arXiv subjects

Lukas Niebel

Publications and source records attributed to Lukas Niebel.

At least 19 recordsLinked to original sources

arXiv2026

A fast dynamo with zero topological entropy

We construct an autonomous Lipschitz fast dynamo on the flat three-torus whose particle flow has zero topological entropy and whose ideal induction equation has no exponential growth. For every sufficiently small positive magnetic diffusivity, the induction operator has an eigenvalue with real part bounded below by a positive constant independent of the diffusivity. A corresponding non-zero real-valued magnetic field satisfies an exact exponential growth law in $\mathrm{L}^2$. For the same velocity field, every ideal solution grows at most linearly in $\mathrm{L}^2$. Moreover, the particle flow and its inverse have Lipschitz constants growing at most linearly in time. The velocity field is real-valued, divergence-free, differentiable everywhere and smooth away from one circle, but is not $\mathrm{C}^1$.

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arXiv2026

Global rigidity of two-dimensional bubbles

We study bubbles, which are stationary hollow vortices with surface tension, in two dimensions. Such objects solve an overdetermined elliptic free boundary problem in an exterior domain, with an additional boundary condition involving curvature and the Neumann trace. Motivated by a conjecture of Crowdy and Wegmann, we prove that every solution prescribed by a Jordan curve is circular for $0 \le \mathrm{We} \le 3$, where $3$ is the first bifurcation value. This range is sharp: the bifurcation branch at $\mathrm{We}=3$ contains non-circular solutions with $\mathrm{We} > 3$ arbitrarily close to $3$. This elliptic problem describes critical points of the sum of the perimeter and the logarithmic potential energy of compact sets. We prove an area-constrained perimeter-capacity inequality and show that, in the associated convexity-constrained variational problem, the unit disc is the unique minimiser up to translation if and only if $0 \le \mathrm{We} \le 3$.

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arXiv2026

Lions' Maximal Regularity Problem for Divergence-Form Differential Operators: Failure at the $\frac{1}{2}$-Hölder Endpoint

In this work we give a counterexample to maximal $\mathrm{L}^2$-regularity in Lions' problem for divergence-form differential operators. On a bounded interval, we construct a bounded, uniformly elliptic, real scalar diffusion coefficient that is $\frac{1}{2}$-Hölder continuous in time with values in spatial $\mathrm{L}^\infty$. It can be chosen arbitrarily close to the constant coefficient of the heat equation. For zero initial data and a forcing term that is continuous in time with square-integrable spatial values, the unique Lions variational solution has a time derivative that is not square integrable in space-time. Thus $\frac{1}{2}$-Hölder continuity alone does not imply maximal $\mathrm{L}^2$-regularity, even for arbitrarily small scalar perturbations of the heat equation. The construction is based on a lacunary family of oscillatory trigonometric modes localised on shrinking time intervals. The spatial profile and the oscillatory modes, together with their first spatial derivatives, vanish at both endpoints. This permits zero extension of the counterexample to the real line. Tensorisation and localisation by parabolic rescaling then yield real symmetric isotropic counterexamples on $\mathbb{R}^d$ and on every bounded domain $Ω\subset\mathbb{R}^d$, for all $d\ge1$.

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arXiv2026

A Liouville theorem for the two-dimensional stationary hypodissipative Navier--Stokes system

We study the two-dimensional stationary incompressible Navier--Stokes equations on $\mathbb R^2$ with fractional dissipation $(-Δ)^s$. In the full range $s \in (0,1)$, we prove that every smooth solution satisfying the natural energy condition $u\in\dot{\mathrm H}^s(\mathbb R^2;\mathbb R^2)$ has $u\equiv0$ and constant pressure. This is a fractional counterpart of the planar finite-Dirichlet theorem of Gilbarg and Weinberger at $s=1$. The proof uses different arguments in three ranges. For $0<s<\frac13$, we combine an $\mathrm L^2$-estimate derived from the equation with a stream-function truncation argument. For $\frac13\leq s\leq\frac23$, we use a localized energy estimate whose boundary terms are supported on expanding annuli. For $\frac23<s<1$, we establish regularity and decay via a Lorentz-space bootstrap and then apply the maximum principle to the vorticity. We also treat the stationary damped Euler system at $s=0$ by combining the Bernoulli identity with a cut-off argument under an annular growth condition that includes $u\in \mathrm L^r(\mathbb R^2)$ for every $1\le r\le2$.

