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

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

2011-09-09

arxiv.org/abs/1109.1929v1

Mathematics

Analysis of PDEs

21 pages

Scientific paper

Let $\mathrm{X}=(X_{1},...,X_{q})$ be a family of real smooth vector fields satisfying H\"{o}mander'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 $X$ :%\[u_{t}^{i}+X_{\alpha}^{\ast}(a_{ij}^{\alpha\beta}(z)X_{\beta}u^{j}%)=g_{i}+X_{\alpha}^{\ast}f_{i}^{\alpha}(z), \] where $\alpha,\beta=1,2,...,q,$ $i,j=1,2,...,N$, $X_{\alpha}^{\ast}$ is the transposed vector field of $X_{\alpha}$, $z=(t,x)\in{\mathbb{R}}^{n+1}$, and coefficients $a_{ij}^{\alpha\beta}(z)$ belong to the space $VMO$ induced by the vector fields $X_{1}, ...,X_{q}$.

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