On linear evolutionary equations with skew symmetric spatial operators

TL;DR

The paper addresses evolution equations in a real Hilbert space driven by a densely defined skew-symmetric spatial operator $A_0$, studying generalized solutions of $u'-A^*u=0$ and their generation by contractive $C_0$-semigroups. It shows that such semigroups arise from $m$-dissipative extensions $B$ of $-A$ with $B\subset A^*$, and that uniqueness of the generalized solution is equivalent to $A$ being maximal (in particular skew-adjoint for full forward/backward well-posedness). The work provides explicit analysis and criteria via deficiency indices $d_\pm(A)$ and the Cayley transform, and applies the theory to transport equations with solenoidal coefficients and to the linearized Euler system, including the case of Lipschitz coefficients ensuring skew-adjointness. Under Lipschitz (and bounded) regularity one obtains skew-adjoint $A$, yielding existence and uniqueness for both forward and backward problems; the results connect to and extend classical DiPerna–Lions–BV theory for transport coefficients.

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.

Theorems & Definitions (30)

• Definition 1.1
• Remark 1.1
• Theorem 2.1
• proof
• Lemma 2.1
• proof
• Theorem 3.1
• Theorem 3.2
• proof
• Theorem 3.3