arXiv subjects

Lukas Niebel

Publications and source records attributed to Lukas Niebel.

23 records · Page 2Linked to original sources

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.

math.AP↗

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.

math.AP↗

arXiv2021

Kinetic maximal $L^p_μ(L^p)$-regularity for the fractional Kolmogorov equation with variable density

We consider the Kolmogorov equation, where the right-hand side is given by a non-local integro-differential operator comparable to the fractional Laplacian in velocity with possibly time, space and velocity dependent density. We prove that this equation admits kinetic maximal $L^p_μ$-regularity under suitable assumptions on the density and on $p$ and $μ$. We apply this result to prove short-time existence of strong $L^p_μ$-solutions to quasilinear fractional kinetic partial differential equations.

math.AP↗

arXiv2020

Kinetic maximal $L^p$-regularity with temporal weights and application to quasilinear kinetic diffusion equations

We introduce the concept of kinetic maximal $L^p$-regularity with temporal weights and prove that this property is satisfied for the (fractional) Kolmogorov equation. We show that solutions are continuous with values in the trace space and prove, in particular, that the trace space can be characterized in terms of anisotropic Besov spaces. We further extend the property of kinetic maximal $L^p_μ$-regularity to the Kolmogorov equation with variable coefficients. Finally, we show how kinetic maximal $L^p_μ$-regularity can be used to obtain local existence of solutions to a class of quasilinear kinetic equations and illustrate our result with a quasilinear kinetic diffusion equation.

math.AP↗

arXiv2020

Kinetic maximal $L^2$-regularity for the (fractional) Kolmogorov equation

We introduce the notion of kinetic maximal $L^2$-regularity with temporal weights for the (fractional) Kolmogorov equation. In particular, we determine the function spaces for the inhomogeneity and the initial value which characterize the regularity of solutions to the fractional Kolmogorov equation in terms of fractional anisotropic Sobolev spaces. It is shown that solutions of the homogeneous (fractional) Kolmogorov equation define a semi-flow in a suitable function space and the property of instantaneous regularization is investigated.

math.AP↗