Local well-posedness of a class of hyperbolic PDE–ODE systems on a bounded interval

2014 ◽  
Vol 11 (04) ◽  
pp. 705-747 ◽  
Author(s):  
Gilbert Peralta ◽  
Georg Propst
Keyword(s):  
Blow Up ◽  
Hyperbolic Systems ◽  
Energy Estimates ◽  
Well Posedness ◽  
Hyperbolic Pde ◽  
Flow Models ◽  
Bounded Interval ◽  
First Order ◽  
Hyperbolicity Region ◽  
Set Up

The well-posedness theory for hyperbolic systems of first-order quasilinear PDE's with ODE's boundary conditions (on a bounded interval) is discussed. Such systems occur in multi-scale blood flow models, as well as valveless pumping and fluid mechanics. The theory is presented in the setting of Sobolev spaces Hm (m ≥ 3 being an integer), which is an appropriate set-up when it comes to proving existence of smooth solutions using energy estimates. A blow-up criterion is also derived, stating that if the maximal time of existence is finite, then the state leaves every compact subset of the hyperbolicity region, or its first-order derivatives blow-up. Finally, we discuss physical examples which fit in the general framework presented.

scholarly journals STRONGLY HYPERBOLIC SYSTEMS IN GENERAL RELATIVITY

2004 ◽  
Vol 01 (02) ◽  
pp. 251-269 ◽  
Author(s):  
OSCAR A. REULA
Keyword(s):  
General Relativity ◽  
Numerical Modeling ◽  
Evolution Equations ◽  
Hyperbolic Systems ◽  
Propagation Speed ◽  
Energy Estimates ◽  
Well Posedness ◽  
Evolution System ◽  
Hyperbolic Case ◽  
First Order

We discuss several topics related to the notion of strong hyperbolicity which are of interest in general relativity. After introducing the concept and showing its relevance we provide some covariant definitions of strong hyperbolicity. We then prove that if a system is strongly hyperbolic with respect to a given hypersurface, then it is also strongly hyperbolic with respect to any nearby surface. We then study for how much these hypersurfaces can be deformed and discuss then causality, namely what the maximal propagation speed in any given direction is. In contrast with the symmetric hyperbolic case, for which the proof of causality is geometrical and direct, relaying in energy estimates, the proof for general strongly hyperbolic systems is indirect for it is based in Holmgren's theorem. To show that the concept is needed in the area of general relativity we discuss two results for which the theory of symmetric hyperbolic systems shows to be insufficient. The first deals with the hyperbolicity analysis of systems which are second order in space derivatives; they include certain versions of the ADM and the BSSN families of equations. This analysis is considerably simplified by introducing pseudo-differential first-order evolution equations. Well-posedness for some members of the latter family systems is established by showing they satisfy the strong hyperbolicity property. Furthermore it is shown that many other systems of such families are only weakly hyperbolic, implying they should not be used for numerical modeling. The second result deals with systems having constraints. The question posed is which hyperbolicity properties, if any, are inherited from the original evolution system by the subsidiary system satisfied by the constraint quantities. The answer is that, subject to some condition on the constraints, if the evolution system is strongly hyperbolic then the subsidiary system is also strongly hyperbolic and the causality properties of both are identical.

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; }

scholarly journals Notes on the Global Well-Posedness for the Maxwell-Navier-Stokes System

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Ensil Kang ◽  
Jihoon Lee
Keyword(s):  
Global Existence ◽  
Stokes System ◽  
Blow Up ◽  
Energy Estimates ◽  
Navier Stokes ◽  
Well Posedness ◽  
Regular Solutions ◽  
2D System ◽  
Blow Up Criterion

Masmoudi (2010) obtained global well-posedness for 2D Maxwell-Navier-Stokes system. In this paper, we reprove global existence of regular solutions to the 2D system by using energy estimates and Brezis-Gallouet inequality. Also we obtain a blow-up criterion for solutions to 3D Maxwell-Navier-Stokes system.

scholarly journals NONLINEAR WELL-POSEDNESS AND RATES OF DECAY IN THERMOELASTICITY WITH SECOND SOUND

2008 ◽  
Vol 05 (01) ◽  
pp. 25-43 ◽  
Author(s):  
REINHARD RACKE ◽  
YA-GUANG WANG
Keyword(s):  
Hyperbolic Systems ◽  
Decay Rates ◽  
Energy Estimates ◽  
Well Posedness ◽  
Time Behavior ◽  
Second Sound ◽  
Small Solution ◽  
Nonlinear Thermoelasticity ◽  
The Cauchy Problem ◽  
Rates Of Decay

The Cauchy problem in nonlinear thermoelasticity with second sound in one space dimension is considered. Due to Cattaneo's law, replacing Fourier's law for heat conduction, the system is hyperbolic. The local well-posedness as a strictly hyperbolic system is investigated first, and then the relation between energy estimates for non-symmetric hyperbolic systems and well-posedness are discussed. For the global small solution, the long time behavior is described and the decay rates of the L2-norm are obtained.

scholarly journals ENERGY ESTIMATES FOR WEAKLY HYPERBOLIC SYSTEMS OF THE FIRST ORDER

2005 ◽  
Vol 07 (06) ◽  
pp. 809-837 ◽  
Author(s):  
MICHAEL DREHER ◽  
INGO WITT
Keyword(s):  
Sobolev Spaces ◽  
Upper Bound ◽  
Hyperbolic Systems ◽  
Elliptic Boundary ◽  
Cauchy Data ◽  
Energy Estimates ◽  
Elliptic Boundary Value ◽  
First Order ◽  
The Cauchy Problem ◽  
Loss Of Regularity

For a class of first-order weakly hyperbolic pseudo-differential systems with finite time degeneracy, well-posedness of the Cauchy problem is proved in an adapted scale of Sobolev spaces. These Sobolev spaces are constructed in correspondence to the hyperbolic operator under consideration, making use of ideas from the theory of elliptic boundary value problems on manifolds with singularities. In addition, an upper bound for the loss of regularity that occurs when passing from the Cauchy data to the solutions is established. In many examples, this upper bound turns out to be sharp.

Mathematical modelling of salt purification by recrystallization. A cross-current flow model and its comparison with the countercurrent arrangement

1986 ◽  
Vol 51 (11) ◽  
pp. 2489-2501
Author(s):  
Benitto Mayrhofer ◽  
Jana Mayrhoferová ◽  
Lubomír Neužil ◽  
Jaroslav Nývlt
Keyword(s):  
Flow Model ◽  
Current Flow ◽  
Countercurrent Flow ◽  
Flow Models ◽  
Multi Stage ◽  
Stage Crystallization ◽  
The Cross ◽  
Current Flows ◽  
Set Up ◽  
Cross Current

A model is derived for a multi-stage crystallization with cross-current flows of the solution and the crystals being purified. The purity of the product is compared with that achieved in the countercurrent arrangement. A suitable function has been set up which allows the cross-current and countercurrent flow models to be compared and reduces substantially the labour of computation for the countercurrent arrangement. Using the recrystallization of KAl(SO4)2.12 H2O as an example, it is shown that, when the cross-current and countercurrent processes are operated at the same output, the countercurrent arrangement is more advantageous because its solvent consumption is lower.

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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