Finite time singularities in the Landau equation with very hard potentials

TL;DR

This work constructs finite-time singularities for the 3D inhomogeneous Landau equation with γ ∈ (√3, 2], by lifting smooth, isentropic Euler implosion profiles into the kinetic setting through a self-similar scaling. The authors develop a rigorous macro-micro stability framework in rescaled variables, introducing weighted Sobolev-type spaces and a local Maxwellian profile to capture the asymptotic hydrodynamic limit to Euler while preserving kinetic dissipation. A finite-codimension stability argument, combining precise coercivity for the macroscopic Euler system and sharp micro-dc dissipativity estimates for the collision operator, yields a blowup that is smooth away from the origin but with C^α growth and controlled L^∞ bounds. The results provide a first example of a collisional kinetic model that is globally well-posed in the homogeneous setting yet admits finite-time singularities in an inhomogeneous context, linking kinetic theory to nonlinear hydrodynamic collapse with potential implications for understanding singularity formation in kinetic-plasma models and fluid dynamics.

Abstract

We consider the inhomogeneous Landau equation with $γ\in (\sqrt{3},2]$ and construct smooth, strictly positive initial data that develop a finite time singularity. The $C^α$-norm of the distribution function blows up for every $α>0$, whereas its $L^{\infty}$-norm remains uniformly bounded. In self-similar variables, the solution becomes asymptotically hydrodynamic - the distribution function converges to a local Maxwellian, while the hydrodynamic fields develop an asymptotically self-similar implosion whose profile coincides with a smooth imploding profile of the compressible Euler equations. To our knowledge, this provides the first example of a collisional kinetic model which is globally well-posed in the homogeneous setting, but admits finite time singularities for inhomogeneous data.

Figures (2)

• Figure 1: A cartoon of the limiting density $\mu( {v})$ with $x \neq 0$ when both $x$ and $v$ are one-dimensional, with $v$ defined in \ref{['eq:def_small_vring']}. We have taken $r=1.26<3-\sqrt{3}$ and set $C _{\bar{\mathsf C}} = C _{\bar{\mathbf{U}}} = 1$, $R _0 \to \infty$ in \ref{['eq:def_small_vring']} and \ref{['def:cRx']}. The two images represent the same function $\mu( {v})$ from two different perspectives (rotated by $90^\circ$ in the horizontal plane).
• Figure 2: A contour plot (top-down view) of the same function $\mu( {v})$ from Figure \ref{['fig:1']}. The red regions represent higher values of $\mu( {v})$, while the blue regions represent values close to zero.

Theorems & Definitions (126)

• Theorem 1.1
• Remark 1.2: Range of $\gamma$
• Remark 1.3: Set of initial data
• Remark 1.4: Tail fattening at $x = 0$
• Remark 1.5: Smoothness away from $x=0$
• Remark 1.6: Limiting solution
• Remark 1.7: $L^{\infty}$ bound $\not \Rightarrow$ smoothness
• Remark 1.8: Local well-posedness
• Remark 2.1: Far-field profiles
• Remark 2.2: Growth rate ${R _s}$