Inhomogeneous Landau–Coulomb Global Regularity (HST All-Moment Data)

Inhomogeneous Landau–Coulomb Global Regularity (HST All-Moment Data)

Definitions and mathematical statement

Write $\langle v\rangle=(1+|v|^2)^{1/2}$. The inhomogeneous Landau equation transports a nonnegative particle density $f(t,x,v)$ by $(\partial_t+v\cdot\nabla_x)f=Q(f,f)$ with the Coulomb matrix $A(z)=a|z|^{-1}(I-z\otimes z/|z|^2)$, $a>0$. The collision operator and the all-polynomial-moment, global Hölder, and uniformly positive local-ball hypotheses are retained exactly in the catalogue quotation below. Let $T_E$ be the maximal continuation time of the classical branch selected in the HST source. The problem is $T_E=\infty$ for every admitted initial datum. The source continuation quantity is \[ \Psi(T)=\mathop{\mathrm{ess\,sup}}_{t\le T,x}\int_{{\mathbb{R}}^3}f(t,x,v)\,\mathrm d v +\mathop{\mathrm{ess\,sup}}_{t\le T,x,v}f(t,x,v). \] Establishing its finiteness on every finite time interval is a possible continuation route, not an assumption granting the conclusion. Total mass over all space need not be finite.

Formulation and concept references: [S1].

Short English statement

Does the specified classical Landau–Coulomb solution continue for all time for every initial density in the stated all-moment class, without assuming smallness or bounded continuation quantities in advance?

Research attempt

Current frontier.

The HST paper is Henderson--Snelson--Tarfulea's local-solution theorem; Theorems 1.2--1.4 and Section 5 give the continuation and uniqueness conventions used in the original entry [S1]. For Coulomb interactions the continuation criterion contains both the uniform local velocity mass and $\|f\|_\infty$. Guillen--Silvestre prove global regularity and monotonicity of Fisher information for the spatially homogeneous Landau equation [E1], Theorems 1.1--1.2. That important result cannot be read as a theorem for the inhomogeneous branch here, and the initial hypotheses of that branch are not replaced by finite total mass in all of phase space.

Attack route.

A maximum-principle route would need the diffusion to offset the local quadratic reaction. A moment route would need to control local spatial concentration, not only conserved total quantities. A transfer of the homogeneous Fisher-information argument must handle commutators with transport. We examine all three at the level of explicit identities and construct admissible smooth test data that break the simplest maximum estimate. The goal is to determine which proposed a priori input could actually close the HST continuation criterion.

Attempt: the Coulomb reaction has a fixed positive sign.

Let $\mathcal A[f](v)=\int A(v-w)f(w)\,dw$, with $x,t$ suppressed. Since $A(z)=aD^2|z|$ in three dimensions, \[ b_i(z):=\partial_jA_{ij}(z)=-2a\frac{z_i}{|z|^3}, \qquad \partial_i b_i=-8\pi a\delta_0 \] in distributions. The coefficient of $\delta_0$ follows from the flux of $z/|z|^3$ through a sphere, which is $4\pi$. Integrating the $w$ derivative in the collision operator and expanding the $v$ divergence cancels its two first-derivative terms. Therefore, for smooth rapidly decaying velocity profiles, \begin{equation}\label{wsref-0-65-71-31-37-39-2d-6e-6f-6e-64-69-76} (\partial_t+v\cdot\nabla_x)f =\mathcal A[f]:D_v^2f+8\pi a f^2. \end{equation} This normalization uses the physical constant in the question. A convention absorbing $8\pi a$ into time would not justify dropping it in this display.

At an attained maximum in $(x,v)$, diffusion is nonpositive and transport vanishes, so the elementary maximum comparison gives only the Riccati upper inequality $M'\le8\pi a M^2$. Where the usual localized supremum comparison is justified this yields \[ M(t)\le\frac{M(0)}{1-8\pi a M(0)t} \quad\text{for }t<(8\pi aM(0))^{-1}. \] This is a short-time upper bound, not proof of blow-up and not a global bound. On the noncompact spatial domain a localization argument is needed to promote pointwise maximum reasoning to a supremum; the countertest below uses spatially homogeneous data and an actual velocity plateau, avoiding that separate issue entirely.

Attempt: admissible plateaus defeat a naive damping lemma.

Choose $G(v)>0$ smooth, rapidly decreasing, and constant equal to $c>0$ on $|v|\le1$. Choose $\chi\in C_c^\infty({\mathbb{R}}^3)$ with $0\le\chi\le1$ and $\chi=1$ near the origin. For small $\varepsilon>0$ let \begin{equation}\label{wsref-0-65-71-31-37-39-2d-73-70-69-6b-65} f_{0,\varepsilon}(x,v)=G(v)+\varepsilon^{-1}\chi(v/\varepsilon), \end{equation} independent of $x$. Each datum satisfies the stated global Hölder and all-polynomial-weight bounds, with constants allowed to depend on $\varepsilon$. The positive local-ball hypothesis holds uniformly in $x$ by taking $v_x=0$ and a small fixed radius; the baseline supplies a fixed lower bound. Thus these are legitimate data for the chosen class, not excluded spikes.

