Order Quasilinear Hyperbolic Systems Research Articles

Numerical study of multiphase hyperbolic models

Baer–Nunziato (BN) type models introduce fundamental and widely used class of multi-fluid and multi-velocity models to describe evolution of multi-phase flows. The original BN model is supposed to describe multi-phase flows at spatial scales substantially larger than the characteristic scale of elements of the dispersed phase. Recently a number of attempts have been performed to apply it to the description of multi-phase flows with resolved, at the large spatial scale, inter-phase boundaries. A number of still not well understood issues arise in this situation, related to the complex interplay of the topics related to (i) justification of the hierarchy of BN-type models for diffuse interface type description of multi-phase flows with resolved inter-phase boundaries; (ii) the questions of “correct” definition of generalized solutions of first order quasi-linear hyperbolic systems and (iii) development of “really” path-conservative and asymptotic preserving schemes for quasilinear hyperbolic systems with stiff relaxation terms. The goal of the paper is to study numerically such interplay when BN7 model is applied to describe 1D two-phase flows. Our main target is the diffuse interface type formulations. The obtained numerical solutions are compared to analytical solutions of the “more equilibrated” BN5 equations and heterogeneous formulations for the standard Euler equation. The goal is to provide factual information for the performed tests. We also try to summarize the obtained results in a systematic way from a practical point of view. To avoid ambiguity in interpretation of the numerical results, we provide detailed description of the algorithms and perform a number of auxiliary numerical tests.

On the development of shocks in fluids

In my 2007 work, I studied the formation of shocks in the context of Eulerian equations of the mechanics of compressible fluids. That work studied the maximal classical development of smooth initial data. The present review article is a presentation of my 2019 work, which addresses the problem of the physical continuation of the solution past the point of shock formation. The problem requires the construction of a hypersurface, the shock hypersurface, originating from a given singular spacelike surface, which is acoustically spacelike as viewed from its past, and the construction of a new solution in the future of the union of a given regular null hypersurface with the shock hypersurface, a solution which extends continuously the prior maximal classical solution across the given regular null hypersurface, but which displays discontinuities in physical variables across the shock hypersurface in accordance with the mass, momentum, and energy conservation laws. Moreover, the shock hypersurface is to be acoustically timelike as viewed from its future. Mathematically, this is a free boundary problem, with nonlinear conditions at the free boundary, for a first order quasilinear hyperbolic system of PDE, with characteristic initial data, which are singular at the past boundary of the initial null hypersurface, that boundary being the singular surface of origin. My 2019 work solved a restricted form of the problem through the introduction of new geometric and analytic methods. This Review is focused on the presentation of these methods.

Formation of finite-time singularities for nonlinear hyperbolic systems with small initial disturbances

This article concerns the formation of finite-time singularities in solutions to quasilinear hyperbolic systems with small initial data. We propose a universal test function method that works for many nonlinear hyperbolic systems arising from physical applications. We first present a simpler proof of the main result in the work of Sideris [Commun. Math. Phys. 101(4), 475–485 (1985)]: the global classical solution is non-existent for compressible Euler equations even for some small initial data. Then, we apply this approach to nonlinear magnetohydrodynamics in two space dimensions. Finally, we consider second order quasilinear hyperbolic systems with quadratic nonlinearity arising from elastodynamics of isotropic hyperelastic materials by ignoring the cubic and higher order terms. Under some restriction on the coefficients of the nonlinear terms that imply genuine nonlinearity, we show that the classical solutions to these equations can still blow up in finite time even if the initial data are small enough.

High-gain observers for a class of 2 × 2 quasilinear hyperbolic systems with 2 different velocities

In this work we extend recently proposed observer designs based on high-gain to a more general first order quasilinear hyperbolic system of balance laws. This class of systems is written in an observable form with two states, two different characteristic velocities and distributed measurement. The exponential stability of the related observation error is fully established by means of Lyapunov-based analysis.

Local Exact Boundary Synchronization for a Kind of First Order Quasilinear Hyperbolic Systems

In this paper, the synchronization for a kind of first order quasilinear hyperbolic system is taken into account. In this system, all the equations share the same positive wave speed. To realize the synchronization, a uniform constructive method is adopted, rather than an iteration process usually used in dealing with nonlinear systems. Furthermore, similar results on the exact boundary synchronization by groups can be obtained for a kind of first order quasilinear hyperbolic system of equations with different positive wave speeds by groups.

