[Paper Review] Estimates in Generalized Morrey Spaces for Weak Solutions to Divergence Degenerate Parabolic Systems

Yan Dong, Maochun Zhu+1 more|arXiv (Cornell University)|Sep 9, 2011

Nonlinear Partial Differential Equations19 references3 citations

TL;DR

This paper establishes gradient estimates in generalized Morrey spaces for weak solutions to divergence-form degenerate parabolic systems associated with Hörmander's vector fields. By leveraging Caccioppoli-type inequalities, reverse Hölder inequalities, and VMO coefficients, the authors prove that the horizontal gradient $Xu$ belongs to generalized Morrey spaces $L_{ ho}^{2, u}$ under suitable integrability and regularity conditions on the data and coefficients.

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

Motivation & Objective

• To establish $L^2$ and generalized Morrey space estimates for the horizontal gradient $Xu$ of weak solutions to degenerate parabolic systems.
• To extend classical Morrey space results from the Euclidean setting to sub-Riemannian settings governed by Hörmander's vector fields.
• To handle systems with coefficients in $VMO$ (vanishing mean oscillation) with respect to the vector field structure.
• To prove that the gradient $Xu$ inherits the integrability of the lower-order data $f_i^\alpha$ and $g_i$ in generalized Morrey spaces.
• To overcome the lack of differentiability due to non-commuting vector fields by adapting techniques from the Euclidean case using sub-Riemannian geometry tools.

Proposed method

• Use of a Caccioppoli-type inequality (Lemma 3.1) to control the $L^2$-norm of $Xu$ in smaller cylinders.
• Application of reverse Hölder inequality in the homogeneous space to obtain higher integrability of $Xu$ (Theorem 3.9).
• Adaptation of techniques from Mcbride (2010) for $VMO$ coefficients in the Euclidean case to the subelliptic setting via results from Xu (2008).
• Employment of cutoff functions and localization techniques to extend local estimates to compact subsets $Q' \Subset Q'' \Subset Q_T$.
• Use of the generalized Morrey norm $\| \cdot \|_{L_{\varphi}^{2,\lambda}}$ with a doubling function $\varphi$ satisfying $r^{\gamma - \lambda}/\varphi^2(r)$ almost increasing.
• Proof relies on energy estimates and interpolation via $\varepsilon$-absorption, combined with the uniform ellipticity and $VMO$ regularity of coefficients $a_{ij}^{\alpha\beta}$.

Experimental results

Research questions

• RQ1 Under what conditions on the data and coefficients does the horizontal gradient $Xu$ of a weak solution to a degenerate parabolic system belong to a generalized Morrey space?
• RQ2 How do the properties of the vector fields $X_1, \dots, X_q$ satisfying Hörmander's condition affect the regularity of solutions in Morrey-type spaces?
• RQ3 Can the classical $L^p$ and Morrey space estimates for parabolic systems in the Euclidean setting be extended to systems driven by Hörmander vector fields?
• RQ4 What role does the $VMO$ condition on the coefficients play in achieving gradient estimates in generalized Morrey spaces?
• RQ5 How does the interplay between the homogeneity degree $Q$ and the Morrey exponent $\lambda$ influence the integrability of $Xu$?

Key findings

• The horizontal gradient $Xu$ belongs to the generalized Morrey space $L_{\varphi}^{2,\lambda}(Q')$ for any $Q' \Subset Q'' \Subset Q_T$, provided $\lambda < \gamma < Q+2$ and $r^{\gamma - \lambda}/\varphi^2(r)$ is almost increasing.
• The following estimate holds: $\|Xu\|_{L_{\varphi}^{2,\lambda}(Q')}^2 \leq c\left(\|Xu\|_{L^2(Q'')} + \|f\|_{L_{\varphi}^{2,\lambda}(Q_T)} + \|g\|_{L_{\varphi}^{2,\lambda}(Q_T)}\right)$, showing stability of the Morrey norm under data and lower-order term control.
• For any $\rho \leq R$ with $Q_R \subset Q_T$, the estimate $\iint_{Q_\rho} |Xu|^2 \, dz \leq c \frac{\rho^{\lambda} \varphi^2(\rho)}{R^{\lambda} \varphi^2(R)} \iint_{Q_R} |Xu|^2 \, dz + c \rho^{\lambda} \varphi^2(\rho) (\|f\|_{L_{\varphi}^{2,\lambda}}^2 + \|g\|_{L_{\varphi}^{2,\lambda}}^2)$ is established, linking local and global norms.
• The proof relies on decomposing $u = w + v$, where $w$ solves the homogeneous system and $v$ solves the inhomogeneous one, enabling separate control of the gradient components.
• The result extends known $L^p$ and Morrey estimates to the subelliptic setting with $VMO$ coefficients, generalizing earlier results in the Euclidean case.
• The method successfully overcomes the lack of differentiability due to non-commuting vector fields by using sub-Riemannian geometry and known results on $VMO$ regularity in this context.