---
title: Fuzzy Landau Equation
url: https://www.emergentmind.com/topics/fuzzy-landau-equation
type: topic
---

# Fuzzy Landau Equation

Searching arXiv for recent papers on the fuzzy Landau equation and closely related variational/solvability results.
The fuzzy Landau equation is a nonlocal kinetic equation in which the spatially inhomogeneous Landau collision operator is modified so that collisions are delocalised in the position variable through a nonnegative interaction kernel. For a one-particle density $f=f(t,x,v)$, the basic form is
\[
\partial_t f+v\cdot\nabla_x f=Q_{\sf fuz}(f,f),
\]
with
\[
Q_{\sf fuz}(f,f)
=\nabla_v\!\cdot\!\int_{\Omega\times\mathbb R^3}
\kappa(x-x_*)\,a(v-v_*)\,
\bigl[f_*\,\nabla_v f-f\,\nabla_{v_*}f_*\bigr]\,
dx_*\,dv_*,
\]
where $f_*=f(t,x_*,v_*)$ and $\Omega=\mathbb T^3$ or $\mathbb R^3$ in the three-dimensional formulation. The current theory places this equation in the GENERIC framework, derives a De Giorgi-type variational characterization, proves global $H$-solutions, establishes global-in-time existence and uniqueness of smooth solutions for moderately soft potentials in a specific regime, identifies new Fisher-information monotonicity, derives the grazing-collision limit from non-cutoff fuzzy Boltzmann equations, and proves conditional uniqueness of weak solutions through explicit $2$-Wasserstein stability estimates [2504.07666, 2507.10288, 2507.08689, 2512.04713, 2603.27509].

## 1. Kinetic definition and the role of spatial delocalisation

The defining modification is the replacement of the spatially local collision rule of the classical inhomogeneous Landau equation by a delocalised interaction. In the classical operator, collisions are fully localized in $x$, corresponding formally to the limit $\kappa\to\delta_0$. In the fuzzy equation, the spatial “fuzziness” is encoded by a kernel $\kappa:\Omega\to[0,\infty)$ with $\int_\Omega \kappa=1$; examples in the literature include smooth kernels and exponentially decaying kernels, while later formulations also allow bounded spatial interaction kernels and smooth Gaussian-type kernels [2504.07666, 2512.04713].

The velocity interaction has the standard Landau tensor structure
\[
a(z)=A(z)\,\Pi_{z^\perp},\qquad
\Pi_{z^\perp}=I-\frac{z\otimes z}{|z|^2}.
\]
For Coulomb-type interactions one takes $A(z)=|z|^{2+\gamma}$ with $\gamma\in(-3,1]$ in the three-dimensional setting, while the solvability theory in general dimension treats either $A(|z|)=|z|^{2+\gamma}$ with $\gamma\in(-\gamma_d,1]$, $\gamma_d=\min(d,4)$, or the milder class
\[
C^{-1}\langle z\rangle^{2+\gamma}\le A(|z|)\le C\,\langle z\rangle^{2+\gamma},\qquad \gamma\in(-\infty,1]
\]
[2504.07666, 2507.10288].

This spatial delocalisation changes the analytic character of the equation. Part II explicitly contrasts the fuzzy model with the classical inhomogeneous Landau equation, noting that fuzziness introduces a genuinely non-local transport–diffusion in $x$ but improves compactness and coercivity in $v$. The same source records two asymptotic regimes: $\kappa^\sigma\to\delta(x-x_*)$ gives the classical inhomogeneous Landau equation, while $\kappa\equiv1$ yields the homogeneous Landau equation [2507.10288].

