A regularity theory for second-order parabolic partial differential equations in weighted mixed norm Sobolev-Zygmund spaces

TL;DR

The article develops an optimal regularity theory for second-order parabolic equations with coefficients measurable in time in the setting of weighted mixed-norm Hölder–Zygmund spaces. By employing Hölder–Zygmund spaces $\Lambda^{\gamma}$, a weighted $A_p$ time-integrability, and a trace theorem, it extends classical Schauder estimates to the critical integer-order regime and accommodates nonzero initial data. The main result proves existence, uniqueness, and a priori estimates for solutions $u\in \mathbf{H}_{p,w}^{\gamma+2}(T)$ given data $u_0\in\Lambda_p^{\gamma+2,w}$ and $f\in\mathbf{\Lambda}_{p,w}^{\gamma}(T)$ under uniform ellipticity and spatial $\Lambda^{\gamma}$-regularity of the coefficients, with an optimal initial-data trace space. Collectively, the work unifies partial Schauder theory with Zygmund regularity in a weighted mixed-norm framework, providing tools potentially applicable to nonlinear parabolic PDEs and models such as fluid dynamics.

Abstract

We develop an optimal regularity theory for parabolic partial differential equations in weighted mixed norm Sobolev-Zygmund spaces. The results extend the classical Schauder estimates to coefficients that are merely measurable in time and to the critical case of integer-order regularity. In addition, nonzero initial data are treated in the optimal trace space via a sharp trace theorem.

Theorems & Definitions (26)

• Definition 1.1: Hölder-Zygmund space
• Definition 1.2: Muckenhoupt's class
• Definition 1.3: Solution space
• Remark 1.4
• Theorem 1.6
• Proposition 2.1
• proof
• Proposition 2.2
• proof
• Lemma 2.3: Partition of unity