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Existence and upper bounds of fundamental solutions to kinetic equations with rough non-local diffusion
Pascal Auscher, Lukas Niebel
We study the fundamental solution operators associated to Kolmogorov equations with symmetric non-local diffusion with rough kernels. We prove that these operators are induced by m…
Existence and uniqueness of weak solutions to quasilinear PDEs with critical data
Pascal Auscher, Sebastian Bechtel
We establish existence and uniqueness of global, bounded weak solutions to quasilinear PDEs with bounded, uniformly continuous initial data and investigate their properties. Moreov…
Kinetic Sobolev Spaces
Pascal Auscher, Lukas Niebel
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 sp…
On Hardy-Littlewood-Sobolev estimates for degenerate Laplacians
Pascal Auscher, Khalid Baadi
We establish norm inequalities for fractional powers of degenerate Laplacians, with degeneracy being determined by weights in the Muckenhoupt class , accompanied…
Fundamental solutions for parabolic equations and systems: universal existence, uniqueness, representation
Pascal Auscher, Khalid Baadi
In this paper, we develop a universal, conceptually simple and systematic method to prove well-posedness to Cauchy problems for weak solutions of parabolic equations with non-smoot…
On well-posedness for parabolic Cauchy problems of Lions type with rough initial data
Pascal Auscher, Hedong Hou
We establish a complete picture for well-posedness of parabolic Cauchy problems with time-independent, uniformly elliptic, bounded measurable complex coefficients. We exhibit a ran…
Fundamental solutions to Kolmogorov-Fokker-Planck equations with rough coefficients: existence, uniqueness, upper estimates
Pascal Auscher, Cyril Imbert, Lukas Niebel
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…
Weak solutions to Kolmogorov-Fokker-Planck equations: regularity, existence and uniqueness
Pascal Auscher, Cyril Imbert, Lukas Niebel
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…