## 2. GENERIC formulation and variational characterization

Duong and He recast the fuzzy Landau equation as a GENERIC system, with state variable $f$ in
\[
\mathcal Z
=\Bigl\{\,f\ge0:\ f\in L^1(\Omega\times\mathbb R^3),\ \int f=1,\ \int f|v|^2<\infty\Bigr\}.
\]
The energy and entropy functionals are
\[
E[f]=\int_{\Omega\times\mathbb R^3}\frac12|v|^2\,f(x,v)\,dx\,dv,
\qquad
S[f]=-\!H[f],\quad
H[f]=\int f\log f\,dx\,dv.
\]
The reversible part is generated by the antisymmetric operator
\[
L_f g
=-\nabla_x\!\cdot\!(f\,\nabla_v g)+\nabla_v\!\cdot\!(f\,\nabla_x g),
\]
equivalently
\[
L_f g=-\nabla_{(x,v)}\!\cdot\!\bigl(f\,\mathbb J\nabla_{(x,v)}g\bigr),\qquad
\mathbb J=\begin{pmatrix}0&I\\ -I&0\end{pmatrix}.
\]
The irreversible part is encoded by a diffusion operator $M_f$ built from the delocalised Landau tensor. Introducing the fuzzy Landau-gradient
\[
\tilde\nabla g(x,v;x_*,v_*)
=\sqrt{A(v-v_*)}\,
\Pi_{(v-v_*)^\perp}
\bigl(\nabla_v g(x,v)-\nabla_{v_*}g(x_*,v_*)\bigr),
\]
one has
\[
M_f g=-\frac12\,\tilde\nabla\cdot\bigl(\kappa\,f\,f_*\,\tilde\nabla g\bigr),
\qquad
\langle g,M_f g\rangle
=\frac12\int \kappa\,f\,f_*\,|\tilde\nabla g|^2\ge0.
\]
The GENERIC equation
\[
\partial_t f=L_f\Bigl(\frac{\delta E}{\delta f}\Bigr)+M_f\Bigl(\frac{\delta S}{\delta f}\Bigr),
\qquad
\frac{\delta E}{\delta f}=\frac12|v|^2,\quad
\frac{\delta S}{\delta f}=-(\log f+1),
\]
recovers exactly the fuzzy Landau evolution. The construction satisfies the standard degeneracies
\[
L_f\,\frac{\delta S}{\delta f}=0,\qquad
M_f\,\frac{\delta E}{\delta f}=0,
\]
together with antisymmetry of the Poisson bracket and formal Jacobi identity [2504.07666].

A second structural layer is a De Giorgi-type variational characterization. For a transport–grazing-rate pair $(f,U)$ satisfying
\[
\partial_t f+v\cdot\nabla_x f+\frac12\,\tilde\nabla\cdot U=0,
\]
the action
\[
A(f,U)=\frac12\int \frac{|U|^2}{f\,f_*\,\kappa\,A}\,dx\,dv\,dx_*\,dv_*
\]
and entropy dissipation
\[
D(f)=\frac12\int \kappa\,f\,f_*\,|\tilde\nabla\log f|^2
\]
enter the functional
\[
J_T(f,U)=H[f(T)]-H[f(0)]
+\frac12\int_0^T D(f)\,dt
+\frac12\int_0^T A(f,U)\,dt\ge0.
\]
Equality holds if and only if
\[
U=-\,f\,f_*\,\kappa\,\tilde\nabla\log f,
\]
which is equivalent to $f$ being a weak, or “$H$-”, solution of the fuzzy Landau equation. This formulation unifies the GENERIC flow and the $H$-theorem in a single min-action principle [2504.07666].

## 3. Conservation laws, entropy production, and global \(H\)-solutions

The GENERIC structure yields the fundamental balance laws. For smooth solutions,
\[
\frac{d}{dt}\int f=0,\qquad
\frac{d}{dt}\int f\,v=0,\qquad
\frac{d}{dt}\int f\,|v|^2=0.
\]
Entropy is nonincreasing:
\[
\frac{d}{dt}H[f(t)]
=-\frac12\int \kappa\,f\,f_*\,|\tilde\nabla\log f|^2\le0.
\]
Part I describes this as automatic enforcement of thermodynamic consistency (1st/2nd laws), correct conservation laws and entropy dissipation, together with a “Lyapunov structure” well suited to long-time asymptotics [2504.07666].

