On one criterion of the uniqueness of generalized solutions for linear transport equations with discontinuous coefficients

TL;DR

This paper investigates the conditions under which generalized solutions to multidimensional transport equations with discontinuous coefficients are unique, linking this to operator properties and establishing criteria for solution existence and uniqueness.

Contribution

It establishes a criterion connecting the uniqueness of generalized solutions to the essential skew-adjointness of a related operator in Hilbert space.

Findings

Generalized solutions satisfy the renormalization property under certain operator conditions.

Existence of a contractive semigroup providing solutions is proven.

A criterion for the uniqueness of solutions is established.

Abstract

We study generalized solutions of multidimensional transport equation with bounded measurable solenoidal field of coefficients $a(x)$. It is shown that any generalized solution satisfies the renormalization property if and only if the operator $a\cdot\nabla u$, $u\in C_0^1(\mathbb{R}^n)$ in the Hilbert space $L^2(\mathbb{R}^n)$ is an essentially skew-adjoint operator, and this is equivalent to the uniqueness of generalized solutions. We also establish existence of a contractive semigroup, which provides generalized solutions, and give a criterion of its uniqueness.