The heat equation under linear conditions on the moments in higher dimensions

TL;DR

This paper extends the well-posedness results of the heat equation with moment-preserving integral conditions from one dimension to higher dimensions, using operator theory and distributional solutions.

Contribution

It introduces a framework for the heat equation with moment conditions in higher dimensions, proving existence and uniqueness via operator realizations and semigroup theory.

Findings

Well-posedness established for higher-dimensional heat equations with moment conditions.

Connection made between Laplacian realizations and Krein--von Neumann extension.

Solutions exist in a distributional sense and are classical for L^2 initial data.

Abstract

We consider the heat equation on the $N$-dimensional cube $(0,1)^N$ and impose different classes of integral conditions, instead of usual boundary ones. Well-posedness results for the heat equation under the condition that the moments of order 0 and 1 are conserved had been known so far only in the case of N=1 -- for which such conditions can be easily interpreted as conservation of mass and barycenter. In this paper we show that in the case of general $N$ the heat equation with such integral conditions is still well-posed, upon suitably relax the notion of solution. Existence and uniqueness of solutions with general initial data in a suitable space of distibutions over $(0,1)^N$ are proved by introducing two appropriate realizations of the Laplacian and checking by form methods that they generate analytic semigroups. The solution thus obtained does however solve the heat equation only in a certain distributional sense. However, it turns out that one of these realizations is tightly related to a well-known object of operator theory, the Krein--von Neumann extension of the Laplacian. This connection also establishes well-posedness in a classical sense, as long as the initial data are $L^2$-functions.