At $(x,0)$ the velocity Hessian and the spatial gradient vanish. For these smooth data the equation at initial time, or its smooth right-hand limit, gives exactly \begin{equation}\label{wsref-0-65-71-31-37-39-2d-70-6f-73-69-74-69-76-65} \partial_t f_{\varepsilon}(0,x,0) =8\pi a(c+\varepsilon^{-1})^2>0. \end{equation} The spike adds only $O(\varepsilon^2)$ to velocity mass, $O(\varepsilon^4)$ to energy, and $O(\varepsilon^2\log(1/\varepsilon))$ to the positive entropy contribution. Every fixed integrated polynomial moment remains bounded in this family. The entropy estimates follow by separating the support of the spike and using its volume $O(\varepsilon^3)$ and height $O(\varepsilon^{-1})$; the baseline has finite entropy. Therefore bounded mass, energy, and entropy alone cannot force a universal pointwise inequality with an instantaneous damping term that dominates the quadratic growth at all high flat maxima.

This does not contradict [E1]: diffusion may act immediately off the plateau and subsequently control the evolution by a nonlocal mechanism. The construction is not evidence of a solution blowing up. It refutes only a proposed instantaneous maximum-damping estimate derived from those coarse quantities. Uniform Hölder or pointwise weighted bounds across this family have deliberately not been asserted.

Attempt: the exact local-mass interpolation.

Let $M=\|f\|_{L^\infty_{x,v}}$ and $F_k=\|\langle v\rangle^k f\|_{L^\infty_{x,v}}$ for a fixed $k>3$. For $R\ge1$, split the velocity integral into a ball and its complement: \[ \sup_x\int f(x,v)\,dv\le CMR^3+C_kF_kR^{3-k}. \] If $M>0$, then $F_k\ge M$ and $R=(F_k/M)^{1/k}\ge1$ is admissible. It follows that \begin{equation}\label{wsref-0-65-71-31-37-39-2d-69-6e-74-65-72-70-6f-6c-61-74-69-6f-6e} \sup_x\int f(x,v)\,dv \le C_k M^{1-3/k}F_k^{3/k}. \end{equation} For $M=0$ the statement is immediate. Thus bounds on $M$ and one such weighted supremum up to a putative finite maximal time would bound $\Psi$ and invoke HST continuation. But boundedness of every weight on each compact subinterval of existence does not provide a bound uniform as that maximal time is approached. Assuming the latter would insert the missing estimate into the hypotheses.

Velocity integration of the equation yields $\partial_t\rho+\nabla_x\cdot j=0$, not a maximum principle for $\rho$. The flux $j=\int vf\,dv$ has no sign in this equation. Hence even a conserved global spatial mass, when finite in a smaller data class, would not bound the spatial supremum in (eq179-interpolation). For the actual class it need not be finite at all.

Attempt: transport obstructs the homogeneous monotonicity.

On a smooth positive rapidly decaying phase-space test solution where the integrals are finite, set $I_v(f)=\int f|\nabla_v\log f|^2\,dx\,dv$. The transport part alone contributes \begin{equation}\label{wsref-0-65-71-31-37-39-2d-63-72-6f-73-73} \left.\frac{d}{dt}I_v(f)\right|_{\partial_t f=-v\cdot\nabla_xf} =-2\int f\,\nabla_v\log f\cdot\nabla_x\log f\,dx\,dv. \end{equation} Indeed $D_t\log f=0$ for free transport, but $D_t\partial_{v_i}\log f=-\partial_{x_i}\log f$ because $[\partial_{v_i},D_t]=\partial_{x_i}$; integration of the transport divergence has no boundary term under these decay assumptions. The right-hand side has no fixed sign. Thus homogeneous monotonicity of $I_v$ does not transfer simply by adding transport. A hypocoercive combination involving spatial derivatives is a plausible repair, but it would require new estimates that control those derivatives and the local mass without assuming the desired continuation bound.

Stress test.

Equation (eq179-cross) is used only as a finite-integral diagnostic, not as a globally defined functional for all HST data. The admissible spatially homogeneous plateau family has infinite total spatial mass and is nonetheless allowed by the target. Mixing these two tests into one unjustified global integration argument would be an error.

The distributional Coulomb divergence, the positive sign of its reaction, and the distinction between integrated moments and weighted supremum bounds have all been tracked. Neither the Riccati comparison nor the plateau growth proves finite-time blow-up. Conversely, the homogeneous global theorem does not supply the missing estimates for the spatial flux or the commutator term.

Outcome.

Outcome: Exploratory approach with unresolved gap.

The attempt derives the exact reaction identity, an admissible countertest to a naive maximum-damping argument, a local-mass interpolation bound, and the transport obstruction to direct Fisher-information monotonicity. These computations identify specific missing estimates rather than a new global regularity theorem. What remains is a uniform finite-time bound closing $\Psi$ for every admitted inhomogeneous datum. No blow-up solution or complete continuation argument is claimed.

Sources

• S1 Inhomogeneous Landau–Coulomb Global Regularity (HST All-Moment Data); exact editorial selection using the primary formulations and hypotheses recorded in the source receipts. https://www.numdam.org/item/10.1016/j.anihpc.2020.04.004.pdf. Catalogue formulation source.
• E1 Nestor Guillen and Luis Silvestre, The Landau equation does not blow up, Acta Mathematica 234$2$ $2025$; arXiv:2311.09420v2, Theorems 1.1--1.2 and Section 11. https://arxiv.org/html/2311.09420v2. Consulted 27 September 2026. The cited theorem is spatially homogeneous; the inhomogeneous transport term is not covered.