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arXiv2026

On Leonardo da Vinci's paradox

Buoyancy points straight upward, yet in the absence of external forces, a rising air bubble may trace a spiral through the water. This phenomenon was already observed by Leonardo da Vinci. We show that the gravity-free, three-dimensional two-phase incompressible Euler equations with surface tension already admit such a motion. For every nonnegative inner-to-outer density ratio and every sufficiently small positive axial Weber number, we construct smooth near-spherical bubbles and drops with compact interfaces and phasewise irrotational flow. They are stationary in a frame translating along and rotating about a fixed axis, while their centroids lie off that axis and therefore trace genuine circular helices in laboratory coordinates. These helical relative equilibria form a symmetry-breaking branch that issues from the axisymmetric family of rectilinearly translating solutions and is generated by the spherical harmonic mode of degree two and azimuthal number one. The mathematical construction exploits an overdetermined free-boundary formulation, its variational structure, and an $\mathrm{SO}(2)$-equivariant Lyapunov-Schmidt reduction. Particular difficulties in the form of resonances arise at a discrete set of density ratios, including the frequently studied vacuum case with vanishing inner density. Complementary to our existence result, in the absence of surface tension and if the inner density is not larger than the outer density we prove that the model admits neither nontrivial translating relative equilibria nor relative equilibria with helical centroid trajectories. As a byproduct, we show that every regular finite-energy stationary vortex sheet of spherical topology for the homogeneous three-dimensional Euler equations, with irrotational flow on both sides, is trivial.

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arXiv2026

Sharp kinetic trace theory

We establish sharp kinetic trace estimates and counterexamples across several velocity models. For half-space position domains, without any common bound on velocity support, we prove the natural trace estimate for both Lebesgue and standard Gaussian velocity measures. Density yields natural trace operators and Green's formula on the corresponding kinetic energy spaces. For bounded spatial domains in $d\ge2$, in the bounded-support Euclidean velocity model and in the spherical velocity model, we identify the sharp boundary regularity threshold for the trace weights $\min\{|v \cdot n|,|v \cdot n|^p\}$, $1\le p<\infty$. Writing $α_p=1/(p+1)$, the estimate holds on every bounded $\mathrm{C}^{1,α}$ domain with $α\geα_p$, and it fails for every $0<α<α_p$ on some strictly convex bounded domain of exact regularity $\mathrm{C}^{1,α}$. In particular, the natural trace ($p=1$) has the regularity threshold $\mathrm{C}^{1,1/2}$. On bounded $\mathrm{C}^{1,1/2}$ domains in $d\ge2$, density yields natural trace operators and Green's formula in the bounded-support Euclidean velocity model with either Lebesgue or standard Gaussian measure, and in the spherical velocity model. Norm-preserving velocity translation rules out the unrestricted Lebesgue trace estimate on every bounded $\mathrm{C}^1$ domain. In the unrestricted Gaussian model, for each $1\le p<2$, we construct counterexamples on every bounded $\mathrm{C}^{1,1}$ domain in dimension $d\ge2$, answering Question 1.8 of Albritton, Armstrong, Mourrat, and Novack (2024) negatively. For $2\le p<\infty$, the Gaussian $ω_2$ estimate and density instead yield $ω_p$-trace operators.

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arXiv2026

Transfer of regularity by kinetic mollification along critical trajectories

We give a trajectory-based proof of transfer-of-regularity estimates à la Bouchut-Hörmander for kinetic equations at the weak scale of local diffusion. The proof is based on kinetic mollification along critical kinetic trajectories, whose endpoint map has precisely the kinetic scaling. Instead of using Fourier multipliers or the Kolmogorov fundamental solution, we work directly in physical space. The key step is an explicit formula for the mollification defect and corresponding critical support, size, and difference estimates for the kernels. These bounds yield the sharp, scale-invariant homogeneous estimates through maximal-function and Littlewood-Paley estimates on the kinetic group.

