@article{Garain2023, 
author = {Prashanta Garain and Kaj Nyström},
title = {On regularity and existence of weak solutions to nonlinear Kolmogorov-Fokker-Planck type equations with rough coefficients},
year = {2023},
journal = {Mathematics in Engineering},
volume = {5},
number = {2},
pages = {1-37},
keywords = {Kolmogorov equation, parabolic, ultraparabolic, hypoelliptic, nonlinear Kolmogorov-Fokker-Planck equations, existence, uniqueness, regularity},
url = {https://www.sciopen.com/article/10.3934/mine.2023043},
doi = {10.3934/mine.2023043},
abstract = {We consider nonlinear Kolmogorov-Fokker-Planck type equations of the form   (∂t+X⋅∇Y)u=∇X⋅(A(∇Xu,X,Y,t)).The function  A=A(ξ,X,Y,t):Rm×Rm×Rm×R→Rm is assumed to be continuous with respect to  ξ, and measurable with respect to  X,Y and  t.  A=A(ξ,X,Y,t) is allowed to be nonlinear but with linear growth. We establish higher integrability and local boundedness of weak sub-solutions, weak Harnack and Harnack inequalities, and Hölder continuity with quantitative estimates. In addition we establish existence and uniqueness of weak solutions to a Dirichlet problem in certain bounded  X,  Y and  t dependent domains.}
}