Boltzmann and PDE: Lecture 2

(22 Jan 2026)

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

Part I: Elliptic operators, local and non-local.

The Laplacian.

The Laplacian operator of a twice-differentiable function $u(x)$ in some domain $D\subset \mathbb{R}^n$ is given by

\[\Delta u(x) = \partial_{x_1x_1}^2u(x) + \ldots + \partial_{x_nx_n}^2u(x)\]

This defines a differential operator in the space of twice differentiable functions There are several equivalent ways of writing this operator, including

\[\Delta u(x) = \text{div}(\nabla u(x)),\]

appropriately called the ‘‘divergence form’’, and

\[\Delta u(x) = \text{tr}(D^2u(x)),\]

which is termed ‘‘the non-divergence form’’. These are two ways of writing the same thing which hide different aspects of the operators. Using the coordinate notation, these two expressions are written as

\[\sum_{i,j=1}^n\partial_{x_i}( \delta_{ij} \partial_{x_j}u)\]

and

\[\sum \limits_{i,j=1}^n \delta_{ij}\partial_{ij}^2u,\]

respectively. Here, $\delta_{ij}$ denotes Kronecker’s delta.

Infinitesimal mean oscillation. There is an elementary but meaningful characterization of $\Delta u(x)$ as a measure of the infinitesimal mean oscillation of $u$ near $x$,

\[\Delta u(x_0) = \lim \limits_{r\to 0^+} \frac{2n}{r^2} \frac{1}{|\partial B_r|}\int_{\partial B_r(x_0)}u(y)-u(x_0)d\sigma(y)\]

This is not hard to see: what is happening here is that as $r$ becomes small the main contribution to the spherical average of $u(y)-u(x_0)$ over $\partial B_r(x_0)$ comes from the quadratic part of $u$ in its Taylor expansion around $x_0$.

For concreteness let us fix $x_0=0$, and let us consider first the case where $u$ is a quadratic form

\[u(x) = \frac{1}{2}(Ax,x) = \frac{1}{2}\sum \limits_{i,j=1}^n a_{ij}x_ix_j\]

In this case, since $u(rx) = r^2u(x)$, we have

\[\frac{1}{r^2}\frac{1}{|\partial B_r|}\int_{\partial B_r(0)}u(y)d\sigma(y) = \frac{1}{|\partial B_1|}\int_{\partial B_1(x)}u(y)d\sigma(y)\]

From here, basic symmetry considerations show that

\[\int_{\mathbb{S}^{n-1}}y_iy_jd\sigma(y) = \left \{ \begin{array}{ll} 0 & \text{ if } i\neq j\\ \frac{1}{n}|\partial B_1| & \text{ if } i=j\end{array}\right.\]

Therefore

\[\frac{1}{r^2}\frac{1}{|\partial B_r|}\int_{\partial B_r(0)}u(y)d\sigma(y) = \sum \limits_{i=1}^n \frac{1}{2n}a_{ii} = \frac{1}{2n} \sum \limits_{i=1}^n a_{ii} = \frac{1}{2n}\Delta u(0)\]

For a general smoooth function $u$, we have

\[u(x) = u(0) + \nabla u(0)\cdot x + \frac{1}{2}(D^2u(0)x,x) + O(|x|^3)\]

From here, one can see that

\[\frac{2n}{r^2} \frac{1}{|\partial B_r|}\int_{\partial B_r(0)}u(y)-u(0)d\sigma(y) = \Delta u(0) + O(r)\]

The Laplacian of $u$ at $x$ measures the average oscillation in an infinitesimal sphere around $x$. From this observation one may guess that $u(x)$ would tend to be larger than its nearby average when $\Delta u(x)\leq 0$ and smaller than its nearby average when $\Delta u(x) = 0$.