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arXiv2026

On the kinetic $p$-Laplace equation with nonlocal diffusion

We study two nonlocal versions of the kinetic $p$-Laplace equation: a Gagliardo-type model defined through differences and a Bessel-type model defined via Fourier multiplication. Using critical kinetic trajectories, we derive representation formulas adapted to the kinetic transport-diffusion geometry and establish homogeneous and scale-invariant kinetic Gagliardo-Nirenberg inequalities for nonlocal diffusion, which yield gain-of-integrability estimates for weak solutions to the kinetic $p$-Laplace equations with nonlocal diffusion.

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arXiv2026

Nonlinear Kinetic Diffusion Equations with $p$-Growth

We establish the local boundedness of (sub-)solutions to nonlinear kinetic diffusion equations with $p$-growth, where the kinetic p-Laplace equation is a prototypical example. A key ingredient is the derivation of kinetic Gagliardo-Nirenberg inequalities, where the Lebesgue norm of a function is estimated in terms of its transport and diffusive directions controlled in different Lebesgue spaces.

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arXiv2026

Spherical rigidity for an exterior overdetermined problem with Neumann data prescribed by mean curvature

We study an overdetermined elliptic free boundary problem for exterior domains in $\mathbb{R}^N$, $N \ge 2$, introduced by F. Morabito [Comm. PDE 46 (2021), 1137-1161]. The overdetermining condition prescribes the Neumann data as a multiple of the boundary mean curvature, with parameter $Γ$, together with a spherical compatibility condition. For $N \ge 3$, we prove rigidity of the spherical solution among star-shaped domains when $Γ\ge N-2$; in the borderline case $Γ= N-2$, the star-shapedness assumption can be removed, and rigidity holds among all bounded domains. The proof combines the Pohozaev identity, geometric identities, and the sharp boundary inequality of Agostiniani and Mazzieri for capacitary potentials. We also obtain rigidity among bounded domains for $Γ\le 0$ via Serrin's moving plane method. In dimension two, the unit disc is the only admissible domain for every $Γ$.

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arXiv2026

Kinetic Sobolev Spaces

We define and study homogeneous kinetic Sobolev spaces adapted to the Kolmogorov equation. We consider both local and non-local diffusion. The spaces are built from the Lebesgue spaces L p for all integrability exponents p $\in$ (1, $\infty$) with regularity assumptions in the transport and diffusive directions according to the scaling of the Kolmogorov equation. The regularity scale accommodates weak and strong solutions. We prove that the proposed spaces satisfy sharp embeddings quantifying the transfer-ofregularity {à} la Bouchut-H{ö}rmander, continuity-in-time in the spirit of Lions and the gainof-integrability of Sobolev and Hardy-Littlewood-Sobolev type. A core tool are mapping properties of the Kolmogorov operator, given by the fundamental solution, established between anisotropic homogeneous Sobolev spaces. To achieve this, we prove L^p boundedness of related singular integral operators, for which we deduce novel kernel estimates by a Littlewood-Paley decomposition and geometric considerations. Moreover, we provide a new uniqueness criterion which allows us to show well-posedness of the Cauchy problem.

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arXiv2025

Weak solutions to Kolmogorov-Fokker-Planck equations: regularity, existence and uniqueness

We prove existence, uniqueness and regularity of weak solutions of Kolmogorov--Fokker--Planck equations with either local or non-local diffusion in the velocity variable and rough diffusion coefficients or kernels. Our results cover the Cauchy problem and allow a broad class of source terms under minimal assumptions. The core of the analysis is a set of sharp kinetic embeddings à la Lions and transfer-of-regularity results à la Bouchut--H\''ormander. We formulate these tools in a homogeneous, scale-invariant form, available for a large range of regularity parameters.

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arXiv2025

Nash's $G$ bound for the Kolmogorov equation

We prove Nash's $G$ bound for the Kolmogorov equation with rough coefficients. Our proof is inspired by the treatment of the parabolic problem by Nash (1958) and Fabes and Stroock (1986). To transfer their ideas to the kinetic setting, we employ critical kinetic trajectories. From Nash's $G$ bound, we recover the sharp lower bound on the fundamental solution and thus provide an alternative proof of the Harnack inequality for the Kolmogorov equation.

