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arXiv2014

Higher gradients estimates in Morrey spaces for weak solutions to linear ultraparabolic equations

The aim of this paper is to consider the linear ultraparabolic equation with bounded and VMO coefficients $a_{ij} (z)$. Assume that the operator $L_0$ obtained by freezing the coefficients $a_{ij}(z)$ at any point ${z_0} \in {\mathbb{R}^{N + 1}}$ is hypoelliptic. We first establish a Caccioppoli type inequality by choosing a cutoff function, a Sobolev type inequality by prosperities of the fundamental solution to $L_0$, and a Poincaré type inequality with a new cutoff function. Then $L^p$ estimate for weak solutions is derived by using the reverse Hölder inequality on homogeneous spaces. Finally, higher Morrey estimates for weak solutions to the above equation are shown by investigating a homogeneous ultraparabolic equation of variable coefficients with a nonhomogeneous boundary value condition, and a nonhomogeneous ultraparabolic equation of variable coefficients with homogeneous boundary value condition.

math.AP↗

arXiv2011

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

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 $X_{1}, ...,X_{q}$.

math.AP↗