(You may see the rest of the lecture notes here).

Part I: Elliptic operators, local or not.

In the previous lecture we learned that

\begin{align} \lim \limits_{s\to 1^-} \text{P.V.} \int_{\mathbb{R}^n}c_{n,s} \frac{u(y)-u(x)}{|x-y|^{n+2s}}dy = \Delta u(x) \end{align}

More generally, one may consider families of kernels $K_\varepsilon(x,y)$ where

1. $K_\varepsilon(x,y)\geq 0$ for a.e. $(x,y) \in \mathbb{R}^{2n}$
2. $K_\varepsilon(x,y) \to 0$ as $\varepsilon\to 0^+$ uniformly in compact subsets of $\{(x,y) \in \mathbb{R}^{2n} \mid x\neq y\}$.

3. For a.e. $x$ we have,

   \[\lim \limits_{\varepsilon \to 0} \int_{\mathbb{R}^{n}}|x-y|^2K_\varepsilon(x,y)dy >0\]

and study the limit as $\varepsilon \to 0^+$ of the operators

\begin{align} L_\varepsilon u(x) = \text{P.V.}\int_{\mathbb{R}^n}(u(y)-u(x))K_\varepsilon(x,y)dy \end{align}

Lemma. Suppose that $A(x)$, $A:\mathbb{R}^n \mapsto \mathbb{R}^{n\times n}_{\text{Sym}}$ and $b:\mathbb{R}^n \to \mathbb{R}^n$ and both functions differentiable in the parameter $x$, and suppose further that $A(x)\geq 0$ for every $x$,. Then, there is a family of kernels $K_\varepsilon(x,y)$ such that the following holds for any $u \in L^\infty(\mathbb{R}^n)$ such that $u$ is $C^2$ in a neighborhood of the point $x$, \begin{align} \lim \limits_{\varepsilon\to 0}L_\varepsilon u(x) = \text{div} \left ( A(x)\nabla u(x)+ u(x)b(x)\right ) \end{align}

This sort of thing is what happens in what is known as the grazing limit. In fact, we will now study a local limit of a specific family of nonlocal operators to arrive at the Landau collision operator. This is a limit of the Boltzmann collision operator in the asymptotic regime where the so-called grazing collisions dominate.

Grazing limits and the Landau equation

Let us recall from Lecture 1 the Boltzmann collisional integral,

\begin{align} \int_{\mathbb{R}^3}\int_{\mathbb{S}^2}(f(v_\sigma)f(w_\sigma)-f(v)f(w))B(v-w,\sigma)d\sigma dw. \end{align}

We learned of an important family of kernels, those given by

\begin{align} B(v-w,\sigma) = b(\cos(\theta))|v-w|^\gamma. \end{align}

We also noted that (as explained by Maxwell) these kernels correspond to particles interacting via an inverse power potential $\phi(r) = r^{-(s-1)}$. In this case the $s$,$\gamma$, and $b$ are related by

\[\gamma = \frac{s-5}{s-1},\;\ b((\cos(\theta))) \approx |\theta|^{-2-\frac{2}{s-1}}\]

The singularity in $b(\cdot)$ is too strong when $s=2$, and the collisional integral is not finite for any $f$ save for constants. The problem is that as $s\to 2$ nearly all of the ‘‘mass’’ in $b(\cdot)$ accumulates near $\theta = 0$. That is, the collisions that are ‘‘grazing’’ are the ones contributing the most to the collisional integral.

In light of this, it makes sense to consider a regime where we have families of kernels where the corresponding collision operators are well defined but which are increasingly concentrated along grazing collisions. More specifically, considering the grazing limit amounts to taking a family $\{b_\varepsilon(\cdot)\}_\varepsilon$ such that the respective collision integral $Q_\varepsilon(f,f)$ is well defined for all regular $f$, and such that as $\varepsilon \to 0$ the functions $b_\varepsilon(\cos(\theta))$ become concentrated around $\theta = 0$.