Part II develops the weak theory in a functional framework based on weighted $L^1$ spaces, mixed $L^p_vL^q_x$ norms, entropy space $L\log L$, velocity moments $M_s(f)$, Fisher information
\[
I(f)=\int \langle v\rangle^\gamma |\nabla_v\sqrt f|^2\,dx\,dv,
\]
and fuzzy entropy dissipation
\[
D(f)=\frac12\int\!\!\!\int
\kappa(x-x_*)\,f\,f_*\,
|\tilde\nabla\log f|^2\,dx\,dv\,dx_*\,dv_*\ge0.
\]
For initial data
\[
f_0\ge0,\qquad
f_0\in L^1_{2,2+\max(0,\gamma)}\cap L\log L(\Omega\times\mathbb R^d),
\]
Duong and He define weak solutions and $H$-solutions through the integrated GENERIC identity
\[
\int f_0\,\varphi(0)
+\int_0^T\!\!\int f(\partial_t\varphi+v\cdot\nabla_x\varphi)
=-\frac12\int_0^T\!\!\int
\kappa\,f\,f_*\,
\tilde\nabla\varphi\cdot\tilde\nabla\log f.
\]
They prove existence of global-in-time $H$-solutions. For such solutions, mass and momentum are conserved, energy satisfies
\[
\int |v|^2f_t\le\int |v|^2f_0,
\]
the spatial second moment obeys
\[
\int |x|^2f_t\le C(T)\int (|x|^2+|v|^2)f_0,
\]
and the entropy inequality holds:
\[
H(f_t)-H(f_0)+\int_0^t D(f_s)\,ds\le0
\]
[2507.10288].

The analytic mechanism behind this existence theory combines several ingredients recorded explicitly in Part II: the Desvillettes-type bound
\[
I(f)\lesssim 1+D(f),
\]
a weighted Sobolev embedding
\[
\|\langle v\rangle^\gamma f\|_{L^1_xL^k_v}\lesssim D(f),\qquad k=\frac d{d-2},
\]
Hardy–Littlewood–Sobolev estimates for soft potentials, coercivity of the diffusion matrix
\[
\bar a(x,v)=\kappa*_x f *_v a
\]
through lower bounds of the form
\[
\xi^T\bar a(x,v)\,\xi
\ge C_{\rm coe}\,\langle v\rangle^\gamma |\xi|^2\,\exp(-C\langle x\rangle)
\]
or
\[
\xi^T\bar a(x,v)\,\xi
\ge C_{\rm coe}\,\langle v\rangle^\gamma |\xi|^2,
\]
and finally approximation, parabolic regularisation in $v$, and a fixed-point argument leading to smooth approximants, a priori bounds, compactness, and passage to the limit [2507.10288].

## 4. Strong well-posedness, ellipticity, and Fisher information

A complementary line of work studies a fuzzy variant on $\mathbb R^3\times\mathbb R^3$ with a smooth, everywhere-positive mollifier $K_\varepsilon(x,y)$ in place of the spatial delta kernel. In this formulation,
\[
Q_\varepsilon(f,f)(x,v)
=\operatorname{div}_v\int_{\mathbb R^3}\int_{\mathbb R^3}
K_\varepsilon(x,y)\,a(v-v_*)\,
\bigl[f(y,v_*)\nabla_v f(x,v)-f(x,v)\nabla_{v_*}f(y,v_*)\bigr]
\,dv_*\,dy,
\]
with
\[
C_1|x-y|^{-\lambda}\le K_\varepsilon(x,y)\le C_2|x-y|^{-\lambda}.
\]
For $\gamma\in(-2,0]$ and integer $s\ge2$, the main theorem gives a unique strong solution provided the initial datum satisfies the stated $L^1$, $L^\infty$, moment, and weighted Sobolev assumptions. The solution obeys the weighted bounds
\[
v^{\,s-|\alpha|-|\beta|}D_v^\alpha D_x^\beta f\in L^\infty(0,T;L^2_{x,v}),
\]
and
\[
v^{\,s+\gamma/2-|\alpha|-|\beta|}D_v^\alpha D_x^\beta \nabla_v f\in L^2(0,T;L^2_{x,v}),
\]
for all multi-indices with $|\alpha|+|\beta|\le s$; mass, momentum, and energy are conserved [2507.08689].