Differential constraints and exact solutions for the ET6 model

In this paper a first order quasilinear hyperbolic system describing a polyatomic gas far from the equilibrium is considered. After giving a classification of all the possible first order differential constraints admitted by the governing equations under interest, classes of exact solutions parameterized in terms of arbitrary functions are determined. That can help in solving initial or boundary value problems. Finally the consistency of the exact solutions characterized during the reduction procedure with the entropy principle is studied.

Exact internal controllability of nodal profile for first order quasilinear hyperbolic systems

For 1-D first order quasilinear hyperbolic systems without zero eigenvalues, based on the theory of semi-global C1 solution, by means of the results on both the exact boundary controllability and the exact boundary controllability of nodal profile, using an extension method, the exact controllability of nodal profile can be obtained purely with the help of internal controls. Moreover, using a perturbation method, by adding suitable internal controls to the part of equations corresponding to zero eigenvalues, the corresponding exact internal controllability of nodal profile for 1-D first order quasilinear hyperbolic systems with zero eigenvalues can be also obtained.

Exact controllability of nodal profile for 1‐D first order quasilinear hyperbolic systems with internal controls

For 1‐D first order quasilinear hyperbolic systems without zero eigenvalues, based on the theory of exact boundary controllability of nodal profile, using an extension method, the exact controllability of nodal profile can be realized in a shorter time by means of additional internal controls acting on suitably small space‐time domains. On the other hand, using a perturbation method, the exact controllability of nodal profile for 1‐D first order quasilinear hyperbolic systems with zero eigenvalues can be realized by additional internal controls to the part of equations corresponding to zero eigenvalues. Furthermore, by adding suitable internal controls to all the equations on suitable domains, the exact controllability of nodal profile for systems with zero eigenvalues can be realized in a shorter time.

Internal Controllability of First Order Quasi-linear Hyperbolic Systems with a Reduced Number of Controls

In this paper we investigate the exact controllability of $n \times n$ first order one-dimensional quasi-linear hyperbolic systems by $m<n$ internal controls that are localized in space in some part of the domain. We distinguish two situations. The first one is when the equations of the system have the same speed. In this case, we can use the method of characteristics and obtain a simple and complete characterization for linear systems. Thanks to a linear test this also provides some sufficient conditions for the local exact controllability around the trajectories of semilinear systems. However, when the speed of the equations is not the same, we see that we encounter the problem of loss of derivatives if we try to control quasi-linear systems with a reduced number of controls. To solve this problem, as in a prior article by Coron and Lissy on a Navier--Stokes control system, we first use the notion of algebraic solvability due to Gromov. However, in contrast with this prior article where a standard fixed...

Exact controllability for first order quasilinear hyperbolic systems with internal controls

Based on the theory of the local exact boundary controllability for first order quasilinear hyperbolic systems, using an extension method, the authors establish the exact controllability in a shorter time by means of internal controls acting on suitable domains. In particular, under certain special but reasonable hypotheses, the exact controllability can be realized only by internal controls, and the control time can be arbitrarily small.

GLOBAL SOLUTIONS OF THE EXPONENTIAL WAVE EQUATION WITH SMALL INITIAL DATA

We study the initial value problem of the exponential wave equation in <TEX>$\math{R}^{n+1}$</TEX> for small initial data. We shows, in the case of <TEX>$n=1$</TEX>, the global existence of solution by applying the formulation of first order quasilinear hyperbolic system which is weakly linearly degenerate. When <TEX>$n{\geq}2$</TEX>, a vector field method is applied to show the stability of a trivial solution <TEX>${\phi}=0$</TEX>.

Finite-Time Stabilization of $2\times2$ Hyperbolic Systems on Tree-Shaped Networks

We investigate the finite-time boundary stabilization of a one-dimensional first order quasilinear hyperbolic system of diagonal form on [0,1]. The dynamics of both boundary controls are governed by a finite-time stable ODE. The solutions of the closed-loop system issuing from small initial data in Lip$([0,1])$ are shown to exist for all times and to reach the null equilibrium state in finite time. When only one boundary feedback law is available, a finite-time stabilization is shown to occur roughly in a twice longer time. The above feedback strategy is then applied to the Saint-Venant system for the regulation of water flows in a network of canals.

