Linear hyperbolic systems on networks: well-posedness and qualitative properties

Abstract

We study hyperbolic systems of one - dimensional partial differential equations under general , possibly non-local boundary conditions. A large class of evolution equations, either on individual 1- dimensional intervals or on general networks , can be reformulated in our rather flexible formalism , which generalizes the classical technique of first - order reduction . We study forward and backward well - posedness ; furthermore , we provide necessary and sufficient conditions on both the boundary conditions and the coefficients arising in the first - order reduction for a given subset of the relevant ambient space to be invariant under the flow that governs the system. Several examples are studied . p, li { white-space: pre-wrap; }

Telegraph systems on networks and port-Hamiltonians. I. Boundary conditions and well-posedness

“…We note that this contrasts with the approach of [26], where there is no a priori separation into the incoming and outgoing data in the boundary conditions at the cost of being confined to dissipative cases in the Hilbert space setting, not covering some standard cases and having not fully explicit representation, see Example 6. We show that our definition covers boundary conditions discussed in [32,26] and can describe more general situations. Next we provide an alternative, purely semigroup-theoretic, proof of the well-posedness of the problem.…”

“…On the other hand, systems of hyperbolic first order equations have not received much attention until recently. Here we note the paper [11] on linearized blood flow, the papers on the momentum operator [17] and recent works on more general hyperbolic problems on graphs such as [32,26]; the latter contains a comprehensive bibliography of the subject.…”

“…Such conditions have appeared in [27,Section 3] or [24] in the context of quantum graphs, where the dynamics on each edge is given by the heat or Schrödinger operators and it is required to construct boundary conditions ensuring the self-adjointness of the problem; in the latter the conditions were formulated in terms of maximal isotropic subspaces of the domain related to the Laplacian. Recently, similar ideas have been used in [32,26], where the authors considered hyperbolic systems at the edges of a network and at each node they constructed general boundary conditions that ensure the dissipativity of the problem in an appropriately weighted L 2 -space. The construction is based on the integration by parts along each edge and ensuring that the end-point terms of the integration by parts are non-negative.…”

“…The construction is based on the integration by parts along each edge and ensuring that the end-point terms of the integration by parts are non-negative. The authors achieve this by assuming that the vertex values of the solution belong to the so-called totally isotropic subspace associated with the quadratic form defined by a symmetrization of the matrix M. Due to this approach, the theory of [32,26] is strongly dependent on the Hilbert space setting.…”

“…Hence, we begin this paper by defining general Kirchhoff's conditions at a vertex and provide conditions under which they determine all outgoing data from this vertex through the incoming ones, as required in [9], and also that they satisfy the solvability assumptions of [19]. We note that this contrasts with the approach of [26], where there is no a priori separation into the incoming and outgoing data in the boundary conditions at the cost of being confined to dissipative cases in the Hilbert space setting, not covering some standard cases and having not fully explicit representation, see Example 6. We show that our definition covers boundary conditions discussed in [32,26] and can describe more general situations.…”