That is, we have

\begin{align} Q_\varepsilon(f,f) = \int_{\mathbb{R}^3}\int_{\mathbb{S}^2}(f(v_\sigma)f(w_\sigma)-f(v)f(w))|v-w|^{\gamma}b_\varepsilon(\sigma \cdot \frac{v-w}{|v-w|}) d\sigma dw. \end{align}

Now, recall that in $Q_\varepsilon(f,f)$ above the velocities $v_\sigma,w_\sigma$ are given in terms of $v,w,$ and $\sigma$ by

\begin{align} v_\sigma = \frac{1}{2}(v+w)+\frac{1}{2}|v-w|\sigma,\;\;w_\sigma = \frac{1}{2}(v+w)-\frac{1}{2}|v-w|\sigma \end{align}

That $b_\varepsilon(\cos(\theta))$ is mostly concentrated around $\theta=0$ is the same as saying that $\sigma$ and $v-w$ are almost parallel, and therefore

\[v_\sigma \approx v,\; w_\sigma \approx w\]

In other words, $Q_\varepsilon(f,f)$ is mostly accounting for such collisions when the velocities after the collision are almost the same as their velocities before the. One may be tempted to say that in the limit, collisions are only changing the velocities infinitesimally. One says that in such collisions the particles are only grazing each other.

To formalize this picture, we consider a family $b_\varepsilon(\cdot)$ such that

\begin{align} & \lim \limits_{\varepsilon \to 0^+} b_\varepsilon(\sigma\cdot\sigma_0) = 0 \text{ locally uniformly in } \{ \sigma \neq \sigma_0 \}, \\ \
& \lim \limits_{\varepsilon \to 0^+} \int_{\mathbb{S}^2} b_\varepsilon(\sigma\cdot \sigma_0)|\sigma-\sigma_0|^2d\sigma > 0\;\;\forall\;\sigma_0 \in \mathbb{S}^2. \end{align}

Proposition (Landau 1936, Rosenbluth-McDonald-Judd 1957) Fix $\gamma \in (-3,0)$ and consider a family $b_\varepsilon(\cdot)$ as above, and define $$ Q_\varepsilon(f,f) = \int_{\mathbb{R}^3}\int_{\mathbb{S}^2}(f(v_\sigma)f(w_\sigma)-f(v)f(w)) b_\varepsilon(\cos(\theta)) |v-w|^\gamma d\sigma dw,$$ Then, for any given $f \in \mathcal{S}(\mathbb{R}^3)$ we have $$\lim \limits_{\varepsilon \to 0}Q_\varepsilon(f,f) = \int_{\mathbb{R}^3}|v-w|^\gamma\text{div}_{v-w}\left ( \Pi(v-w) \nabla_{v-w}(f(v)f(w)) \right ) dw $$ where for $z\neq 0$ we use $\Pi(z)$ to denote the orthogonal projection onto the orthogonal complement of $z$.

