Degenerate Hyperbolic Systems Research Articles

Indirect boundary stabilization of strongly coupled degenerate hyperbolic systems

Indirect boundary stabilization of strongly coupled degenerate hyperbolic systems

Sonic-supersonic solutions for the two-dimensional steady compressible multiphase flow equations

This paper is concerned with the sonic-supersonic structures for the two-dimensional steady compressible inviscid multiphase flow equations. We construct a local classical supersonic solution near a given smooth sonic curve. This problem is originated from the transonic channel multiphase flows, which are one kind of the most important problems in mathematical fluid dynamics. In order to overcome the difficulties caused by the parabolic degeneracy near sonic and the multivariable dependence of pressure, we adopt the mixed variables of the pressure and angle functions and derive the characteristic decompositions of these quantities. In terms of the angle coordinate system, the multiphase flow equations can be transformed into a new degenerate hyperbolic system with an explicit singularity-regularity structure. We verify the convergence of the iterative sequence generated by the new system and then return the solution to the original physical variables. As a by-product, we obtain the existence of sonic-supersonic solutions for the steady full Euler equations with non-polytropic gases.

On Boundary Null Controllability of Strongly Degenerate Hyperbolic Systems on Star-Shaped Planar Network

In this paper we discuss the problem of boundary exact null controllability for weakly and strongly degenerate linear wave equation defined on star-shaped planar network. The network is represented by a singular measure in a bounded planar domain. The novelty of this article lies in the degeneration of the leading coefficient representative of the material properties at the common node of network. We discuss the existence of weak and strong solutions to the degenerate hyperbolic problem and establish the corresponding controllability properties.

Long Time Existence for the Bosonic Membrane in the Light Cone Gauge

This paper aims to establish the well-posedness on time interval $$[0,\varepsilon ^{-\frac{1}{2}}T]$$ of the classical initial problem for the bosonic membrane in the light cone gauge. Here $$\varepsilon $$ is the small parameter which measures the nonlinear effects. The bosonic membrane is timelike submanifolds with vanishing mean curvature. Since the initial Riemannian metric may be degenerate, the corresponding equation can be reduced to a quasi-linear degenerate hyperbolic system of second order with an area preserving constraint via a Hamiltonian reduction.

Modeling of Wave Propagation in Degenerate Hyperbolic Systems

Degeneration in a small spatial coordinate of the original initial---boundary-value problem for hyperbolic equations is considered in order to construct simpler models. The problem is investigated in a domain whose one spatial scale is much less than the other scales. This makes it possible to expand the desired functions in power series and hence to reduce the dimension of the problem. However, this comes at the cost of degeneration of the spectrum of the original three-dimensional problem.

Long-time behavior and stability of entropy solutions for linearly degenerate hyperbolic systems of rich type

We study linearly degenerate hyperbolic systems of rich type in one space dimension. It is showed that such a system admits exact traveling wave solutions after a finite time, provided that the initial data are Riemann type outside a space interval. We prove the convergence of entropy solutions toward traveling waves in the $L^1$ norm as the time goes to infinity. The traveling waves are determined explicitly in terms of the initial data and the system. We also obtain the stability of entropy solutions in $L^1$. Applications concern physical models such as the generalized extremal surface equations, the Born-Infeld system and augmented Born-Infeld system.

Delta Shock Waves for a Linearly Degenerate Hyperbolic System of Conservation Laws of Keyfitz-Kranzer Type

This paper is devoted to the study of delta shock waves for a hyperbolic system of conservation laws of Keyfitz-Kranzer type with two linearly degenerate characteristics. The Riemann problem is solved constructively. The Riemann solutions include exactly two kinds. One consists of two (or just one) contact discontinuities, while the other contains a delta shock wave. Under suitable generalized Rankine-Hugoniot relation and entropy condition, the existence and uniqueness of delta shock solution are established. These analytical results match well the numerical ones. Finally, two kinds of interactions of elementary waves are discussed.

Blow-up of periodic solutions to linearly degenerate quasilinear hyperbolic systems

An example is given to show that for the Cauchy problem of n×n homogeneous linearly degenerate quasilinear hyperbolic systems with periodic initial data, when n>2, C1 solutions may blow up in a finite time, no matter how small and smooth the initial data are.

