On linear evolutionary equations with skew symmetric spatial operators

TL;DR

This paper investigates generalized solutions of linear evolutionary equations with skew-symmetric operators, establishing existence, uniqueness, and applications to transport and Euler equations under certain regularity conditions.

Contribution

It introduces criteria for existence and uniqueness of solutions for skew-symmetric operator-based equations, including conditions for skew-adjointness and applications to fluid dynamics.

Findings

Existence of a contractive semigroup for solutions

Criteria for uniqueness of generalized solutions

Conditions under which operators are skew-adjoint

Abstract

We study generalized solutions of an evolutionary equation related to a densely defined skew-symmetric operator in a real Hilbert space. We establish existence of a contractive semigroup, which provides generalized solutions, and find criteria of uniqueness of generalized solutions. Some applications are given including the transport equations and the linearised Euler equations with solenoidal (and generally discontinuous) coefficients. Under some additional regularity assumption on the coefficients we prove that the corresponding spatial operators are skew-adjoint, which implies existence and uniqueness of generalized solutions for both the forward and the backward Cauchy problem.