Proof. The hard work of the proof will be to compute the following limit for every pair of velocities $v,w$, \begin{align} \lim \limits_{\varepsilon\to0^+}\int_{\mathbb{S}^2}(f(v_\sigma)f(w_\sigma)-f(v)f(w)) b_\varepsilon(\cos(\theta)) d\sigma \end{align} We are going to show this limit is always equal to \begin{align} \text{div}_{v-w}\left (\Pi(v-w)(\nabla_v-\nabla_w)(f(v)f(w)) \right ) \end{align} Now, we fix $v$ and $w$ with $v\neq w$, and let us write \begin{align} z = \frac{1}{2}(v+w),\; r = |v-w|, \text{ and } \sigma_0 = \frac{v-w}{|v-w|} \end{align} Then, let us denote by $F(\sigma)$ the function \begin{align} F(\sigma) = f(z+r\sigma)f(z-r\sigma) \end{align} In terms of $F$, and using the notation $z,r,\sigma_0$, we have \begin{align} \int_{\mathbb{S}^2}(f(v_\sigma)f(w_\sigma)-f(v)f(w)) b_\varepsilon(\cos(\theta)) d\sigma = \int_{\mathbb{S}^2}(F(\sigma)-F(\sigma_0))b_\varepsilon(\sigma\cdot \sigma_0)d\sigma \end{align} The integral on the right is an integro-differential elliptic Laplacian on the sphere (the same kind of operator as the fractional Laplacian on the sphere), and the limit $\varepsilon \to 0^+$ yields a local limit. Now, what happens as $\varepsilon \to 0^+$? First, note that for any $\delta>0$ we have \begin{align} \lim \limits_{\varepsilon\to 0^+}\int_{\mathbb{S}^2}(F(\sigma)-F(\sigma_0))b_\varepsilon(\sigma\cdot \sigma_0)d\sigma = \lim \limits_{\varepsilon\to 0^+}\int_{\mathbb{S}^2 \cap B_\delta(\sigma_0)}(F(\sigma)-F(\sigma_0))b_\varepsilon(\sigma\cdot \sigma_0)d\sigma \end{align} As such, we may take $\delta$ small and approximate $F(\sigma)$ in terms of a Taylor expansion at $\sigma_0$ (this being a Taylor expansion of $F$ as a function in $\mathbb{R}^3$), \begin{align} & F(\sigma) - F(\sigma_0) \\\ \\\ & = \nabla F(\sigma_0)\cdot (\sigma-\sigma_0) + \frac{1}{2}(D^2F(\sigma_0)(\sigma-\sigma_0),\sigma-\sigma_0)+o(|\sigma-\sigma_0|^2) \end{align} From here, arguing analogously as with the fractional Laplacian in $\mathbb{R}^n$, we conclude that \begin{align} & \lim \limits_{\varepsilon\to 0^+}\int_{\mathbb{S}^2}(F(\sigma)-F(\sigma_0))b_\varepsilon(\sigma\cdot \sigma_0)d\sigma \\\ \\\ & = \lim \limits_{\varepsilon\to 0^+}\int_{\mathbb{S}^2 \cap B_\delta(\sigma_0)}(\nabla F(\sigma),\sigma-\sigma_0) b_\varepsilon(\sigma\cdot \sigma_0)d\sigma \\\ \\\ & + \frac{1}{2}\lim \limits_{\varepsilon\to 0^+}\int_{\mathbb{S}^2 \cap B_\delta(\sigma_0)}(D^2F(\sigma)(\sigma-\sigma_0),\sigma-\sigma_0)b_\varepsilon(\sigma\cdot \sigma_0)d\sigma \end{align} The lemma now follows by computing the limit as $\varepsilon \to 0^+$ of \begin{align} \int_{\mathbb{S}^2}(\sigma-\sigma_0)b_\varepsilon(\sigma\cdot\sigma_0)d\sigma \end{align} and \begin{align} \int_{\mathbb{S}^2}(\sigma-\sigma_0)\otimes (\sigma-\sigma_0) b_\varepsilon(\sigma\cdot\sigma_0)d\sigma \end{align}

Exercise. Show that for any smooth function $F(v,w)$ we have \begin{align} & |v-w|^{2+\gamma}\text{div}_{v-w}\left (\Pi(v-w)(\nabla_v-\nabla_w)(F(v,w)) \right ) \\\ \\\ & = \text{div}_{v-w}\left (|v-w|^{2+\gamma}\Pi(v-w)(\nabla_v-\nabla_w)(F(v,w)) \right ) \end{align}

The Landau equation is simply the Boltzmann equation where instead of Boltzmann’s collisional integral $Q$ one uses Landau’s operator $Q_L$,

\begin{align} \partial_t f + v\cdot \nabla_x f & = Q_L(f,f) \\ \\ & = \text{div}_v \left ( c\int_{\mathbb{R}^3}|v-w|^{2+\gamma} \Pi(v-w) \nabla_{v-w}(f(v)f(w)) dw\right ) \end{align}

This can also be written as

\[\partial_t f + v\cdot \nabla_x f = \text{div}_v \left (A_f\nabla_v f-f \text{div}_v(A_f) \right )\]

where

\[A_f = c\int_{\mathbb{R}^3}|v-w|^{2+\gamma} \Pi(v-w)f(w) dw\]

