Approximation of the solution of transport-diffusion equation in Hölder space

In this paper, approximate solutions for the transport-diffusion equation in $\mathbb{R}^d$ and their limit function are considered and it is proved that the limit function belongs to the Hölder space corresponding to the regularity of given functions and satisfies the equation. More precisely, we construct these approximate solutions by using the heat kernel and the translation corresponding to the transport on each step of time discretization. Under the assumption of the boundedness of given functions and their partial derivatives with respect to the space variables up to the $m$-th order ($m > 2$) and of the $\alpha$-Hölder continuity of their $m$-th derivatives ($2/3 < \alpha \leqslant 1$; if $\alpha = 1$, it means the Lipschitz condition), we first establish suitable estimates of the approximate solutions and then, using these estimates, we prove their convergence to a function which satisfies the equation and the $\alpha$-Hölder continuity of the $m$-th derivatives with respect to the space variables of the limit function. Note that these estimates do not depend on the coefficient of diffusion, so they can be used even in the case where the coefficient of diffusion tends to 0.