Part IV: The nonlinear Cauchy problem

In the last lecture we introduced the lifted equation. We showed how the evolution of some quantities along the Landau equation flow can be understood by understanding the flow of ```their lift’’ along the lifted equation flow.

The main theorem then, is the following.

Theorem III. Suppose that \begin{align} \Lambda(\alpha) := \left ( \sup \frac{|r\alpha'(r)|}{2\alpha(r)}\right )^2 \leq \frac{11}{2} \end{align} Then for any solution of the lifted equation we have \begin{align} \frac{d}{dt}I(F(t)) \leq 0 \end{align} In particular, this will be the case if $\gamma(r) = cr^\gamma$ for $\gamma \in [-3,1]$.

Basic structure of the lifted equation

Now we focus solely on the equation \begin{align} \partial_t F & = Q(F) \text{ in } \mathbb{R}^{n}\times \mathbb{R}^n\times (0,\infty)\\ \
F_{\mid t = 0} & = f(v)f(w) \end{align}

The start of our analysis comes from working in the right coordinate system. Consider the following change of variables in $\mathbb{R}^{2n}$: $(v,w) \mapsto (z,r,\sigma)$,

\begin{align} z = \frac{1}{2}(v+w),\; r= \frac{1}{2}|v-w|,\; \sigma = \frac{v-w}{|v-w|} \end{align} We will also use the variable \begin{align} u = \frac{1}{2}(v-w) \in \mathbb{R}^n \end{align} Observe that \begin{align} v = z + u = z + r\sigma\\ \
w = z - u = z - r\sigma \end{align}

Proposition. \begin{align} Q(F)(v,w) = \alpha(r) \left (\Delta_{\mathbb{S}^{n-1}} \right )_\sigma F(z,r,\sigma) \end{align}

Proof. \begin{align} Q(F)(v,w) = \text{div}_{v-w}\left (\alpha a \nabla_{v-w}F \right ) \end{align} Expanding, \begin{align} Q(F)(v,w) & = \alpha \text{div}_{v-w}\left (a \nabla_{v-w}F \right ) + (\nabla_{v-w}\alpha,a \nabla_{v-w}F ) \\ & = \alpha \text{div}_{v-w}\left (a \nabla_{v-w}F \right ). \end{align} \begin{align} \text{div}_u(a(u)\nabla_u F) = (\Delta_{\mathbb{S}^{n-1}})_\sigma F \end{align}

This lemma gives us a clear picture of what the lifted equation really is. We see that a solution to the lifted equation is the same as a function \begin{align} F:\mathbb{R}^3\times (0,\infty) \times \mathbb{S}^2\times \times (0,\infty) \to \mathbb{R} \end{align} which solves for every $z$ and $r$ fixed the heat equation on the sphere with diffusivity constant $\alpha(r)$ \begin{align} \partial_t F(z,r,\sigma,t) = \alpha(r)(\Delta_{\mathbb{S}^{n-1}})_\sigma F(z,r,\sigma,t) \text{ in } \mathbb{S}^{n-1}\times (0,\infty) \end{align} In particular, we have a simple representation for $F(z,r,\sigma,t)$ in terms of $H(t,\sigma,\sigma’)$, the heat kernel on the sphere. We have \begin{align} F(z,r,\sigma,t) = \int_{\mathbb{S}^2}H(\alpha(r)t,\sigma,\sigma’)F_{\text{in}}(z,r,\sigma’)d\sigma’ \end{align}

The Fisher information in the $(z,r,\sigma)$ coordinates

It will be convenient to write the Fisher information directly in terms of $(z,r,\sigma)$. First of, we clearly have \begin{align} |\nabla_{v,w}F|^2 = |\nabla_z F|^2+|\nabla_u F|^2 \end{align} Then we express $|\nabla_uF|^2$ in terms of $r$ and $\sigma$, and arrive at

\begin{align} |\nabla_{v,w}F|^2 = |\nabla_zF|^2 +r^{-2}|\nabla_\sigma F|^2 + (\partial_rF)^2 \end{align}

It follows then that \begin{align} I(F) = \int_{0}^\infty\int_{\mathbb{R}^n}\left \{\int_{\mathbb{S}^{n-1}} |\nabla_z \ln F|^2 F d\sigma + \int_{\mathbb{S}^{n-1}} |\nabla_\sigma \ln F|^2Fd\sigma + \int_{\mathbb{S}^{n-1}} (\partial_r \ln F)^2Fd\sigma \right \}dzdr \end{align}

