On evolutionary equations related to skew-symmetric spatial operators
Evgeny Yu. Panov
TL;DR
The paper addresses generalized solutions of evolution equations generated by densely defined skew-symmetric operators and connects this framework to transport equations with solenoidal coefficients. It develops contractive semigroups of generalized solutions by classifying $m$-dissipative extensions of $-A$ via the Cayley transform and deficiency indices, yielding precise uniqueness criteria. A key result is that a generalized solution semigroup is unique if and only if the deficiency index $d_-$ of $A$ vanishes, with explicit constructions when $d_+=0$ or $d_-=0$. The operator-theoretic framework ties abstract well-posedness to transport phenomena and reconciles with known results of DiPerna–Lions and Ambrosio for Sobolev or BV coefficient fields, while clarifying sources of nonuniqueness in more general settings.
Abstract
We study generalized solutions of an evolutionary equation related to some densely defined skew-symmetric operator in a real Hilbert space. We establish existence of a contractive semigroup, which provides generalized solutions, and suggest a criteria of uniqueness of this semigroup. We also find a stronger criteria of uniqueness of generalized solutions. Applications to transport equations with solenoidal (and generally discontinuous) coefficients are given.