The central a priori mechanism is a global ellipticity effect created by delocalisation. Defining
\[
A[f](x,v)=\int_{\mathbb R^6}\kappa(x-y)\,a(v-v_*)\,f(y,v_*)\,dy\,dv_*,
\]
the paper proves that there exists $\alpha>0$ such that
\[
\xi\cdot A[f](x,v)\,\xi\ge \alpha |x|^{-\lambda}|v|^\gamma |\xi|^2
\]
for all $x,v,\xi$. The source emphasizes that because $K_\varepsilon>0$ everywhere, any two particles “feel” a weak collisional interaction regardless of spatial separation, and that this forces an immediate gain of ellipticity in $v$ even where $f$ would be vacuum [2507.08689].

The same paper establishes additional Lyapunov structure through Fisher information. For $\kappa\equiv1$ and $\gamma\in[-3,1]$, any smooth solution satisfies
\[
\frac{d}{dt}I_x(f(t))\le0,
\qquad
I_x(f)=\int |\nabla_x\sqrt f|^2\,dx\,dv,
\]
and
\[
\sup_{0\le t\le T}\bigl(I_v(f(t))+I_x(f(t))\bigr)\le C(T;f^{in}),
\qquad
I_v(f)=\int |\nabla_v\sqrt f|^2\,dx\,dv.
\]
The proof uses a doubled-variable representation with
\[
F(x,v,y,w)=f(x,v)f(y,w),
\]
which converts the Landau operator into a diffusion-type operator on $\mathbb R^{12}$. In addition, for Maxwell molecules $(\gamma=0)$ the mixed Fisher functional
\[
J(f)=\int |(t\nabla_x+\nabla_v)f|^2/f\,dx\,dv
\]
satisfies $dJ/dt\le0$ [2507.08689].

## 5. Grazing-collision limit and asymptotic connections

The fuzzy Landau equation arises as the grazing-collision limit of non-cutoff fuzzy Boltzmann equations. In the Boltzmann model, particles at $(x,v)$ and $(x_*,v_*)$ undergo delocalised elastic collisions with collision operator
\[
Q(f,f)=\int_{\mathbb R^d_{x_*}\times\mathbb R^d_{v_*}\times S^{d-1}}
\kappa(x-x_*)\,B(|v-v_*|,\sigma)\,(f'f_*'-ff_*)\,
dx_*\,dv_*\,d\sigma,
\]
where
\[
B(|v-v_*|,\sigma)=A_0(|v-v_*|)\,b(\theta),
\qquad
A_0(|v-v_*|)=|v-v_*|^\gamma,\quad \gamma\in[-2,1].
\]
The non-cutoff regime is described by an angular singularity
\[
\beta(\theta):=\sin^{d-2}\theta\,b(\theta)\gtrsim \theta^{-1-\nu},
\qquad \nu\in(0,2),
\qquad
\int_0^{\pi/2}\beta(\theta)\,\theta^2\,d\theta<\infty.
\]
To isolate grazing events one rescales
\[
\beta^\varepsilon(\theta)=\frac{\pi^3}{\varepsilon^3}\,\beta\!\left(\frac{\pi\theta}{\varepsilon}\right),
\]
so that $\operatorname{supp}\beta^\varepsilon\subset[0,\varepsilon/2]$ and $\int\beta^\varepsilon(\theta)\theta^2\,d\theta$ is fixed. Formally, as $\varepsilon\to0$, the Boltzmann operator converges to the fuzzy Landau operator with
\[
A(|v-v_*|)=A_0(|v-v_*|)\,|v-v_*|^2
\]
[2512.04713].

The variational theory persists across this limit. Duong, Golubkov, and He formulate the fuzzy Boltzmann equation through a non-quadratic dual dissipation pair $(\mathfrak R,\mathfrak R^*)$, while the fuzzy Landau equation corresponds to a quadratic dissipation pair. The associated action functionals satisfy
\[
\mathcal J_B(f,U)\ge0,\qquad \mathcal J_L(f,U)\ge0,
\]
with equality characterizing solutions in each case. Their main theorem states that if $f^\varepsilon$ are $H$-solutions of the fuzzy Boltzmann equation under the stated moment–entropy bounds and uniform controls on $\kappa^\varepsilon$, then
\[
f^\varepsilon\to f \quad \text{narrowly},
\qquad
\sqrt{f^\varepsilon}\to\sqrt f \quad \text{strongly in }L^2,
\]
and the limit $f$ is an $H$-solution of the fuzzy Landau equation. The proof proceeds through compactness, velocity averaging and fractional diffusion from the angular singularity, together with a $\Gamma$-liminf argument for the actions and dissipation functionals [2512.04713].