Mean oscillation over all scales. One could try to do an average that sees all spatial scales with some weighting and not just the limit $r\to 0$,

\[\int_0^\infty K(r)\left ( \int_{\partial B_1(0)}u(x_0+r\sigma )-u(x_0)d\sigma \right ) dr\]

where $K(r)$ is some non-negative weight function.

\[\int_0^\infty\int_{\partial B_1(0)}(u(x_0+r\sigma)-u(x_0))r^{-(n-1)}K(r)r^{n-1}d\sigma dr\]

Then, using the formula for integration in spherical coordinates the above can be rewritten as

\[\int_{\mathbb{R}^n}(u(x_0+y)-u(x))|y|^{-(n-1)}K(|y|)dy\]

If, for instance, $K(r) = c r^{-1-s}$ for some $s\in(0,1)$, the resulting operator is

\[c\int_{\mathbb{R}^n}(u(y)-u(x))|x-y|^{-n-s}dy\]

For the right choice of $c$ ($c=c(n,s)$, see below) this operator corresponds to a fractional power of the Laplacian

\[L_su(x) := -(-\Delta)^su(x)\]

The most traditional definition of the fractional Laplacian $(-\Delta)^su$ is in terms of the Fourier transform: $(-\Delta)^s$ is the linear operator determined by the relation

\[\mathcal{F}\left ((-\Delta)^su \right )(\xi) = |\xi|^{2s} \mathcal{F}(u)(\xi),\]

valid for all Schwartz functions $u$ in $\mathbb{R}^n$ – here $\mathcal{F}(u)$ denotes the Fourier transform of $u$.

\[L_su(x) = -(-\Delta)^su(x) = c_{n,s} \int_{\mathbb{R}^n}(u(y)-u(x))|x-y|^{-n-2s}dy\]

The integral is understood in the principal value sense, meaning

\[L_su(x) = c_{n,s} \lim \limits_{\varepsilon\to 0^+}\int_{\mathbb{R}^n\setminus B_\varepsilon(x)}(u(y)-u(x))|x-y|^{-n-2s}dy\]

The constant $c(n,s)$ can be computed using classical Fourier analysis, we have (see, for instance, Landkof’s Foundations of modern potential theory)

\[c(n,s) = \frac{2^{2s}\Gamma\left (\frac{n+2s}{2}\right )}{\pi^{\frac{n}{2}}|\Gamma(-s)|}.\]

Non-local operators

The fractional Laplacian is the canonical example of a non-local elliptic operator. More generally, a typical non-local elliptic operator has the form

\[Lu(x) = \int_{\mathbb{R}^n}(u(y)-u(x))K(x,y)dy\]

where $K(x,y)$ is a non-negative function that typically may be quite singular near the diagonal $x=y$. This expression still captures some average oscillation of $u$ about the points $x$, but it might not be as symmetric an average as in the case of the fractional power of the Laplacian.

The adjective non-local refers to the fact that the value of $Lu$ at $x$ is affected by the values of $x$ at points that are far from $x$. The adjective elliptic is related to the fact that $K$ is non-negative.

There are two ways in which the positivity of $K$ makes $L$ have properties one associates with the traditional linear elliptic operator:

If $K(x,y)=K(y,x)$ then for any smooth enough function $u$ we have $-\int_{\mathbb{R}^n} u(x)Lu(x)dx \geq 0$ – for instance, for the standard Laplacian $\Delta$ this expression is the square of the $L^2$ norm of $\nabla u$.
The operator $L$ has the Global Comparison Property: this says that if $u,v:\mathbb{R}^n\to \mathbb{R}$ are smooth enough, $u\leq v$ everywhere in $\mathbb{R}^n$, and $u(x_0)=v(x_0)$ at some $x_0$, then $Lu(x_0) \leq Lv(x_0)$.

For the first part of the course we are going to learn the theory behind equations of the form

\[\int_{\mathbb{R}^n} (u(y)-u(x))K(x,y)dy = f(x),\]

as well as their corresponding time evolutions

\[\partial_t u(x) = \int_{\mathbb{R}^n} (u(t,y)-u(t,x))K(t,x,y)dy + f(t,x)\]

These extend (and in an asymptotic sense, contain) all linear elliptic and parabolic equations.

The Laplacian as a limit of non-local operators

The following computation is elementary but very instructive.

Proposition. If $u:\mathbb{R}^n\to\mathbb{R}$ is a function in $L^\infty(\mathbb{R}^n) \cap C^2(\mathbb{R}^n)$, then for every $x \in \mathbb{R}^n$ we have $$\lim \limits_{s\to 1^-}-(-\Delta)^su(x) = \Delta u(x) $$

Proof.

