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

Part I: Elliptic operators, local or not.

In the previous lectures we have established the principle that the analysis of linear equations of the form

\begin{align} \partial_t f + v\cdot \nabla_x f = Q(g,f) \end{align} (for a given function $g(x,v,t)$) is of the utmost importance for the study of the nonlinear equations in kinetic theory (Boltzmann, Landau, etc).

We discussed this idea in some detail for a “local” equation, namely

\begin{align} \partial_tf = \text{div}(a_f\nabla f),\;\text{ where } a_f(x,t) = 1 + \int_{B_1(x)}f(y,t)dy \end{align}

Linear equations

As we have been discussing, there are two kinds of linear equations we will care about, parabolic equations

\begin{align} \partial_tf = Lf,\; f= f(t,v) \end{align}

and kinetic equations

\begin{align} \partial_t f + v\cdot \nabla_x f= Lf,\; f=f(t,x,v), \end{align}

where in both cases $Lf$ is a linear elliptic operator (local or not) acting on the $v$ variable. We will follow the convention of referring to $v$ as the velocity variable and $x$ as the spatial variable, although there are kinetic equations where the variables represent other physical quantities. Solutions to kinetic equations which are independent with respect to the spatial variable are then solutions to a standard parabolic equation in $v$.

The specific linear operators $L$ appearing above are the same ones we discussed in the previous two lectures, they include:

• The Laplacian and its fractional powers \begin{align} Lf = \Delta_v f,\;\; Lf = -(-\Delta_v)^sf \;s\in(0,1), \end{align}
• The generator of the Fokker-Planck equation \begin{align} Lf = \text{div}_v(\nabla_vf + f\nabla_v\phi) \end{align} where $\phi = \phi(v)$ is typically a convex function.
• Local operators with variable coefficients \begin{align} Lf = \text{div}_v(A\nabla_v f + f b) \end{align} where $A= A(t,x,v)$, $b=b(t,x,bv)$ note that here the coefficients may depend on $t$ and $x$ as well.
• Non-local operators with variable coefficients, including for instance \begin{align} Lf = \int_{\mathbb{R}^n}(f(\cdot,w)-f(\cdot,v))K(v,w)dw + fU \end{align} where $U$ is some potential function, and $K(v,w)\geq 0$.

In the last item one may also allow for $K$ and $U$ to vary with $t$ and $x$. This will be necessary for analyzing linearizations of the inhomogeneous Boltzmann equation, since we want to cover any $L$ given by

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

for some non-negative function $g= g(t,x,v)$. It is in our interest to understand as much about these linear equations \begin{align} \partial_t f + v\cdot \nabla_x f = Q(g,f) \end{align} under minimal assumptions on $g$, as that understanding could then be applied to solutions of the nonlinear equation (i.e. when $g=f$). In particular, it is important we obtain estimates on these equations without assuming much regularity on the coefficients, which corresponds to not requiring much regularity from $g$ itself.

Linearity and fundamental solutions

The oldest approach to solving linear parabolic or kinetic equations is through a fundamental solution – if it is available. This is well known for the heat equation

\begin{align} \partial_tf = \Delta f \end{align}

and for the Kolmogorov equation (including its nonlocal version)

\begin{align} \partial_tf + v\cdot \nabla_x f = -(-\Delta_v)^s f,\;\; s\in (0,1]. \end{align}

For the heat equation of course one of the first things one learns in PDE course is the derivation of the heat kernel, \begin{align} K_t(x,y) = \frac{1}{(4\pi t^2)^{\frac{n}{2}}}e^{-\frac{|x-y|^2}{4t}} \end{align} which is the fundamental solution for the heat equation. If one is given an initial data $f_{\text{in}}$ that is locally integrable and does not grow too fast at infinity then one can define for $t>0$ the function \begin{align} f(t,x) = \int_{\mathbb{R}^n}f_{\text{in}}(y)K_t(x,y)dy, \end{align} which will, for $t>0$ be a smooth solution to the heat equation which takes the initial data $f_{\text{in}}$ in some (possibly weak sense if $f_{\text{in}}$ is not continuous) as $t\to 0^+$.

