scholarly journals Recent Advances on Boundary Conditions for Equations in Nonequilibrium Thermodynamics

Symmetry ◽  
2021 ◽  
Vol 13 (9) ◽  
pp. 1710
Author(s):  
Wen-An Yong ◽  
Yizhou Zhou
Keyword(s):  
Boundary Conditions ◽  
Structural Stability ◽  
Stability Condition ◽  
Hyperbolic Systems ◽  
Moment Closure ◽  
First Order ◽  
Nonequilibrium Phenomena ◽  
Classical Models ◽  
Spatial Domains ◽  
Kreiss Condition

This paper is concerned with modeling nonequilibrium phenomena in spatial domains with boundaries. The resultant models consist of hyperbolic systems of first-order partial differential equations with boundary conditions (BCs). Taking a linearized moment closure system as an example, we show that the structural stability condition and the uniform Kreiss condition do not automatically guarantee the compatibility of the models with the corresponding classical models. This motivated the generalized Kreiss condition (GKC)—a strengthened version of the uniform Kreiss condition. Under the GKC and the structural stability condition, we show how to derive the reduced BCs for the equilibrium systems as the classical models. For linearized problems, the validity of the reduced BCs can be rigorously verified. Furthermore, we use a simple example to show how thus far developed theory can be used to construct proper BCs for equations modeling nonequilibrium phenomena in spatial domains with boundaries.

scholarly journals Linear Hyperbolic Systems on Networks: well-posedness and qualitative properties

2020 ◽  
Author(s):  
Marjeta Kramar ◽  
Delio Mugnolo ◽  
Serge Nicaise
Keyword(s):  
Boundary Conditions ◽  
Evolution Equations ◽  
Hyperbolic Systems ◽  
Sufficient Conditions ◽  
Order Reduction ◽  
Well Posedness ◽  
Qualitative Properties ◽  
Local Boundary ◽  
First Order ◽  
Necessary And Sufficient

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; }

ON BOUNDARY CONDITIONS FOR FIRST-ORDER SYMMETRIC HYPERBOLIC SYSTEMS WITH CONSTRAINTS

2013 ◽  
Vol 10 (04) ◽  
pp. 725-734 ◽  
Author(s):  
NICOLAE TARFULEA
Keyword(s):  
Boundary Conditions ◽  
Wave Equations ◽  
Hyperbolic Systems ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
First Order ◽  
The Cauchy Problem ◽  
Initial Boundary Value ◽  
Well Posed

The Cauchy problem for many first-order symmetric hyperbolic (FOSH) systems is constraint preserving, i.e. the solution satisfies certain spatial differential constraints whenever the initial data does. Frequently, artificial space cut-offs are performed for such evolution systems, usually out of the necessity for finite computational domains. However, it may easily happen that boundary conditions at the artificial boundary for such a system lead to an initial boundary value problem which, while well-posed, does not preserve the constraints. Here we consider the problem of finding constraint-preserving boundary conditions for constrained FOSH systems in the well-posed class of maximal non-negative boundary conditions. Based on a characterization of maximal non-negative boundary conditions, we discuss a systematic technique for finding such boundary conditions that preserve the constraints, pending that the constraints satisfy a FOSH system themselves. We exemplify this technique by analyzing a system of wave equations in a first-order formulation subject to divergence constraints.

A Note on the One-Side Exact Boundary Observability for Quasilinear Hyperbolic Systems

2008 ◽  
Vol 15 (3) ◽  
pp. 571-580
Author(s):  
Tatsien Li ◽  
Bopeng Rao ◽  
Zhiqiang Wang
Keyword(s):  
Boundary Conditions ◽  
Hyperbolic Systems ◽  
Exact Boundary Controllability ◽  
Boundary Controllability ◽  
Quasilinear Hyperbolic Systems ◽  
First Order ◽  
Exact Boundary ◽  
The One ◽  
Boundary Observability ◽  
Exact Boundary Observability

Abstract The known theory on the one-side exact boundary observability for first order quasilinear hyperbolic systems requires that the unknown variables are suitably coupled or satisfy the Group Property in boundary conditions on the non-observation side (see [Tatsien, C. R. Math. Acad. Sci. 342: 937–942, 2006]–[Tatsien, ESAIM Control Optim. Calc. Var.], [Russell, SIAM Rev. 20: 639–739, 1978]). In this paper we illustrate, with an inspiring example, that the one-side exact boundary observability can be realized by means of a suitable coupling of the unknown variables in quasilinear hyperbolic system itself instead of in boundary conditions. Moreover, an implicit duality between the one-side exact boundary controllability and one-side exact boundary observability is also revealed in this situation.

scholarly journals Telegraph systems on networks and port-Hamiltonians. Ⅱ. Network realizability