Looking at the various factors in the for $c(n,s)$, we note that as $s\to 1$ \begin{align} \frac{\Gamma\left (\frac{n+2s}{2} \right )}{\pi^{n/2}} \to \left ( \frac{\pi^{n/2}}{\Gamma\left (\frac{n}{2}+1 \right )}\right )^{-1} = \omega_n^{-1}, \end{align} where $\omega_n$ denotes the volume of the unit ball in $\mathbb{R}^n$. Using that $\Gamma$ has a pole at $-1$, we conclude that as $s\to 1$ we have \begin{align} c(n,s) = \frac{2^{2s}\Gamma\left (\frac{n+2s}{2}\right )}{\pi^{\frac{n}{2}}|\Gamma(-s)|} = c_n(1-s)+o(1-s) \end{align} With this asymptotic behavior at hand we analyze the integral. First, we estimate the contributions to the integral far from $x=y$. For a fixed $\rho>0$ we have $$\left |c_{n,s}\int_{\mathbb{R}^n \setminus B_\rho(x)}(u(y)-u(x))|x-y|^{-n-2s}dy\right |\leq c(1-s)\|u\|_\infty \int_{\mathbb{R}^n\setminus B_\rho(x)}|x-y|^{-n-2s}dy$$ It is elementary that \begin{align} \int_{\mathbb{R}^n\setminus B_\rho(x)}|x-y|^{-n-2s}dy = \omega_n \frac{1}{2s}\rho^{-2s} \end{align} We conclude that for any $\rho>0$, $$ \lim\limits_{s\to 1^-}\left |c_{n,s}\int_{\mathbb{R}^n \setminus B_r(x)}(u(y)-u(x))|x-y|^{-n-2s}dy\right | = 0.$$ Now we study the contributions to the integral coming from $B_\rho(x)$. We use that $u$ is twice differentiable near $x$, writing its second order expansion, $$u(y)-u(x) = \nabla u(x)\cdot (y-x) + \frac{1}{2}(D^2u(x)(y-x),y-x) + o(|x-y|^2)$$ Observe that for any $\rho>\varepsilon>0$ we have by spherical symmetry, $$\int_{B_\rho(x)\setminus B_\varepsilon(x)}\nabla u(x)\cdot (y-x) |x-y|^{-n-2s}dy = 0 \;\forall\;\varepsilon>0.$$ The remainder term contribution also vanishes (which can be seen using the form for the remainder in the second order Taylor expansion). Therefore, for every $\rho>\varepsilon>0$ we have $$\lim\limits_{s\to 1^-}-(-\Delta)^su(x) = \lim\limits_{s\to 1^-} \frac{c_{n,s}}{2}\int_{B_\rho(x)\setminus B_\varepsilon(x)}(D^2u(x)(y-x),y-x)|x-y|^{-n-2s}dy $$ We change variables, writing $h = y-x$, so that $$ \int_{B_\rho(x)\setminus B_\varepsilon(x)}(D^2u(x)(y-x),y-x)|x-y|^{-n-2s}dy \\ = \int_{B_\rho(0)\setminus B_\varepsilon(0)}(D^2u(x))h,h)|h|^{-n-2s}dh$$ Writing this in spherical coordinates we have $$ = \int_\varepsilon^\rho \int_{\partial B_1}(D^2u(x)ry,ry)r^{-n-2s} r^{n-1}d\sigma(y)dr$$ The integrand is homogeneous in $r$, so we have $$ = \left ( \int_\varepsilon^\rho r^{1-2s}dr \right )\left ( \int_{\partial B_1}(D^2u(x)y,y) d\sigma(y)\right )$$ $$ = \frac{1}{2(1-s)}\left ( \rho^{2(1-s)}-\varepsilon^{2(1-s)}\right )\frac{1}{2n}|\partial B_1| \Delta u(x) $$ We now use the form of $c(n,s)$ and its behavior as $s\to 1$ and conlcude that $$ \lim\limits_{s\to 1^-}-(-\Delta)^su(x)= \lim\limits_{s\to 1^-} \lim\limits_{\varepsilon \to 0^+} \frac{c_{n,s}}{2}\int_{B_r(x)\setminus B_\varepsilon(x)}(D^2u(x)h,h)|h|^{-n-2s}dh = \Delta u(x),$$ as we wanted.