When $\gamma = -3$ and $c$ is a certain normalization constant ($c= (8\pi)^{-1})$) this amounts to

\[\partial_t f+v\cdot \nabla_x f = \text{tr}(A_fD^2f) + f^2\]

\begin{align} \partial_t u = Lu
\end{align}

Parabolic and kinetic equations

We have seen that the study of the Boltzmann and Landau equations entails the study of equations which look like the Kolmogorov equation, \begin{align} \partial_t f + v\cdot \nabla_x f = \Delta_v f \end{align} or the kinetic Fokker-Planck equation. \begin{align} \partial_t f + v\cdot \nabla_x f = \text{div}( \nabla_v f + v f). \end{align} One may also look at the the kinetic Fokker-Planck equation with variable coefficients, \begin{align} \partial_t f + v\cdot \nabla_x f = \text{div}(A(t,x,v)\nabla_v f + f b(t,x,v)) \end{align} the fractional Kolmogorov equation, \begin{align} \partial_t f + v\cdot \nabla_x f = -(-\Delta_v)^s f \end{align} as well as more general integro-differential equations, \begin{align} \partial_t f + v\cdot \nabla_x f = \int_{\mathbb{R}^3}(f(w)-f(v))K_f(t,x,v,w)dw + U(t,x,v)f. \end{align} In this course we will first study such linear equations. With good enough estimates for the linear theory we will be able to build solutions to nonlinear equations.

Moreover, one can treat a solution to some nonlinear equations like Landau and Boltzmann as solving some implicit linear equation, the more we can say about linear equations without assuming much about their coefficients (for instance, not assuming even any continuity) the more we can say about solutions to nonlinear equations. This is a simple but foundational idea in nonlinear PDE, one of the most famous (and one of the first) applications of this idea can be found in the parallel solutions to Hilbert’s 19th problem by De Giorgi and Nash.

A toy model

Consider the following initial value problem: to find, given $f_{\text{in}} \in L^1(\mathbb{R}^n)\cap C^\infty(\mathbb{R}^n)$ with $f_{\text{in}} \geq 0$ everywhere, a function $f:\mathbb{R}^n\times [0,\infty)\to \mathbb{R}$ solving

\begin{align} \partial_t f & = \text{div}( a_f \nabla f) \text{ for } t>0, \\ \
f & = f_{\text{in}} \text{ for } t=0 \end{align}

where, given a function $f$, we define $a_f$ by \begin{align} a_f(x,t) = 1 + \int_{B_1(x)}f(y,t)dy \end{align}

We have the following existence theorem.

Theorem. Given a non-negative function $f_{in} \in L^1(\mathbb{R}^n)\cap C^\infty(\mathbb{R}^n)$ there is a unique smooth function $f=f(x,t)$ defined for all $t\geq 0$ and solving the initial value problem above.

Sketch of the solution. We start by defining a bilinear operator, \begin{align} Q_{\text{toy}}(g,f) := \text{div}( a_g \nabla f) \end{align}

Consider $g:\mathbb{R}^n\times [0,\infty) \to \mathbb{R}$, a non-negative function such that $g(\cdot,t) \in L^1(\mathbb{R}^n)$ for every $t$, then the linear theory for parabolic equations (as we will develop in the next few lectures) guarantees there is a unique function $f$ such that

\begin{align} \partial_t f & = Q(g,f) \text{ for } t>0,\\ \
f & = f_{\text{in}} \text{ for } t=0. \end{align}

In light of this, we generate a sequence of functions $f_k(x,t)$ defined in $\mathbb{R}^n\times [0,\infty)$ as follows: first, define $f_0(x,t)$ by

\begin{align} f_0(x,t) = f_{\text{in}}(x) \text{ for all } x\in\mathbb{R}^n, t\geq 0. \end{align}

Now for $k\geq 1$ if we have already defined $f_{k-1}$, define $f_k$ as the unique solution of the initial value problem \begin{align} \partial_t f_{k} & = Q(f_{k-1},f_{k}) \text{ for } t>0,\\ \
f_{k} & = f_{k-1} \text{ for } t=0. \end{align}

Two basic observations about the sequence $f_k$:

1. The functions $f_k$ are all non-negative.
2. The total mass is fixed, that is for every $t>0$ and every $k$ we have

\begin{align} \ ||f(t)\ ||_{L^1(\mathbb{R}^n)} = \int_{\mathbb{R}^n}f(x,t)dx = \int_{\mathbb{R}^n}f_{\text{in}}(x)dx = \ ||f_{\text{in}}\ ||_{L^1(\mathbb{R}^n)}. \end{align}

With the whole sequence thus generated, the goal is to show that as $k\to \infty$ the functions $f_k$ all start to resemble one another and converge to a function $f(x,t)$, which will then solve the nonlinear initial value problem.