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Jacek Banasiak ◽  
Adam Błoch
Keyword(s):  
Boundary Conditions ◽  
Adjacency Matrix ◽  
Hyperbolic Equations ◽  
Hyperbolic Systems ◽  
General Linear ◽  
First Order ◽  
Metric Graph ◽  
Potential Map ◽  
The Matrix ◽  
Network Language

<p style='text-indent:20px;'>Hyperbolic systems on networks often can be written as systems of first order equations on an interval, coupled by transmission conditions at the endpoints, also called port-Hamiltonians. However, general results for the latter have been difficult to interpret in the network language. The aim of this paper is to derive conditions under which a port-Hamiltonian with general linear Kirchhoff's boundary conditions can be written as a system of <inline-formula><tex-math id="M1">\begin{document}$ 2\times 2 $\end{document}</tex-math></inline-formula> hyperbolic equations on a metric graph <inline-formula><tex-math id="M2">\begin{document}$ \Gamma $\end{document}</tex-math></inline-formula>. This is achieved by interpreting the matrix of the boundary conditions as a potential map of vertex connections of <inline-formula><tex-math id="M3">\begin{document}$ \Gamma $\end{document}</tex-math></inline-formula> and then showing that, under the derived assumptions, that matrix can be used to determine the adjacency matrix of <inline-formula><tex-math id="M4">\begin{document}$ \Gamma $\end{document}</tex-math></inline-formula>.</p>

Chaotic Oscillations of Solutions of First Order Hyperbolic Systems in 1D with Nonlinear Boundary Conditions

2014 ◽  
Vol 24 (05) ◽  
pp. 1450072 ◽  
Author(s):  
Xiongping Dai ◽  
Tingwen Huang ◽  
Yu Huang ◽  
Goong Chen
Keyword(s):  
Boundary Conditions ◽  
Chaos Theory ◽  
Nonlinear Boundary ◽  
Dimensional Space ◽  
Hyperbolic Systems ◽  
Nonlinear Boundary Conditions ◽  
Chaotic Oscillations ◽  
One Dimensional ◽  
First Order ◽  
Delay Effects

We study chaotic oscillations of solutions of a first order hyperbolic system in one-dimensional space, where the governing equation is linear but the boundary condition contains nonlinearity with nonlocal and possibly time-delay effects. The main thrust of the paper is the advancement of existing chaos theory to multicomponent hyperbolic PDEs that allows a unified treatment of a general class of nonlinear, nonlocal and time-delayed boundary conditions where components of waves travel with several different speeds.

On the Boundary Value Problem in A Dihedral Angle for Normally Hyperbolic Systems of First Order

1998 ◽  
Vol 5 (2) ◽  
pp. 121-138
Author(s):  
O. Jokhadze
Keyword(s):  
Partial Differential Equations ◽  
Boundary Value Problem ◽  
Principal Part ◽  
Hyperbolic Systems ◽  
Boundary Value ◽  
Three Dimensional ◽  
Order System ◽  
First Order ◽  
Dimensional Boundary ◽  
Class Of Functions

Abstract Some structural properties as well as a general three-dimensional boundary value problem for normally hyperbolic systems of partial differential equations of first order are studied. A condition is given which enables one to reduce the system under consideration to a first-order system with the spliced principal part. It is shown that the initial problem is correct in a certain class of functions if some conditions are fulfilled.

scholarly journals Intermittent heat instabilities in an air plume

2016 ◽  
Vol 23 (4) ◽  
pp. 319-330
Author(s):  
Jean-Louis Le Mouël ◽  
Vladimir G. Kossobokov ◽  
Frederic Perrier ◽  
Pierre Morat
Keyword(s):  
Temperature Sensors ◽  
Time Varying ◽  
First Order ◽  
Spatial Function ◽  
Universal Feature ◽  
Classical Models ◽  
Set Up ◽  
Plume Flow ◽  
Temperature Time ◽  
Classical Shape

Abstract. We report the results of heating experiments carried out in an abandoned limestone quarry close to Paris, in an isolated room of a volume of about 400 m3. A heat source made of a metallic resistor of power 100 W was installed on the floor of the room, at distance from the walls. High-quality temperature sensors, with a response time of 20 s, were fixed on a 2 m long bar. In a series of 24 h heating experiments the bar had been set up horizontally at different heights or vertically along the axis of the plume to record changes in temperature distribution with a sampling time varying from 20 to 120 s. When taken in averages over 24 h, the temperatures present the classical shape of steady-state plumes, as described by classical models. On the contrary, the temperature time series show a rich dynamic plume flow with intermittent trains of oscillations, spatially coherent, of large amplitude and a period around 400 s, separated by intervals of relative quiescence whose duration can reach several hours. To our knowledge, no specific theory is available to explain this behavior, which appears to be a chaotic interaction between a turbulent plume and a stratified environment. The observed behavior, with first-order factorization of a smooth spatial function with a global temporal intermittent function, could be a universal feature of some turbulent plumes in geophysical environments.

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