Boltzmann and PDE: Lecture 1

(20 Jan 2026)

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

Philosophy of the course

The Boltzmann equation is an integro-differential PDE that is a foundational part of modern statistical mechanics. The study of this equation has been the impetus for important developments in functional and harmonic analysis, stochastic processes, and of course in the broader theory of partial differential equations.

I am taking as a principle that recent breakthroughs in the study of the Boltzmann equation provide a helpful framework to introduce many important techniques in modern PDE to newcomers in the field. Studying the Cauchy problem for the Boltzmann equation in particular provides a concrete and exciting goal that touches on current research directions and serves as motivation to learn about such topic as De Giorgi-Nash-Moser theory, hypoelliptic equations, integro-differential equations, the Fokker-Planck and Kolmogorov equations, comparison principles, log-Sobolev inequalities, and more. For this reason I believe this course will be of benefit to students with a broad variety of interests and not just those interested in kinetic theory.

An organizational objective of this course is to fully cover two recent results in particular. The first, coming from a series of works by Cyril Imbert and Luis Silvestre, in particular their Harnack inequality (2019, Journal of the European Mathematical Society) shows that any potential singularities for the Boltzmann equation must be of such a nature as to be detectable at the macroscopic scale of fluid dynamics. The second is a result in my joint work with Luis Silvestre (2025, Acta Mathematica), ruling out singularities altogether for the Landau equation in the spatially homogeneous regime, a result that was later expanded to the homogeneous Boltzmann equation by Cyril Imbert, Luis Silvestre, and Cedric Villani (2025, Inventiones Mathematicae). As regularity theory will play a central role, the notes from Cyril Imbert’s recent course on kinetic regularity theory will be an important reference for this course.

Today kinetic equations is a highly active and lively research field, I hope the students will acquire a solid perspective of the field and feel encouraged to join the ongoing research efforts in this area.

From Newton to Boltzmann

If ones takes an initial mass density $f_{\text{in}}(x,v)$ for particles at locations $x$ and $v$, and those particles are moving free from forces and without interacting between each other, then the mass density of those particles at different times defines a function $f(t,x,v)$ solving the linear transport equation \begin{align} \partial_t f + v\cdot \nabla_x f = 0 \text{ in } \mathbb{R}\times \mathbb{R}^3\times \mathbb{R}^3. \end{align} If in addition the particles are moving under the effect of some, possibly time-changing force field $E(t,x)$ the function $f(t,x,v)$ will instead solve the equation \begin{align} \partial_t f + v\cdot \nabla_x f + E \cdot \nabla_v f = 0 \text{ in } \mathbb{R}\times \mathbb{R}^3\times \mathbb{R}^3. \tag{L} \end{align} This equation is known as Liouville’s equation. Liouville’s equation is then a special case of linear transport equation corresponding to the Eulerian description of Newton’s law $\ddot x = E(t,x)$. This equation is apt if the ensemble of particles described by $f(t,x,v)$ are not interacting between themselves or if such interactions are negligible in the regime under consideration.

In the 1870’s, Boltzmann and Maxwell established one of the cornerstones of statistical mechanics by deriving an equation that enriches Liouville’s equation in a manner that effectively accounts for interactions between the particles. This equation is known as Boltzmann’s equation, and in its simplest form it models the evolution of the mass density ($f=f(t,x,v)$) for a monoatomic gas.

The Boltzmann equation for a monoatomic gas is then as follows, \begin{align} \partial_tf + v\cdot \nabla_x f = \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, \tag{B} \end{align} where, given $v,w\in\mathbb{R}^3$ and $\sigma \in \mathbb{S}^2$, \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} This captures the effects of binary collisions (all other collisions are neglected) on the evolution of $f(t,x,v)$. It has become traditional to rewrite the collisional integral on the right as $Q(f,f)$, which emphasizes it being a quadratic expression in $f$. Then, one often writes \begin{align}\label{ch1 e:basic Boltzmann} \partial_tf + v\cdot \nabla_x f = Q(f,f) \end{align} The kernel $B(v-w,\sigma)$ typically has the form \begin{align} B(u,\sigma) = b( \hat u\cdot \sigma)|v-w|^\gamma,\;\ \hat u = u/|u|. \end{align} This kernel structure is directly related to the type of collision/interaction between particles, and this was first understood by Maxwell. In fact, Maxwell showed that if individual particles interact between themselves according to an inverse square potential \begin{align} \phi(r) = \frac{1}{r^{s-1}} \text{ for some } s, \end{align} then the corresponding $B$ will be as above, with \begin{align} \gamma = \frac{s-5}{s-1} \text{ and } b(\cos(\theta)) \approx \frac{1}{\theta^{2+\frac{2}{1-s}}} \end{align} Note that this relates $\gamma$ and $b(\cdot)$ for this particular model – however, it is of interest to consider $b(\cdot)$ and $\gamma$ with other values unrelated to $s$ above.

