标题： 关于具有反对称空间算子的线性演化方程 显示英文标题

标题： On linear evolutionary equations with skew symmetric spatial operators

摘要： 我们研究与实希尔伯特空间中一个稠密定义的斜对称算子相关的演化方程的广义解。 我们建立了提供广义解的压缩半群的存在性，并找到了广义解唯一性的条件。 给出了一些应用，包括传输方程和线性化欧拉方程，其系数为无散（且通常不连续）的。 在系数的一些额外正则性假设下，我们证明了相应的空间算子是斜自伴的，这意味着对于正向和反向的柯西问题，广义解都存在且唯一。 显示英文摘要

摘要： 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.