There is a similar formula for the Kolmogorov equation: given some initial data $f_{\text{in}}(x,v)$ that is locally integrable and is not too large at infinity one can define for $t>0$ \begin{align} f(t,x,v) = \int_{\mathbb{R}^n}\int_{\mathbb{R}^n}f_{\text{in}}(y,w)K_t^{(s)}(x,v,y,w)dwdy \end{align} and this will be a smooth solution solution to the (fractional if $s<1$) Kolmogorov equation for positive times. The Kolmogorov kernel has an explicit expression not unlike that for the heat equation, and can be obtained for instance by taking the Fourier transform in the Kolmogorov equation and solving the resulting transport equation. <!– where \begin{align} K_t^{(s)}(x,v,y,w) = J^{(s)}(t,x-y-tw,v-w) \end{align}

\begin{align} J^{(s)}(t,x,v) = c_d\frac{1}{t^{d+\frac{d}{2s}}}J_0^{(s)}(\frac{x}{t^{1+\frac{1}{2s}}} ,\frac{v}{t^{\frac{1}{2s}}} ) \end{align}

\begin{align} \hat J_0^{(s)}(\xi,\eta)= e^{-\int_0^1|\eta-\tau \xi|d\tau} \end{align} –>

What about equations with variable coefficients? How do we show existence of a solution – or of a fundamental solution – when it is not clear how to get an explicit formula? If solutions exists, how much do they regularize from their initial data as time progresses?

Remember: for our overaching goal of attacking nonlinear problems, we wanto to be able to answer these questions (to the extent that is possible) for as broad a class of variable coefficients / kernels as possible.

The problem of existence.

We will study the variational/integral approach to this problem, specifically one based on Hilbert space methods. This theory was developed in the mid 20th century in a series of works by Hille, Yosida, Lax, Milgram, among others.

The setting is that of a finding a function $f:\mathbb{R}^n\times (0,\infty)\to\mathbb{R}$ such that $u$ has some prescribed initial data and solves $\partial_tf = Lf$, where $L$ is generally not bounded on the space where $u$ lives, but nevertheless satisfies some sort of weak positivity condition.

Warm up: the heat equation in $\mathbb{R}^n$.

Ultimately, we are going to use this Hilbert space approach to show the existence of a unique weak solution to very general parabolic / kinetic linear PDE. However, for the sake of exposition –and to better appreciate the power of the method in its full generality– we are going to review first the main ideas in a simpler setting, that of the heat equation in $\mathbb{R}^n$. Then we will iterate the discussion and consider more general situations – adding variable coefficients, drifts, non-local terms, etc.

Suppose $f:[0,\infty)\times \mathbb{R}^n\to \mathbb{R}$ is a smooth solution of $\partial_tf = \Delta f$ with $f(x,0)=f_{\text{in}}(x)$. Then,

\begin{align} \frac{1}{2}\frac{d}{dt}\int_{\mathbb{R}^n} f^2dx & = \int_{\mathbb{R}^n}f\partial_t fdx \\ \
& = \int_{\mathbb{R}^n}f\Delta fdx = -\int_{\mathbb{R}^n}|\nabla f(t,x)|^2dx. \end{align} Therefore, for any pair of times $t_2>t_1>0$ we have

\begin{align} \frac{1}{2}||f(t_2)||_2^2 + \int_{t_1}^{t_2}\int_{\mathbb{R}^n}|\nabla f(t,x)|^2dxdt = \frac{1}{2}||f(t_1)||_2^2 \end{align}

We see then that not only is $||f(t)||_2$ decrasing with time, but $f$ enjoys some regularity averaged over space and time, that is

\begin{align} \int_{0}^{\infty}||f(t)||_{\dot H^1}^2 \;dt \leq \frac{1}{2}||f_{\text{in}}||_2^2. \end{align}

This regularization is a broad phenomenon, to take a nonlocal example: if $f$ solves the fractional heat equation $\partial_tf = -(-\Delta)^sf$, a similar computation yields the identity

\begin{align} \frac{1}{2}||f(t_2)||_2^2 + \frac{c_{n,s}}{2}\int_{t_1}^{t_2}\int_{\mathbb{R}^n}\int_{\mathbb{R}^n}\frac{(f(x,t)-f(y,t))^2}{|x-y|^{n+2s}}dxdydt = \frac{1}{2}||f(t_1)||_2^2 \end{align}

This identity follows just as for the local heat equation, with the minor difference that the role of integration by parts is now played by its nonlocal counterpart (there is a discrete analogue of this for the Laplacian in a graph, and in that context the identity is commonly referred to as summation by parts). The following exercise explains why this is so.

