Navier–Stokes Equations on Weighted Graphs

Physica D: Nonlinear Phenomena

We derive a class of equations describing low Reynolds number steady flows of incompressible and viscous fluids in networks made of straight channels, with several sources and sinks, and adaptive conductivities. The flow is controlled by the fluxes at sources and sinks. The network is represented by a graph and the adaptive conductivities describe the transverse channel elasticities, mirroring several network structures found in physics and biology. Minimising the dissipated energy per unit time, we have found an explicit form for the adaptation equations and, asymptotically in time, a steady state tree geometry for the graph connecting sources and sinks is reached. A phase transition tuned by an order parameter for the adapted steady sate graph has been found.

We introduce a notion of generalized Navier-Stokes flows on manifolds, that extends to the viscous case the one defined by Brenier. Their kinetic energy extends the kinetic energy for classical Brownian flows, defined as the L2 norm of their drift. We prove that there exists a generalized flow which realizes the infimum of kinetic energies among all generalized flows with prescribed initial and final configuration. Finally we construct generalized flows with prescribed drift and kinetic energy smaller than the L2 norm of the drift.

Journal of Geometry and Physics

We study properties of the solutions to Navier-Stokes system on compact Riemannian manifolds. The motivation for such a formulation comes from atmospheric models as well as some thin film flows on curved surfaces. There are different choices of the diffusion operator which have been used in previous studies, and we make a few comments why the choice adopted below seems to us the correct one. This choice leads to the conclusion that Killing vector fields are essential in analyzing the qualitative properties of the flow. We give several results illustrating this and analyze also the linearized version of Navier-Stokes system which is interesting in numerical applications. Finally we consider the 2 dimensional case which has specific characteristics, and treat also the Coriolis effect which is essential in atmospheric flows.

Physica A: Statistical Mechanics and its Applications, 2013

A remarkable feature of fluid dynamics is its relationship with classical dynamics and statistical mechanics. This has motivated in the past mathematical investigations concerning, in a special way, the "derivation" based on kinetic theory, and in particular the Boltzmann equation, of the incompressible Navier-Stokes equations (INSE). However, the connection determined in this way is usually merely asymptotic (i.e., it can be reached only for suitable limit functions) and therefore presents difficulties of its own. This feature has suggested the search of an alternative approach, based on the construction of a suitable inverse kinetic theory (IKT; , which can avoid them. IKT, in fact, permits to achieve an exact representation of the fluid equations by identifying them with appropriate moment equations of a suitable (inverse) kinetic equation.

viXra, 2015

The main purpose of this paper is to extend Banach spaces on topological graphs with operator actions and show all of these extensions are also Banach space with a bijection with a bijection between linear continuous functionals and elements, which enables one to solve linear functional equations in such extended space, particularly, solve algebraic, differential or integral equations on a topological graph, find multi-space solutions on equations, for instance, the Einstein’s gravitational equations.

New Directions in Mathematical Fluid Mechanics, 2020

Mechanics is a forum for the publication of high quality monographs, or collections of works, on the mathematical theory of fluid mechanics, with special regards to the Navier-Stokes equations. Its mathematical aims and scope are similar to those of the Journal of Mathematical Fluid Mechanics. In particular, mathematical aspects of computational methods and of applications to science and engineering are welcome as an important part of the theory. So also are works in related areas of mathematics that have a direct bearing on fluid mechanics.

2006

The aim of the present work is two fold. In one hand it shows to mathematicians how the apparently pure mathematical concepts can be applied to the efficient solution of problems in structural mechanics. In the other hand it illustrates to engineers the important role of mathematical concepts for the solution of engineering problems. In this paper a number of applications of graph theory in structural mechanics are presented. These applications simplify the analysis of structures and make their optimal analysis feasible. For each case, the main problem is stated and then the formulation together with illustrative examples is presented.

2010

Let V be a Banach space over a field F. A − → G-flow is a graph − → G embedded in a topological space S associated with an injective mappings L : u v → L(u v) ∈ V such that L(u v) = −L(v u) for ∀(u, v) ∈ X − → G holding with conservation laws u∈N G (v) L (v u) = 0 for ∀v ∈ V − → G , where u v denotes the semi-arc of (u, v) ∈ X − → G , which is an abstract model, also a mathematical object for things embedded in a topological space, or matters happened in the world. The main purpose of this paper is to extend Banach spaces on topological graphs with operator actions and show all of these extensions are also Banach space with unique correspondence in elements on linear continuous functionals, which enables one to solve linear functional equations in such extended space, particularly, solve algebraic, differential or integral equations on a topological graph, i.e., find multi-space solutions for equations, for instance, the Einstein's gravitational equations. A generalization of so...

In this paper we present a unifying geometric and compositional framework for modeling complex physical network dynamics as port-Hamiltonian systems on open graphs. Basic idea is to associate with the incidence matrix of the graph a Dirac structure relating the flow and effort variables associated to the edges, internal vertices, as well as boundary vertices of the graph, and to formulate energy-storing or energy-dissipating relations between the flow and effort variables of the edges and internal vertices. This allows for state variables associated to the edges, and formalizes the interconnection of networks. Examples from different origins such as consensus algorithms are shown to share the same structure. It is shown how the identified Hamiltonian structure offers systematic tools for the analysis of the resulting dynamics. * A.J. van der Schaft is with the Johann 20 Or the composition of the effort-continuous graph Dirac structure with {(f0, e 0 ) ∈ Λ0 × Λ 0 | f0 = 0}.

Analysis & PDE

We define and study the Airy operator on star graphs. 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 Kortewegde 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 solved here 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.