(\partial _{t}m+\nabla _{x}\rho =-2\nabla _{x}e-\nabla _{x}\cdot \Theta \lbrack (I-P)f]\), \(\partial _{t}e=-\frac{1}{3}\nabla _{x}\cdot m-\frac{1}{6}\nabla _{x}\cdot \Lambda \lbrack (I-P)f]\), where the high-order moment functions \(\Theta \lbrack g]=(\Theta _{ij}[g])_{d\times d}\) and \(\Lambda \lbrack g]=(\Lambda _{i}[g])_{d}\) are defined as \(\Theta _{ij}[g]=\left\langle (v_{i}v_{j}-1)\sqrt{\mu },g\right\rangle _{L_{v}^{2}}\), \(\Lambda _{i}[g]=\left\langle (\left\vert v\right\vert ^{2}-(d+2)v_{i}) \sqrt{\mu },g\right\rangle _{L_{v}^{2}}\), and \(P\) is the projection onto the kernel of the linearized operator \(\mathcal{L}\). They slightly modify the energy functional on which they prove linear decay estimates. \N\NThe main point of the proof in the case of \(\mathbb{R}^{d}\) is a bootstrap estimate from which the conclusion can be reached using a straightforward regularization argument.