Exercise. Show that if $f,g$ are Schwartz functions in $\mathbb{R}^n$, then \begin{align} \int_{\mathbb{R}^n} f(x)(-\Delta)^sg(x)dx = \frac{c_{n,s}}{2}\int_{\mathbb{R}^n}\int_{\mathbb{R}^n}\frac{(f(x)-f(y))(g(x)-g(y))}{|x-y|^{n+2s}}dxdy \end{align} where $c_{n,s}$ is the normalization constant used in the definition of the fractional Laplacian (as discussed in Lecture 2). Hint: Consider a bounded continuous function $K(x,y)$ such that $K(x,y)=K(y,x)$, use Fubini's theorem and the symmetry of $K$ to show that \begin{align} \int_{\mathbb{R}^n}\int_{\mathbb{R}^n}f(x)(g(y)-g(x))K(x,y)dydx = \int_{\mathbb{R}^n}\int_{\mathbb{R}^n}f(y)(g(x)-g(y))K(x,y)dydx \end{align} What expression do you obtain if you average the expression on the left and on the right?

Exercise. Use the previous exercise and a density argument to show the previous identity \begin{align} \int_{\mathbb{R}^n} f(x)(-\Delta)^sg(x)dx = \frac{c_{n,s}}{2}\int_{\mathbb{R}^n}\int_{\mathbb{R}^n}\frac{(f(x)-f(y))(g(x)-g(y))}{|x-y|^{n+2s}}dxdy \end{align} still holds when $g \in \mathcal{S}(\mathbb{R}^n)$ and $f \in H^s(\mathbb{R}^n)$.

The corresponding inequality now involves the fractional order Sobolev space $\dot H^s$,

\begin{align} \int_{0}^{\infty}||f(t)||_{\dot H^s}^2 \;dt \leq \frac{1}{2}||f_{\text{in}}||_2^2. \end{align}

Therefore you are still “gaining” some regularity, this time in a fractional Sobolev space.

What was important in the computation $\partial_tf = Lf$ for the two operators $L$ considered above was that

\begin{align} -(f,Lf) \geq 0. \end{align}

Hille-Yosida for the heat equation in $\mathbb{R}^n$.

The Hille-Yosida theorem was an important progress for the theory of evolution PDE, it was one of the first instances of a general theory of solutions to a time dependent PDE that does not rely on explicit formulas. The theorem extends beyond the framework of Hilbert spaces, but for our present discussion it is best to focus on the Hilbert space setting.

We will not state the formal Hille-Yosida theorem for the heat equation. However, a version of their theorem boils down to the statement that if certain operator satisfies a certain positivity property, then even if that operator is unbounded* one can solve the Cauchy problem for $\partial_tf = Lf$.

*if the operator were bounded, one could solve the equation as if it were a standard ODE in a Banach space using Picard’s theorem.

The basic observation. Fix $h>0$. For any $w\in L^2(\mathbb{R}^n)$ there is a unique $u \in L^2(\mathbb{R}^n)$ such that $\Delta u \in L^2(\mathbb{R}^n)$ (this means that $\Delta u$, defined first a priori as a distribution, happens to be a function in $L^2(\mathbb{R}^n)$) and \begin{align} u-h\Delta u = w \end{align} Moreover, this function $u$ is such that $\|\|u\|\|_{L^2(\mathbb{R}^n)} \leq \|\|w\|\|_{L^2(\mathbb{R}^n)}$.

(We will not prove this basic observation about the Laplacian in this lecture, we will come back to it later)

We will use the above observation together with the tools from Hilbert space theory (in particular, weak compactness of bounded sets in a Hilbert space) to prove the following theorem.

Theorem. Given $f_0 \in L^2(\mathbb{R}^n)$ there exists a unique weak solution to the Cauchy problem \begin{align} \partial_t f & = \Delta f \text{ in } \mathbb{R}^n\times (0,\infty) \\ \\ f(x,0) & = f_{\text{in}}(x) \end{align} The initial data is achieved in the sense that $f(t) \mapsto f_{\text{in}}$ in the sense of $L^2$ weak convergence as $t\to 0^+$.

