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Discrete Heat Equation with irregular thermal conductivity and tempered distributional data

Marianna Chatzakou, Aparajita Dasgupta, Michael Ruzhansky, Abhilash Tushir

TL;DR

This work develops a rigorous framework for the semi-classical heat equation on the lattice $\hbar\mathbb{Z}^n$ with time-dependent and distributional coefficients. By combining weighted $\ell^{2}_{s}(\hbar\mathbb{Z}^n)$ well-posedness results for classical data with a robust very weak solution theory for distributional data, the authors address solvability and stability in the presence of singular coefficients. They show that, under suitable Sobolev regularity, solutions on the semi-classical lattice converge to the Euclidean solutions on $\mathbb{R}^n$ as $\hbar\to0$, thereby bridging discrete and continuous models. The results extend to tempered distributions and establish consistency between the very weak and classical frameworks, providing a versatile toolkit for parabolic equations with irregular coefficients on lattices and their continuum limits.

Abstract

In this paper, we consider a semi-classical version of the nonhomogeneous heat equation with singular time-dependent coefficients on the lattice $\hbar \mathbb{Z}^n$. We establish the well-posedeness of such Cauchy equations in the classical sense when regular coefficients are considered, and analyse how the notion of very weak solution adapts in such equations when distributional coefficients are regarded. We prove the well-posedness of both the classical and the very weak solution in the weighted spaces $\ell^{2}_{s}(\hbar \mathbb{Z}^n)$, $s \in \mathbb{R}$, which is enough to prove the well-posedness in the space of tempered distributions $\mathcal{S}'(\hbar \mathbb{Z}^n)$. Notably, when $s=0$, we show that for $\hbar \rightarrow 0$, the classical (resp. very weak) solution of the heat equation in the Euclidean setting $\mathbb{R}^n$ is recaptured by the classical (resp. very weak) solution of it in the semi-classical setting $\hbar \mathbb{Z}^n$.

Discrete Heat Equation with irregular thermal conductivity and tempered distributional data

TL;DR

This work develops a rigorous framework for the semi-classical heat equation on the lattice with time-dependent and distributional coefficients. By combining weighted well-posedness results for classical data with a robust very weak solution theory for distributional data, the authors address solvability and stability in the presence of singular coefficients. They show that, under suitable Sobolev regularity, solutions on the semi-classical lattice converge to the Euclidean solutions on as , thereby bridging discrete and continuous models. The results extend to tempered distributions and establish consistency between the very weak and classical frameworks, providing a versatile toolkit for parabolic equations with irregular coefficients on lattices and their continuum limits.

Abstract

In this paper, we consider a semi-classical version of the nonhomogeneous heat equation with singular time-dependent coefficients on the lattice . We establish the well-posedeness of such Cauchy equations in the classical sense when regular coefficients are considered, and analyse how the notion of very weak solution adapts in such equations when distributional coefficients are regarded. We prove the well-posedness of both the classical and the very weak solution in the weighted spaces , , which is enough to prove the well-posedness in the space of tempered distributions . Notably, when , we show that for , the classical (resp. very weak) solution of the heat equation in the Euclidean setting is recaptured by the classical (resp. very weak) solution of it in the semi-classical setting .
Paper Structure (10 sections, 8 theorems, 114 equations)

This paper contains 10 sections, 8 theorems, 114 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Preliminaries
  4. Spaces of functions and distributions on the lattice $\hbar\mathbb{Z}^{n}$
  5. Space of periodic functions and distributions on the torus $\mathbb{T}_{\hbar}^{n}$
  6. Related semi-classical Fourier analysis
  7. The discrete fractional Laplacian
  8. Proof of the main results
  9. Semi-classical limit $\hbar\to 0$
  10. Remarks

Key Result

Theorem 1.1

Let $m\in \mathbb{R}$ and $f\in L^{2}([0,T];H^{m}(\mathbb{R}^{n}))$. Assume that $a \in L_1^{\infty}([0, T])$ satisfies $\inf\limits_{t \in[0, T]} a(t)=a_0>0$ and $b \in L^{\infty}([0, T])$. If the initial Cauchy data $u_{0} \in H^{m}(\mathbb{R}^{n})$, then the Cauchy problem heatpdeEuc has a unique for all $t\in[0,T]$, where the positive constant $C_{T,a,b}$ is given by

Theorems & Definitions (19)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 2.1: Classical solution
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4: Existence
  • Definition 2.5
  • Remark 2.6
  • Theorem 2.7: Uniqueness
  • Theorem 2.8: Consistency
  • ...and 9 more