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On solvability of parabolic equations with singular coefficients in odd mixed-norm Morrey-Sobolev spaces

N. V. Krylov

TL;DR

This work develops solvability theory for second-order parabolic equations in the whole space with time-measurable and space–BMO coefficients within odd mixed-norm Morrey–Sobolev spaces ${\sf E}^{1,2}_{q,p,\beta}$. By introducing the time-first mixed-norm ${\sf L}_{q,p}$ framework and Morrey refinements ${\sf E}_{q,p,\beta}$, the authors establish a priori estimates and existence/uniqueness results under small mean oscillation of the main coefficient and small Morrey norm of the lower-order term, using weighted harmonic analysis and localization. The paper handles rough singularities in the first-order terms and extends results to fully variable coefficients, with a pathway toward Itô stochastic applications. The combination of Fefferman–Stein type bounds, $A_p$ weights, and a continuity method yields robust solvability in this novel Morrey–Sobolev setting. These results advance solvability theory for parabolic equations with singular coefficients beyond classical Lebesgue/Sobolev frameworks and establish tools for stochastic and higher-order generalizations.

Abstract

We prove an existence and uniqueness theorem for second-order parabolic equations in the whole space with constant zeroth-order coefficient in mixed-norm Morrey-Sobolev spaces. The main coefficient $a$ is assumed to be measurable in $t$ and BMO in $x$ and the first-order coefficients $b$ are in an appropriate mixed-norm Morrey classes (thus admitting rather rough singularities). The mixed-norm Morrey-Sobolev spaces are ``odd'' in the sense that the interior integration in the formula defining the norm is performed with respect to $t$ and not to $x$ as is customary.

On solvability of parabolic equations with singular coefficients in odd mixed-norm Morrey-Sobolev spaces

TL;DR

This work develops solvability theory for second-order parabolic equations in the whole space with time-measurable and space–BMO coefficients within odd mixed-norm Morrey–Sobolev spaces . By introducing the time-first mixed-norm framework and Morrey refinements , the authors establish a priori estimates and existence/uniqueness results under small mean oscillation of the main coefficient and small Morrey norm of the lower-order term, using weighted harmonic analysis and localization. The paper handles rough singularities in the first-order terms and extends results to fully variable coefficients, with a pathway toward Itô stochastic applications. The combination of Fefferman–Stein type bounds, weights, and a continuity method yields robust solvability in this novel Morrey–Sobolev setting. These results advance solvability theory for parabolic equations with singular coefficients beyond classical Lebesgue/Sobolev frameworks and establish tools for stochastic and higher-order generalizations.

Abstract

We prove an existence and uniqueness theorem for second-order parabolic equations in the whole space with constant zeroth-order coefficient in mixed-norm Morrey-Sobolev spaces. The main coefficient is assumed to be measurable in and BMO in and the first-order coefficients are in an appropriate mixed-norm Morrey classes (thus admitting rather rough singularities). The mixed-norm Morrey-Sobolev spaces are ``odd'' in the sense that the interior integration in the formula defining the norm is performed with respect to and not to as is customary.

Paper Structure

This paper contains 5 sections, 25 theorems, 191 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. the case of main coefficients depending only on $t$
  4. Variable main coefficients and Sobolev spaces
  5. Localizing (\ref{['4.30,2']}), (\ref{['4.3,5']}) and the proof of Theorem \ref{['theorem 5.8,20']}

Key Result

Theorem 2.1

There exist such that, if $\lambda\geq \lambda_{0}\rho_{b}^{-2}$ and then for any $u\in {\sf{E}}^{1,2}_{q,p,\beta}$ where Furthermore, for any $f\in {\sf{E}}_{q,p,\beta}$ and $\lambda\geq\lambda_{0}\rho_{b}^{-2}$ there exists a unique $u\in {\sf{E}}^{1,2}_{q,p,\beta}$ such that in $\mathbb{R}^{d+1}$

Theorems & Definitions (35)

  • Theorem 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Remark 2.6
  • Lemma 3.1
  • Remark 3.2
  • Remark 3.3
  • Theorem 3.4
  • ...and 25 more