One can enrich the equation (B) model by considering multiple types of particles – and accordingly, multiple densities and coupled evolution equations, one may also couple this system with other models. For instance one can consider equations of the form \begin{align} \partial_tf + v\cdot \nabla_x f + E\cdot \nabla_x f= Q(f,f) \end{align} where $E = E(t,x)$ is a fixed time changing vector field. There is also the non-linear Vlasov-Poisson-Boltzmann equation which arises if we one asks $E(t,x) = -\nabla_x U(t,x)$, where $U$ is a total potential corresponding to the $f$ at time $t$, that is \begin{align} U(x,t) = \int_{\mathbb{R}^3}\Phi(x-y)\rho(t,x)dx \end{align} where $\rho(t,x) = \int_{\mathbb{R}^3}f(t,x,v)dv$ is the density of all particles at location $xs$.

However, the analysis of all such models are dependent on our understanding for the canonical model (B) for a monoatomic gas, so we will mostly focus on (B) as a canonical equation.

The Cauchy problem: global existence versus finite time breakdown.

The Boltzmann equation is an important model in and of itself, but if even if you are not initially interested in its study it does serve as a MacGuffin for the study of the PDE. As such, a chief motivating question for us will be (the still unresolved) quesstion of whether the Boltzmann equation admits for any initial smooth data a unique, smooth solution for all positive, or whether there is initially smooth data that ends in finite time singularities.

Problem. Given any initial smooth mass density $f_{\text{in}}(x,v)$, show there is a unique smooth function $f(t,x,v)$ defined for all $t>0$ and all $x,v\in\mathbb{R}^3$ such that $f(0,x,v) = f_{\text{in}}(x,v)$ and solving \begin{align} \partial_tf + v\cdot \nabla_x f = Q(f,f) \text{ in } (0,\infty)\times \mathbb{R}^3 \times \mathbb{R}^3, \end{align} or, find an initially smooth $f_{\text{in}}(x,v)$ and a solution to the equation above in some interval $(0,T_{\text{sing}})\times \mathbb{R}^3\times \mathbb{R}^3$ such that $f(t,x,v)$ develops a singularity as $t$ approaches $T_{\text{sing}}$?.

The problem has been studied by mathematicians in one form or another since at least the 1930’s. It was first Torsten Carleman who studied the Cauchy problem in the setting known as the space-homogeneous regime, there $f$ is independent of $x$ and so one asks for $f=f(t,v)$ solving

\begin{align} \partial_tf = Q(f,f) \text{ in } (0,T)\times \mathbb{R}^3. \end{align} Carleman was able to construct solutions for a special class of regular kernels $B$.

The analysis in general is quite difficult, reflecting the richness of phenomena modeled by (B). In fact, one could argue that the most challenging issue today is understanding $v\cdot \nabla_x$ in \begin{align} \partial_t f + v\cdot \nabla_x f= Q(f,f), \end{align} and what role it plays in propagating the effects of $Q(f,f)$, which is better understood on its own right now (more on this below).

Two recent results

In recent years there has been considerable progress in understanding the Cauchy problem, although we are still far from understanding the full problem.

Among the many results, we are going to highlight two that are particularly relevant for what we will study this semester. For the first one, we need to introduce the following fields associated to a function $f(t,x,v)$, they are the mass density, the macroscopic velocity, the energy density, and the entropy density, and they are defined by the following relations

\begin{align} \rho(t,x) = \int_{\mathbb{R}^3}f(t,x,v)dv \end{align}

\begin{align} \rho(t,x)u(t,x) = \int_{\mathbb{R}^3}f(t,x,v)vdv \end{align}

\begin{align} \rho(t,x)e(t,x) = \int_{\mathbb{R}^3}f(t,x,v)|v|^2dv \end{align}

\begin{align} \rho(t,x) h(t,x) = \int_{\mathbb{R}^3}(f\log f)(t,x,v)dv \end{align}

The following result is not exactly stated in this way in the literature, in particular, in the literature it is stated as an a priori estimate without the existence claim. However the version below follows from a combination of recent results, with the key role played by a series of works by Imbert and Silvestre, starting with their 2019 result.