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arXiv2025

On a kinetic Poincaré inequality and beyond

In this article, we give a trajectorial proof of a kinetic Poincaré inequality which plays an important role in the De Giorgi-Nash-Moser theory for kinetic equations. The present work improves a result due to J. Guerand and C. Mouhot [10] in several directions. We use kinetic trajectories along the vector fields $\partial_t + v \cdot \nabla_x$ and $\partial_{v_i}$, $i = 1,\dots, d$ and do not rely on higher-order commutators such as $[\partial_{v_i},\partial_t + v \cdot \nabla_x] = \partial_{x_i}$ or on the fundamental solution. The presented method also applies to more general hypoelliptic equations. We illustrate this by studying a Kolmogorov equation with $k$ steps.

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arXiv2025

Critical trajectories in kinetic geometry

We construct critical trajectories in kinetic geometry, i.e. curves in $\mathbb{R}^{1+2n}$ that are: tangential to the vector fields $\partial_t+v\cdot \nabla_x$ and $\nabla_v$, connecting any two given points, respecting the underlying kinetic scaling, and with the property, that the singularity of the $v$-tangent vector near the starting point equates the degeneracy of the dependency of the curve velocity in terms of the endpoint velocity. The construction is based on Newton's laws of motion, where the ansatz for the forcing of the kinetic trajectory is the superposition of functions combining the correct power scaling with desynchronised logarithmic oscillations. These critical trajectories provide a robust and versatile ''almost exponential map'' that allows to prove several functional analytic estimates. We introduce a notion of kinetic mollification and, as an application, deduce the kinetic Sobolev inequality with optimal exponent without relying on the fundamental solution. Moreover, we establish a universal estimate for the logarithm of positive supersolutions to the Kolmogorov equation with rough coefficients inspired by the work of Moser (1961, 1964) on elliptic and parabolic problems. Combining this estimate with De Giorgi-Moser iterations and a lemma due to Bombieri and Giusti, we give an alternative proof of the (weak) Harnack inequality for the Kolmogorov equation with rough coefficients, following the ideas of Moser (1971). Our result gives the optimal range of exponents in the weak Harnack inequality and the optimal (geometric) dependency of the Harnack constant on the bounds of the diffusion matrix.

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arXiv2025

Steady bubbles and drops in inviscid fluids

We construct steady non-spherical bubbles and drops, which are traveling wave solutions to the axisymmetric two-phase Euler equations with surface tension, whose inner phase is a bounded connected domain. The solutions have a uniform vorticity distribution in this inner phase and they have a vortex sheet on its surface. Our construction relies on a perturbative approach around an explicit spherical solution, given by Hill's vortex enclosed by a spherical vortex sheet. The construction is sensitive to the Weber numbers describing the flow. At critical Weber numbers, we perform a bifurcation analysis utilizing the Crandall-Rabinowitz theorem in Sobolev spaces on the 2-sphere. Away from these critical numbers, our construction relies on the implicit function theorem. Our results imply that the model containing surface tension is richer than the ordinary one-phase Euler equations, in the sense that for the latter, Hill's spherical vortex is unique (modulo translations) among all axisymmetric simply connected uniform vortices of a given circulation.

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arXiv2024

Fundamental solutions to Kolmogorov-Fokker-Planck equations with rough coefficients: existence, uniqueness, upper estimates

We show the existence and uniqueness of fundamental solution operators to Kolmo\-gorov-Fokker-Planck equations with rough (measurable) coefficients and local or integral diffusion on finite and infinite time strips. In the local case, that is to say when the diffusion operator is of differential type, we prove $Ł^2$ decay using Davies' method and the conservation property. We also prove that the existence of a generalized fundamental solution with the expected pointwise Gaussian upper bound is equivalent to Moser's $Ł^2-Ł^\infty$ estimates for local weak solutions to the equation and its adjoint. When coefficients are real, this gives the existence and uniqueness of such a generalized fundamental solution and a new and natural way to obtain pointwise decay.

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arXiv2022

A trajectorial interpretation of Moser's proof of the Harnack inequality

In 1971 Moser published a simplified version of his proof of the parabolic Harnack inequality. The core new ingredient is a fundamental lemma due to Bombieri and Giusti, which combines an $L^p-L^\infty$-estimate with a weak $L^1$-estimate for the logarithm of supersolutions. In this note, we give a novel proof of this weak $L^1$-estimate. The presented argument uses parabolic trajectories and does not use any Poincaré inequality. Moreover, the proposed argument gives a geometric interpretation of Moser's result and could allow transferring Moser's method to other equations.

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