We will not go over the proof that the $f_k$ are converging to a single function $f$ here, but simply show how the rest of the proof follows once this step is achieved. Suffice to say that a relatively standard energy estimate makes it possible to show that there is a function $f(x,t)$ such that for any $T>0$ we have

\begin{align} \lim \limits_{n\to \infty}\|f_k-f\|_{L^\infty(0,T,L^2(\mathbb{R}^n))} = 0. \end{align}

We will learn this “relatively standard energy estimate” in full detail later on when we finish the local existence existence theory for the Boltzmann and Landau equations. The issue at hand now is to show that $f(x,t)$ is smooth, and that it solves the nonlinear problem. To do this, we will use the regularity theory for parabolic equations to show that the functions $\{f_k \}_k$ are uniformly Holder continuous together with their derivatives.

Continuity estimate for $f_k$. Observe that for any $g\geq 0$ we have

\begin{align} 1 \leq a_g \leq 1 + \||g\||_{L^1(\mathbb{R}^n)}. \end{align} It follows that for every $k$ and every $x\in\mathbb{R}^n,t\geq 0$,

\begin{align} 1 \leq a_{f_k}(x,t) \leq 1 + \||f_{\text{in}}\||_{L^1(\mathbb{R}^n)}. \end{align}

As we will see, from this condition alone De Giorgi-Nash-Moser theory guarantees the functions $\{f_k\}_k$ are locally uniformly (uniformly in $k$!) Holder continuous with respect to $x$ and $t$ . This regularity estimate starts a process that is known informally as elliptic regularization or bootstrapping.

Continuity estimate for the derivatives of $f_k$. The bootstrapping process refers to using a regularity estimate of the functions $\{f_k\}_k$ in order to conclude the coefficients of the equation it solves are more regular, which enables one to control higher derivatives of the functions themselves, a process that we can repeat indefintely. For now the Holder regularity of solutions guarantees that the sequence of diffusion coefficients $\{a_{f_k}\}_k$ are also uniformly Holder continuous in $x$ and $t$, and under such a situation Schauder theory for linear parabolic equations with Holder continuous coefficients guarantees that the spatial derivatives of $\{f_k\}_k$ are uniformly Holder continuous in $x$ and $t$.

All of the above ultimately leads to the following fact: There is a $\alpha \in (0,1)$ depending only on $f_{\text{in}}$ and the dimension $n$ and numbers $C_1,C_2,C_3>0$ depending only on $f_{\text{in}}$, the dimension $n$, and a compact set $K \subset \mathbb{R}^n\times (0,\infty)$ such that

\begin{align} [f]_{C^\alpha(K)} & \leq C_1\\ \
[D^2_xf]_{C^\alpha(K)} & \leq C_2\\ \
[\partial_tf]_{C^\alpha(K)} & \leq C_3
\end{align}

Therefore, the functions $f_k$ and their derivatives of order $2$ in space and of order $1$ in time are uniformly equibounded and uniformly equicontinuous in compact sets of $\mathbb{R}^n \times (0,\infty)$. By the Arzela-Ascoli theorem, they must converge locally uniformly respectively to $f$ and its corresponding derivatives. This convergence is strong enough to ensure that as $k\to\infty$ the following convergence happens pointwise in all of $\mathbb{R}^n\times (0,\infty)$:

\begin{align} \partial_tf_k \to \partial_t f,\;\; Q(f_{k-1},f_k) \to Q(f,f) \end{align}

from where we conclude that

\begin{align} \partial_t f = Q(f,f), \end{align}

a little further work is done (also in the form of estimates, but now using some the initial regularity $f_{\text{in}}$) to show that just as for every function $f_k$ in the sequence, the limit function $f$ is continuous as $t\to 0$ and is equal to $f_{\text{in}}$ at $t=0$. In this manner the global solution to the Cauchy problem is constructed.