星图上的Airy型发展方程
Airy-type evolution equations on star graphs
摘要：
本文定义并研究了星图上的Airy算子。Airy算子是一个三阶微分算子，在不同的背景下产生，但我们主要关注的是它的作用，作为线性部分的Korteweg-de Vries方程，通常研究一条线或半线。处理和解决的第一个问题是它的正确定义，具有不同的特征，作为一个星形图上的斜伴随算子，一组在一个公共顶点连接的线，例如，分支通道的网络。一个必要条件是图是平衡的，即在顶点处有相同数量的进出边。最简单的例子是在顶点有一个点相互作用的直线。在这些情况下，艾里动力学由酉群或等距群（在真实情况下）给出。特别地，该分析提供了给出动量的边界条件的完整分类（即，解的$L^2$ -范数）在图上保持演化。第二个更普遍的问题在这里解决的是条件下的艾里算子产生的压缩半群的特征。在这种情况下，不平衡的星形图是允许的。在酉和收缩动力学中，如果质量守恒（即，解决方案的整体性）进一步施加。上述适定性结果可作为分析非线性波在分支结构上传播的基础。
英文摘要：
In the present paper the Airy operator on star graphs is defined and studied. The Airy operator is a third order differential operator arising in different contexts, but our main concern is related to its role as the linear part of the Korteweg-de Vries equation, usually studied on a line or a half-line. The first problem treated and solved is its correct definition, with different characterizations, as a skew-adjoint operator on a star graph, a set of lines connecting at a common vertex representing, for example, a network of branching channels. A necessary condition turns out to be that the graph is balanced, i.e. there is the same number of ingoing and outgoing edges at the vertex. The simplest example is that of the line with a point interaction at the vertex. In these cases the Airy dynamics is given by a unitary or isometric (in the real case) group. In particular the analysis provides the complete classification of boundary conditions giving momentum (i.e., $L^2$-norm of the solution) preserving evolution on the graph. A second more general problem here solved is the characterization of conditions under which the Airy operator generates a contraction semigroup. In this case unbalanced star graphs are allowed. In both unitary and contraction dynamics, restrictions on admissible boundary conditions occur if conservation of mass (i.e., integral of the solution) is further imposed. The above well posedness results can be considered preliminary to the analysis of nonlinear wave propagation on branching structures.
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