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On well-posedness for non-autonomous parabolic Cauchy problems with rough initial data

Hedong Hou

TL;DR

This paper develops a comprehensive well-posedness theory for non-autonomous parabolic Cauchy problems in divergence form with rough, complex-valued coefficients $A(t,x)$. By leveraging weighted tent spaces and homogeneous Hardy–Sobolev Besov-type spaces, it ties initial data regularity to time-weighted gradient control of weak solutions, and constructs robust propagator and Duhamel frameworks with off-diagonal decay estimates. The main contributions include existence, uniqueness, and explicit representation formulas for homogeneous and inhomogeneous problems, Lions-type equations, and a precise trace/continuity theory in tents spaces, extended to homogeneous Besov spaces via interpolation. The results significantly advance the understanding of parabolic systems with rough coefficients and rough initial data, offering sharp, scalable tools for analysis and potential stochastic applications. Collectively, the work provides a complete, modular picture of well-posedness across a broad spectrum of initial data spaces and time-dependent coefficient matrices.

Abstract

We establish a complete picture for existence, uniqueness, and representation of weak solutions to non-autonomous parabolic Cauchy problems of divergence type. The coefficients are only assumed to be uniformly elliptic, bounded, measurable, and complex-valued, without any additional regularity or symmetry conditions. The initial data are tempered distributions taken in homogeneous Hardy--Sobolev spaces $\dot{H}^{s,p}$, and source terms belong to certain scales of weighted tent spaces. Weak solutions are constructed with their gradients in weighted tent spaces $T^{p}_{s/2}$. Analogous results are also exhibited for initial data in homogeneous Besov spaces $\dot{B}^{s}_{p,p}$.

On well-posedness for non-autonomous parabolic Cauchy problems with rough initial data

TL;DR

This paper develops a comprehensive well-posedness theory for non-autonomous parabolic Cauchy problems in divergence form with rough, complex-valued coefficients . By leveraging weighted tent spaces and homogeneous Hardy–Sobolev Besov-type spaces, it ties initial data regularity to time-weighted gradient control of weak solutions, and constructs robust propagator and Duhamel frameworks with off-diagonal decay estimates. The main contributions include existence, uniqueness, and explicit representation formulas for homogeneous and inhomogeneous problems, Lions-type equations, and a precise trace/continuity theory in tents spaces, extended to homogeneous Besov spaces via interpolation. The results significantly advance the understanding of parabolic systems with rough coefficients and rough initial data, offering sharp, scalable tools for analysis and potential stochastic applications. Collectively, the work provides a complete, modular picture of well-posedness across a broad spectrum of initial data spaces and time-dependent coefficient matrices.

Abstract

We establish a complete picture for existence, uniqueness, and representation of weak solutions to non-autonomous parabolic Cauchy problems of divergence type. The coefficients are only assumed to be uniformly elliptic, bounded, measurable, and complex-valued, without any additional regularity or symmetry conditions. The initial data are tempered distributions taken in homogeneous Hardy--Sobolev spaces , and source terms belong to certain scales of weighted tent spaces. Weak solutions are constructed with their gradients in weighted tent spaces . Analogous results are also exhibited for initial data in homogeneous Besov spaces .
Paper Structure (37 sections, 20 theorems, 184 equations, 2 figures)

This paper contains 37 sections, 20 theorems, 184 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Organization
  4. Notation
  5. Acknowledgments
  6. Function spaces
  7. Homogeneous Hardy--Sobolev spaces
  8. Tent spaces
  9. Slice spaces
  10. Divergence of tent spaces
  11. Basic facts of weak solutions
  12. Energy inequality
  13. Propagators
  14. Inhomogeneous Cauchy problem
  15. Lions' equation
  16. ...and 22 more sections

Key Result

theorem 1

There exists $\beta_A \in [-1,-1/2)$ only depending on the ellipticity of $A$ and the dimension $n$ so that when $\beta_A<\beta<0$ and for any $u_0 \in \DotH^{2\beta+1,p}$, there is a unique global weak solution $u$ to the homogeneous Cauchy problem with $\nabla u \in T^p_{\beta+1/2}$. The equivalence holds that Moreover, $u$ belongs to $C([0,\infty);\scrS')$.

Figures (2)

  • Figure 1: Region of well-posedness
  • Figure 2: Quotient map of $\Phi_A$

Theorems & Definitions (48)

  • theorem 1: Well-posedness of the homogeneous Cauchy problem
  • theorem 2: Representation
  • theorem 3: Well-posedness of the inhomogeneous Cauchy problem
  • theorem 4: Well-posedness of the Lions equation
  • theorem 5: Well-posedness of non-autonomous parabolic Cauchy problems of divergence type
  • definition 1: Homogeneous Hardy--Sobolev spaces
  • lemma 1
  • proof
  • definition 2: Weak solutions
  • lemma 2: Caccioppoli's inequality
  • ...and 38 more