Theorem. There is a maximal interval of existence $T_{\max} \in (0,\infty]$ such that there is a unique smooth solution to the Cauchy problem in $\mathbb{T}^3\times \mathbb{R}^3\times [0,T_{\max})$ and if $T_{\max}<\infty$ no solution exists in a larger interval. Moreover, when $Q(f,f)$ models moderately soft potentials (for very soft potentials one must an additional integrability condition), the following continuation criterion holds: if $T>0$ is such that there are constants $m_0,M_0,E_0,$ and $H_0$ such that for all $(t,x) \in [0,T]\times \mathbb{T}^3$ we have \begin{align} m_0 \leq \rho(t,x) \leq M_0,\;\;e(t,x) \leq E_0,\;\;h(t,x) \leq H_0 \end{align} then $T<T_{\max}$, i.e. the flow goes on classically in a longer time interval.

One way to read the above result is that if a solution of the Boltzmann equation $f(t,x,v)$ develops singularities in finite time, then those singularities will be observed at the macroscopic scale, i.e. at the level of the $\rho(t,x), e(t,x), h(t,x)$.

The study on the existence or not of initially smooth solutions that blow up in finite time is an active frontier of the theory. The above result

More recently, the case of spatially homogeneous solutions has been resolved, building up on the local existence theory and the discovery of a new monotone quantity for the Landau equation in my work with Silvestre (2025), and subsequently for the Boltzmann equation in work of Imbert, Silvestre, and Villani (2025).

Theorem. Assume $f_{\text{in}}$ is constant in $x$, $f_{\text{in}}(x,v) = f_{\text{in}}(v)$, and that it is a smooth function with fast decay at infinity. Assume $Q(f,f)$ is the Landau collision operator with $\gamma \in [-3,0]$ or the Boltzmann collision operator with $\gamma \in (-3,0)$. Then, there is a unique global in time solution of \begin{align} \partial_t f & = Q(f,f) \text{ in } \mathbb{R}^3\times [0,\infty) \end{align} \begin{align} f(0,v) & = f_{\text{in}}(v) \text{ in } \mathbb{R}^3 \end{align}

In this course we will learn the tools from different branches of analysis that made these results possible. These include a robust theory of linear parabolic equations, both local and integro-differential; functional inequalities; regularity theorems following De Giorgi-Nash-Moser and their extension to kinetic equations; and last but not least a lifting procedure to analyze the Fisher information in the space homogeneous regime. The operating philosophy here is that the recent progress for the Boltzmann equation provides a good framework to learn many important techniques in modern PDE.

The role of (non-local) parabolic equations

We end this first lecture by further clarifying how non-local parabolic equations are of prime importance to the study of the Boltzmann equation. The Boltzmann collisional integral $Q(f,f)$ defines in an obvious way a bilinear operator over pairs of smooth functions $f,g:\mathbb{R}^3\to\mathbb{R}$, namely

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

For a fixed function $g$, the operator $f\mapsto Q(g,f)$ is a nonlocal elliptic operator. Later in the semester we will study the structure of this operator in some detail, arriving at the following description: given $g$, there is a kernel $K_g(v,w)$ and a function $U_g(v)$ such that

\begin{align} Q(g,f) = \int_{\mathbb{R}^d}(f(w)-f(v))K_g(v,w)\;dw + f(v)U_g(v) \end{align}

Moreover, $U_g$ is computed from $g$ via convolution against an explicit kernel, and for a non-trivial set of pairs $(v,w)$ (depending on $g$) the kernel $K_g(v,w)$ satisfies
\begin{align} K_g(v,w) \geq c_0 |v-w|^{-3-\alpha} \end{align}

The kernel on the right corresponds to the kernel for the fractional Laplacian $-(-\Delta)^{\alpha/2}$ in $\mathbb{R}^3$. This means that the analysis of the Boltzmann (B) equation has a lot linear integro-differential equation (known as the fractional Kolmogorov equation)

\begin{align} \partial_t f + v\cdot \nabla_x = -(-\Delta_v)^{\alpha/2}f \end{align}

Put differently, the Boltzmann equation is a nonlinear analogue of this equation, the “coefficients” of the underlying nonlocal equation depending from the solution itself in a nonlocal way.