Finite-time stabilization of hyperbolic systems over a bounded interval

We investigate the finite-time boundary stabilization of a 1-D first order quasilinear hyperbolic system of diagonal form on [0,1]. The dynamics of both boundary controls are governed by a finite-time stable ODE. The solutions of the closed-loop system issuing from small initial data in Lip([0,1]) are shown to exist for all times and to reach the null equilibrium state in finite time. A finite-time stabilization is also shown to occur when using only one boundary control. The above feedback strategy is then applied to the Saint-Venant system for the regulation of water flows in a canal. Some explicit numerical scheme is provided to numerically investigate the finite-time stabilization of the system with one (resp. two) boundary control(s).

Exact solutions to magnetogasdynamics using Lie point symmetries

In the present work, we find some exact solutions to the first order quasilinear hyperbolic system of partial differential equations (PDEs), governing the one dimensional unsteady flow of inviscid and perfectly conducting compressible fluid, subjected to a transverse magnetic field. For this, Lie group analysis is used to identify a finite number of generators that leave the given system of PDEs invariant. Out of these generators, two commuting generators are constructed involving some arbitrary constants. With the help of canonical variables associated with these two generators, the assigned system of PDEs is reduced to an autonomous system whose simple solutions provide nontrivial solutions of the original system. Using this exact solution, we discuss the evolutionary behavior of weak discontinuities.

Lipschitz continuous solutions to the Cauchy problem for quasi-linear hyperbolic systems

Lipschitz continuous solutions to the Cauchy problem for 1-D first order quasilinear hyperbolic systems are considered. Based on the methods of approximation and integral equations, the author gives two definitions of Lipschitz solutions to the Cauchy problem and proves the existence and uniqueness of solutions.

On the non-linear stability of scalar field cosmologies

We review recent work on the stability of flat spatially homogeneous and isotropic backgrounds with a self-interacting scalar field. We derive a first order quasi-linear symmetric hyperbolic system for the Einstein-nonlinear-scalar field system. Then, using the linearized system, we show how to obtain necessary and sufficient conditions which ensure the exponential decay to zero of small non-linear perturbations.

Strong (weak) exact controllability and strong (weak) exact observability for quasilinear hyperbolic systems

In this paper, the authors define the strong (weak) exact boundary controllability and the strong (weak) exact boundary observability for first order quasilinear hyperbolic systems, and study their properties and the relationship between them.

Exact boundary observability for a kind of second order quasilinear hyperbolic systems and its applications

In this paper, by means of a constructive method based on the existence and uniqueness of the semi-global C 2 solution, we establish the local exact boundary observability for a kind of second order quasilinear hyperbolic systems. As an application, we obtain the one-sided local exact boundary observability for a kind of first order quasilinear hyperbolic systems of diagonal form with boundary conditions in which the diagonal variables corresponding to the positive eigenvalues and those corresponding to the negative eigenvalues are decoupled.

Exact boundary controllability and observability for first order quasilinear hyperbolic systems with a kind of nonlocal boundary conditions

In this paper we establish the theory on the semiglobal classicalsolution to first order quasilinear hyperbolic systems with a kindof nonlocal boundary conditions, and based on this, thecorresponding exact boundary controllability and observability areobtained by a constructive method. Moreover, with the linearizedSaint-Venant system and the 1-D linear wave equation as examples, weshow that the number of both boundary controls and boundaryobservations can not be reduced, and consequently, we conclude thatthe exact boundary controllability for a hyperbolic system in anetwork with loop can not be realized generically.

Exact Boundary Controllability for 1-D Quasilinear Hyperbolic Systems with a Vanishing Characteristic Speed

The general theory on exact boundary controllability for general first order quasilinear hyperbolic systems requires that the characteristic speeds of the system do not vanish. This paper deals with exact boundary controllability, when this is not the case. Some important models are also shown as applications of the main result. The strategy uses the return method, which allows one in certain situations to recover nonzero characteristic speeds.