As we mentioned before there are more general non-local operators,

\[Lu(x) = \int_{\mathbb{R}^n}(u(y)-u(x))K(x,y)dy,\]

so one could consider a family of kernels $K_\varepsilon(x,y)$ that produce operators $L_\varepsilon$ converging to other local differential operators.

Indeed, if the family $K_\varepsilon(x,y)$ is such that

\[\lim \limits_{\varepsilon \to 0^+} \int_{\mathbb{R}^n\setminus B_\delta(x)}K_\varepsilon(x,y)dy = 0 \text{ for any } \delta>0,\; x\in\mathbb{R}^n,\]

and

\[\lim \limits_{\varepsilon \to 0^+} \int_{B_\delta(x)}|x-y|^2K_\varepsilon(x,y)dy > 0 \text{ for any } \delta>0,\; x\in\mathbb{R}^n,\]

Then, one would expect that if the limit exists

\[\lim\limits_{s\to 0} \int_{\mathbb{R}^n} (u(y)-u(x))K_s(x,y)dy = Lu(x),\]

that the resulting operator $L$ will be a drift-diffusion operator of the form

\[Lu(x) = \text{tr}(A(x)D^2u(x)) + b(x)\cdot \nabla u(x).\]

In fact, any such operator given by reasonably regular diffusion matrix $A(x)$ and vector field $b(x)$ can be obtained by such a limit.

Exercise. Given a positive semi-definite matrix $A(x)$ (continuous in $x$) and vector field $b(x)$ (also continuous) construct a family of kernels $K_\varepsilon(x,y)$ such that 1. $K_\varepsilon(x,y)\geq 0$ for all $x,y$, and $K_\varepsilon$ is bounded in the set $\{|x-y|\geq \delta\}$ for any $\delta>0$. 2. If $L_\varepsilon u(x) := \int_{\mathbb{R}^n}(u(y)-u(x))K_\varepsilon(x,y)dy$, then for any $u \in C^2(\mathbb{R}^n)\cap L^\infty(\mathbb{R}^n)$ we have $\lim \limits_{\varepsilon \to 0^+} L_\varepsilon u(x) = \text{tr}(A(x)D^2u(x)) + b(x)\cdot \nabla u(x)$.

Hint. Consider first the limits of operators given by kernels of the form

\[K_\varepsilon(x,y) = \varepsilon |M(x)(x-y)|^{-n-(2-\varepsilon)}\]

for a given positive definite matrix field $M(x)$ (note that for such kernels we generally have $K_\varepsilon(x,y) \not = K_\varepsilon(y,x))$.

Next time: Grazing limits as local limits

Today, January 22nd, is Landau’s birthday, so it is altogether proper to mention the grazing limits and the equation they give rise to and which is named after him.

In the next lecture we will study such limits in detail and introduce the Landau equation. The limits arise from considering families of collisional integrals

\[Q_\varepsilon(f,f) = \int_{\mathbb{R}^3}\int_{\mathbb{S}^2}(f(v_\sigma)f(w_\sigma)-f(v)f(w))B_\varepsilon d\sigma dw,\]

with $B_\varepsilon = |v-w|^\gamma b_\varepsilon(\cos(\theta))$ and $b_\varepsilon(\cos \theta)$ concentrating at $\theta = 0$ as $\varepsilon \to 0$. This means that the only collisions being accounted for in $Q_\varepsilon(f,f)$ for $\varepsilon$ small are those with $\sigma$ such that

\[\frac{v_\sigma-w_\sigma}{|v_\sigma-w_\sigma|}\cdot \frac{v-w}{|v-w} \approx 1\]

If we recall the formula for the post collisional velocities

\begin{align} v_\sigma & = \tfrac{1}{2}(v+w) + \tfrac{1}{2}|v-w|\sigma \end{align} \begin{align}
w_\sigma & = \tfrac{1}{2}(v+w) - \tfrac{1}{2}|v-w|\sigma
\end{align}