This asymptotic result aligns with the broader program already indicated in the variational paper: the framework is stated to connect smoothly to the grazing-collision limit from Boltzmann to Landau and to the homogeneous fuz or $\delta$-local limits to classical Landau [2504.07666].

## 6. Uniqueness, Wasserstein stability, and terminology

A later development addresses uniqueness of weak solutions in a singular-potential regime. For $\gamma\in[-3,-2]$, the fuzzy Landau equation is written on full phase space $\mathbb R^3\times\mathbb R^3$ as
\[
\partial_t f+v\cdot\nabla_x f
=\operatorname{div}_v\!\int_{\mathbb R^6}
\kappa(x-x_*)\,\Phi(v-v_*)\,(\nabla_v-\nabla_{v_*})
\bigl(f(t,x,v)f(t,x_*,v_*)\bigr)\,dx_*\,dv_*,
\]
with
\[
\Phi(z)=|z|^{2+\gamma}\mathbb P(z),
\qquad
\mathbb P(z)=\mathrm{Id}-\frac{z\otimes z}{|z|^2},
\qquad
b(z)=\operatorname{div}_z\Phi(z)=-2|z|^\gamma z.
\]
A weak solution is required to belong to $\mathcal P_2(\mathbb R^6)$ with finite second moment and
\[
f\in L^1([0,T];L^q_vL^1_x(\mathbb R^6)),
\qquad
q>\frac{3}{4+\gamma}.
\]
If, in addition,
\[
f,g\in L^1([0,T];L^p_vL^1_x(\mathbb R^6)),
\qquad
p=\frac{3}{3+\gamma}\in[3,\infty],
\]
and $\sqrt\kappa$ is bounded and Lipschitz, then the main theorem yields
\[
\sup_{0\le t\le T} d_2^2(f(t),g(t))
\le \omega\!\bigl(d_2^2(f(0),g(0))\bigr),
\]
for a continuous increasing modulus $\omega$ with $\omega(0)=0$. The proof gives the explicit Osgood-type differential inequality
\[
\frac{d}{dt}d_2^2(f(t),g(t))
\le C\,m(t)\,\Psi\!\bigl(d_2^2(f(t),g(t))\bigr),
\]
where
\[
\Psi(r)=\max\{r,-r\ln r\},
\qquad
m(t)=\max\{\|f(t)\|_{L^p_vL^1_x},\|g(t)\|_{L^p_vL^1_x}\}.
\]
This yields conditional uniqueness in the class of weak solutions with finite second moment and the stated $L^1_tL^p_vL^1_x$ integrability [2603.27509].

Two proof strategies are developed. The first builds on the stochastic coupling method of Fournier and Guerin for the homogeneous Landau equation, reformulated analytically through a coupling $\Pi_t\in\Gamma(f(t),g(t))$ and a corresponding linear PDE or McKean–Vlasov-type SDE. The second is an alternative argument based on the symmetrization technique of Guillen and Silvestre, producing comparable stability estimates after a doubled-variable integration-by-parts and commutator analysis. The same framework also covers a wider class of nonlinear Fokker–Planck equations with singular kernels, including the $2$D incompressible Euler equations, the Vlasov–Poisson system, and the Patlak–Keller–Segel model [2603.27509].

A common terminological confusion is that “fuzzy” in the fuzzy Landau equation refers to spatial delocalisation of collisions by the kernel $\kappa(x-x_*)$. This usage is distinct from “fuzzy” in “fuzzy dark matter,” where the term appears in the study of a quantum Landau equation for bosonic particles in gravitational interaction, developed in an infinite and spatially homogeneous system or a local approximation [2012.12858]. The two contexts both involve Landau-type kinetic operators, conservation laws, and an $H$-theorem, but they arise from different modelling assumptions and represent different physical regimes.

Source: https://www.emergentmind.com/topics/fuzzy-landau-equation