Maxwell molecules revisited

For Maxwell molecules, that is when $\alpha(r) \equiv 1$, we have the representation

\begin{align} F(z,r,\sigma,t) = \int_{\mathbb{S}^{n-1}}H(t,\sigma,\sigma’)F_{\text{in}}(z,r,\sigma’)d\sigma’ \end{align} From here it is immediate that \begin{align} \nabla_z F(z,r,\sigma,t) = \int_{\mathbb{S}^{n-1}}H(t,\sigma,\sigma’)\nabla_z F_{\text{in}}(z,r,\sigma’)d\sigma’ \end{align} This, by convexity yields for every $z,r$ the inequality \begin{align} \int_{\mathbb{S}^{n-1}}|\nabla_z \ln F|^2F(z,r,\sigma,t)d\sigma \leq \int_{\mathbb{S}^{n-1}}\int_{\mathbb{S}^{n-1}}H(t,\sigma,\sigma’)|\nabla_z \ln F_{\text{in}}|^2F_{\text{in}}(z,r,\sigma’)d\sigma’d\sigma \end{align} Now exchanging the integration order and using that \begin{align} \int_{\mathbb{S}^{n-1}}H(t,\sigma,\sigma’)d\sigma = 1 \;\;\forall\; t>0,\sigma’\in\mathbb{S}^{n-1}, \end{align} we have \begin{align} \int_{\mathbb{S}^{n-1}}\int_{\mathbb{S}^{n-1}}H(t,\sigma,\sigma’)|\nabla_z \ln F_{\text{in}}|^2F_{\text{in}}(z,r,\sigma’)d\sigma’d\sigma = \int_{\mathbb{S}^{n-1}}|\nabla_z \ln F_{\text{in}}|^2F_{\text{in}}(z,r,\sigma’)\;d\sigma’ \end{align} Therefore, for all $t>0$, and $z\in\mathbb{R}^n$, $r>0$, we have \begin{align} \int_{\mathbb{S}^{n-1}}|\nabla_z \ln F|^2F(z,r,\sigma,t)d\sigma \leq \int_{\mathbb{S}^{n-1}}|\nabla_z \ln F_{\text{in}}|^2F_{\text{in}}(z,r,\sigma’)\;d\sigma’ \end{align} Arguing similarly with $\sigma$ and $r>0$, and adding the resulting inequalities we conclude that \begin{align} I(F(t)) \leq I(F_{\text{in}})\;\;t>0. \end{align}

The case of non-constant $\alpha(r)$

The argument we just went through hinges on the invariance of the equation with respect to $r$, namely, if $F(z,r,\sigma)$ solves the family of equations, then for any $h>0$ we have that $\tilde F(z,r,\sigma) := F(z,r+h,\sigma)$ also solves the equation.

\begin{align} \frac{d}{dt}I(F) \leq -\int_{\mathbb{R}^n}\int_0^\infty \left \{ \int_{\mathbb{S}^{n-1}}\alpha \Gamma_2(\ln F) F - r^2((\sqrt{\alpha}’))^2|\nabla_\sigma \ln F|^2F\;d\sigma \right \}dzdr \end{align}

The Bakry-Emery condition

The inequality we need is \begin{align} r^2((\sqrt{\alpha})’)^2\int_{\mathbb{S}^{n-1}}|\nabla_\sigma \ln f(\sigma)|^2f(\sigma)\sigma \leq \alpha(r) \int_{\mathbb{S}^{n-1}} \Gamma_2(\ln f(\sigma)) f(\sigma)d\sigma \end{align} If we rearrange this, \begin{align} \frac{r^2\alpha’(r)^2}{4\alpha(r)^2}\int_{\mathbb{S}^{n-1}}|\nabla_\sigma \ln f(\sigma)|^2f(\sigma)\sigma \leq \int_{\mathbb{S}^{n-1}} \Gamma_2(\ln f(\sigma)) f(\sigma)d\sigma \end{align} Now, we write \begin{align} \Lambda(\alpha) := \sup_{r>0}\frac{r^2\alpha’(r)^2}{4\alpha(r)^2} = \frac{1}{4}\left (\sup_{r>0} \frac{r\alpha’(r)}{\alpha(r)}\right )^2 \end{align} What we need to prove then is \begin{align} \Lambda(\alpha)\int_{\mathbb{S}^{n-1}}|\nabla_\sigma \ln f(\sigma)|^2f(\sigma)\sigma \leq \int_{\mathbb{S}^{n-1}} \Gamma_2(\ln f(\sigma)) f(\sigma)d\sigma \end{align}

Now we specialize to the case $n=3$.