Estimates in Generalized Morrey Spaces for Weak Solutions to Divergence Degenerate Parabolic Systems

Abstract

Let $\mathrm{X}=(X_{1},...,X_{q})$ be a family of real smooth vector fields satisfying Hömander's condition. The purpose of this paper is to establish gradient estimates in generalized Morrey spaces for weak solutions of the divergence degenerate parabolic system related to $X$ :%\[u_{t}^{i}+X_α^{\ast}(a_{ij}^{αβ}(z)X_βu^{j}%)=g_{i}+X_α^{\ast}f_{i}^α(z), \] where $α,β=1,2,...,q,$ $i,j=1,2,...,N$, $X_α^{\ast}$ is the transposed vector field of $X_α$, $z=(t,x)\in{\mathbb{R}}^{n+1}$, and coefficients $a_{ij}^{αβ}(z)$ belong to the space $VMO$ induced by the vector fields $X_{1}, ...,X_{q}$.

21 pages