Definition. A function $f:[0,\infty)\times \mathbb{R}^n\to\mathbb{R}$ is said to be a weak solution of $\partial_t f= \Delta f$ if first, the following inclusions hold \begin{align} & f \in L^2_{\text{loc}}((0,\infty),H^1(\mathbb{R}^n)) \cap L^\infty((0,\infty),L^2(\mathbb{R}^n)) \\ \
& \partial_t f \in L^2((0,\infty),(H^1(\mathbb{R}^n))^*), \end{align}

and second, if for any smooth test function $\phi(t,x)$ with compact support in $[0,\infty)\times\mathbb{R}^n$ and a.e. pair of times $0<t_1<t_2$ we have

\begin{align} \int_{\mathbb{R}^n}f(t_2,x)\phi(t_2,x) dx + \int_{t_1}^{t_2}\int_{\mathbb{R}^n}(f,-\partial_t \phi) + (\nabla f,\nabla \phi)\;dxdt = \int_{\mathbb{R}^n}f(t_1,x)\phi(t_1,x) dx \end{align}

Remark. For the heat equation the actual Hille-Yosida theorem guarantees that $f(t)$ approaches $f_{\text{in}}$ in the $L^2$ norm as $t\to 0$. However, recall that what we are doing is to describe a method that works for much more general equations where the Hille-Yosida theorem does not exactly apply, even if this method does not provide the strongest statements known in the special case of the heat equation (where one has constant coefficients, among other nice structural conditions).

Exercise. Use integration by parts to show that if $f(t,x)$ is a smooth solution to the heat quation then for any smooth compactly supported test function $\phi$ we have \begin{align} \int_{\mathbb{R}^n}f(t_2,x)\phi(t_2,x) dx + \int_{t_1}^{t_2}\int_{\mathbb{R}^n}(f,-\partial_t \phi) + (\nabla f,\nabla \phi)\;dxdt = \int_{\mathbb{R}^n}f(t_1,x)\phi(t_1,x) dx \end{align} and conclude that any smooth solution is a weak solution.

Fix $h>0$.

We are going to build the solution by solving a semi-discrete version of the equation. First, we generate a sequence of functions $f_k(x)$ as follows: we set $f_0(x) = f_{\text{in}}(x)$, and for larger $k$ we define $f_k$ in terms of $f_{k-1}$ by setting it as the unique function solving

\begin{align} f_{k}(x)-f_{k-1}(x) = h\Delta f_k(x) \end{align}

That is, we are doing implicit Euler in time. Observe that the above equation can be rearranged to read

\begin{align} f_{k}(x)-h\Delta f_k(x) = f_{k-1}(x) \end{align}

and thus the basic observation about $\Delta$ guarantees that $f_k$ is uniquely determined by $f_{k-1}$. This defines an infinite sequence of functions.

Now, we build a function in space and time that is a “step function” in time. For $h>0$, define $f^{(h)}:(0,\infty)\times \mathbb{R}^n\to\mathbb{R}$ by setting \begin{align} f^{(h)}(t,x) = f_k(x) \text{ for } t\in (kh,(k+1)h) \end{align}

In the next lecture, we will gather properties of the sequence $f_k$ and of $f^{(h)}$ and prove the existence theorem. We leave them stated and retake them next time,

1. The $L^2$ norm is monotone decreasing in $k$, \begin{align} ||f_{k+1}|| \leq ||f_k|| \leq ||f_{\text{in}}|| \end{align}
2. For every $m\in\mathbb{N}$ we have the following inequality \begin{align} \int_0^{mh}\int_{\mathbb{R}^n}|\nabla f^{(h)}(t,x)|^2dxdt = h\sum \limits_{k=0}^m||\nabla f_k||^2 \leq \frac{1}{2}||f_{\text{in}}||^2 \end{align}
3. If we define the operator $\partial^{(-h)}_t$ by $\partial^{(-h)}_t\phi(t,x) = (\phi(t,x)-\phi(t-h,x))/h$, then $f^{(h)}$ is a weak solution of \begin{align} \partial_t^{(-h)}f^{(h)} = \Delta f^{(h)} \;\text{ in } (h,\infty)\times \mathbb{R}^n. \end{align}