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Heat Equation driven by mixed local-nonlocal operators with non-regular space-dependent coefficients

Arshyn Altybay, Michael Ruzhansky

TL;DR

The paper addresses the heat equation driven by a mixed local–nonlocal diffusion operator with spatially irregular coefficients. It combines energy methods for the regular-coefficient case with a Friedrichs-regularisation approach to define very weak solutions for distributional coefficients and initial data, proving existence, uniqueness up to negligibility, and consistency with the classical theory. The main contributions are a rigorous very weak framework for parabolic problems with hybrid diffusion and singular coefficients, accompanied by explicit energy estimates and stability results. This work provides a robust foundation for analysis and numerical treatment of diffusion models in heterogeneous media that feature strong spatial irregularities.

Abstract

In this paper, we study the Cauchy problem for a heat equation governed by a mixed local--nonlocal diffusion operator with spatially irregular coefficients. We first establish classical well-posedness in an energy framework for bounded, measurable coefficients that satisfy uniform positivity, and we derive an a priori estimate ensuring uniqueness and continuous dependence on the initial data. We then extend the notion of solution to distributional coefficients and initial data by a Friedrichs-type regularisation procedure. Within this very weak framework, we establish the existence and uniqueness of solution nets and prove consistency with the classical weak solution whenever the coefficients are regular.

Heat Equation driven by mixed local-nonlocal operators with non-regular space-dependent coefficients

TL;DR

The paper addresses the heat equation driven by a mixed local–nonlocal diffusion operator with spatially irregular coefficients. It combines energy methods for the regular-coefficient case with a Friedrichs-regularisation approach to define very weak solutions for distributional coefficients and initial data, proving existence, uniqueness up to negligibility, and consistency with the classical theory. The main contributions are a rigorous very weak framework for parabolic problems with hybrid diffusion and singular coefficients, accompanied by explicit energy estimates and stability results. This work provides a robust foundation for analysis and numerical treatment of diffusion models in heterogeneous media that feature strong spatial irregularities.

Abstract

In this paper, we study the Cauchy problem for a heat equation governed by a mixed local--nonlocal diffusion operator with spatially irregular coefficients. We first establish classical well-posedness in an energy framework for bounded, measurable coefficients that satisfy uniform positivity, and we derive an a priori estimate ensuring uniqueness and continuous dependence on the initial data. We then extend the notion of solution to distributional coefficients and initial data by a Friedrichs-type regularisation procedure. Within this very weak framework, we establish the existence and uniqueness of solution nets and prove consistency with the classical weak solution whenever the coefficients are regular.
Paper Structure (10 sections, 10 theorems, 102 equations)

This paper contains 10 sections, 10 theorems, 102 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Well-posedness of the problem in the regular case
  4. Energy estimate in the regular case
  5. Very weak well-posedness
  6. Spatial regularisation and moderate nets
  7. Existence of very weak solutions
  8. Uniqueness
  9. Consistency with classical theory
  10. Conclusion

Key Result

Lemma 2.1

For $f\in L^2(\mathbb R^d)$ we have $\|f\|_{L^2}^2=\frac{1}{(2\pi)^d}\|\hat{f}\|_{L^2}^2$.

Theorems & Definitions (26)

  • Definition 1: Riesz fractional Laplacian in $\mathbb R^d$
  • Definition 2: Fourier transform
  • Lemma 2.1: Plancherel identity
  • Definition 3: Gagliardo seminorm
  • Lemma 2.2: Equivalence of fractional norms
  • Lemma 2.3: Fractional integration by parts
  • Lemma 2.4: Fourier domination for $0<s<1$
  • Definition 4: Gelfand triple and duality
  • Lemma 2.5: Lions--Magenes
  • Definition 5: Energy bilinear form
  • ...and 16 more