A new approach to the solution of quasilinear nonelliptic first-order systems of inhomogeneous PDEs in many dimensions is presented. It is based on a version of the conditional symmetry and Riemann invariant methods. We discuss in detail the necessary and sufficient conditions for the existence of rank-2 and rank-3 solutions expressible in terms of Riemann invariants. We perform the analysis using the Cayley-Hamilton theorem for a certain algebraic system associated with the initial system. The problem of finding such solutions has been reduced to expanding a set of trace conditions on wave vectors and their profiles which are expressible in terms of Riemann invariants. A couple of theorems useful for the construction of such solutions are given. These theoretical considerations are illustrated by the example of inhomogeneous equations of fluid dynamics which describe motion of an ideal fluid subjected to gravitational and Coriolis forces. Several new rank-2 solutions are obtained. Some physical interpretation of these results is given.
Existence of Riemannian invariants for integrable systems of hydrodynamic type
School of Mathematics, Loughborough University（拉夫堡大学数学学院）；Institute of Mathematics and Mathematical Modeling（数学与数学建模研究所）；Faculty of Mechanics and Mathematics and Center for Fundamental and Applied Mathematics, Moscow State University（莫斯科国立大学力学数学系及基础与应用数学中心）；Institut für Mathematik, Friedrich Schiller Universität Jena（耶拿弗里德里希·席勒大学数学研究所）
Hydrodynamic limit of symmetric exclusion processes in inhomogeneous media
在\cite{J} M. Jara提出了一种方法，将对称排斥过程的流体动力学极限的证明简化为均匀化问题，作为该领域最近工作的统一方法，如\cite{N}，\cite{F1}，\cite{F2}和\cite{FJL}。虽然在\cite{J}中没有说明，但在\cite{N}，\cite{F1}中已经（以不同的方式）得到了将流体动力学极限简化为均匀化问题。这两个问题之间的这种可供选择的、非常简单的关系可以追溯到K的一个概念。Nagy \cite{N}在\cite{F1} [B节]中针对$\bbZ^d$上的排除过程进行了说明，并且如\cite{F2}中所强调的，是完全通用的。上述关系式已应用于\cite{N}、\cite{F1}、\cite{F2}和\cite{FJL}，并可应用于\cite{J}中提到的其他对称排斥过程。在这个简短的说明中，我们简要回顾了完整的一般设置中的这种统一方法。最后，我们回顾一下如何均匀化问题已经解决了上述以前的作品。
Hydrodynamics of Multi-Species Driven Diffusive Systems with Open Boundaries: A Two-Tasep Study
Laboratoire de Physique et Chimie Th\'eoriques, CNRS UMR 7019, Universit\'e de Lorraine, Boulevard des Aiguillettes -- B.P. 70239, F-54506 Vandoeuvre-les-Nancy, France