ODE singularity for blocked linearly degenerate quasilinear hyperbolic systems

The mechanism of singularity formation is discussed for a kind of blocked quasilinear hyperbolic system with linearly degenerate characteristics, so that the ODE singularity can be shown for some kinds of complete reducible systems and, in particular, all the results in Li et al. (Math. Meth. Appl. Sci. 2008; 31:193–227) can be proved without the original assumption on the part richness. Copyright © 2011 John Wiley & Sons, Ltd.

Entropy solutions for linearly degenerate hyperbolic systems of rich type

Consider a linearly degenerate hyperbolic system of rich type. Assuming that each eigenvalue of the system has a constant multiplicity, we construct a representation formula of entropy solutions in L ∞ to the Cauchy problem. This formula depends on the solution of an autonomous system of ordinary differential equations taking x as parameter. We prove that for smooth initial data, the Cauchy problem for such an autonomous system admits a unique global solution. By using this formula together with classical compactness arguments, we give a very simple proof on the global existence of entropy solutions. Moreover, in a particular case of the system, we obtain an another explicit expression and the uniqueness of the entropy solution. Applications include the one-dimensional Born–Infeld system and linear Lagrangian systems.

Semilinear behavior for totally linearly degenerate hyperbolic systems with relaxation

We investigate totally linearly degenerate hyperbolic systems with relaxation. We aim to study their semilinear behavior, which means that the local smooth solutions cannot develop shocks, and the global existence is controlled by the supremum bound of the solution. In this paper we study two specific examples: the Suliciu-type and the Kerr–Debye-type models. For the Suliciu model, which arises from the numerical approximation of isentropic flows, the semilinear behavior is obtained using pointwise estimates of the gradient. For the Kerr–Debye systems, which arise in nonlinear optics, we show the semilinear behavior via energy methods. For the original Kerr–Debye model, thanks to the special form of the interaction terms, we can show the global existence of smooth solutions.

Global weakly discontinuous solutions to quasilinear hyperbolic systems of conservation laws with damping with a kind of non-smooth initial data

For the Cauchy problem with a kind of non-smooth initial data for weakly linearly degenerate hyperbolic systems of conservation laws with the linear damping term, we prove the existence and uniqueness of global weakly discontinuous solution u = u(t, x) containing only n weak discontinuities with small amplitude on t ≥ 0, and this solution possesses a global structure similar to that of the similarity solution $$u = U(\frac{x}{t})$$ of the corresponding homogeneous Riemann problem. As an application of our result, we obtain the existence and uniqueness of global weakly discontinuous solution, continuous and piecewise C 1 solution with discontinuous first order derivatives, of the flow equations of a model class of fluids with viscosity induced by fading memory.

Stability analysis of a degenerate hyperbolic system modelling a heat exchanger

Mathematical modelling of a heat exchanger in a carbon dioxide heat pump, an evaporator, is considered. A reduced model, called the the zero Mach-number limit, is derived from the Euler equations of compressible liquid flow through elimination of time scales associated with sound waves. The well-posedness of the resulting partial differential-algebraic equation (PDAE) is investigated by analysis of a frozen coefficient linearisation as well as by numerical experiments. The linear stability analysis is done through transformation to a canonical form with one hyperbolic component and one parabolic block of dimension 2. Using this canonical form it is seen how to prescribe boundary and initial data and an energy estimate is derived. Numerical experiments on the nonlinear PDAE using a finite difference spatial discretisation support the linear stability analysis.

Cauchy problem for weakly linearly degenerate hyperbolic systems in diagonal form

We study the Cauchy problem for quasi-linear strictly hyperbolic systems of diagonal form with weakly linearly degenerate characteristics. The global existence and uniqueness of C 1 solution with initial data ϕ in C b 1( R)∩W 1,1( R) are proved when the L 1( R) norm of ϕ or the C 0( R) norm of ϕ′ is small. We also give some examples to show that if the initial data are not in W 1,1( R) or not small, then the solution to the Cauchy problem may blow up in a finite time.

Boundary Value Problems for Degenerate Hyperbolic Systems of First Order Equations

In this paper we discuss some boundary value problems for degenerate hyperbolic complex equations of first order in a simply connected domain, in which the boundary value problems include the Riemann-Hilbert problem and the Cauchy problem. We first give the representation of solutions of the boundary value problems for the equations, and then prove the uniqueness and existence of solutions for the problems. In [A.V. Bitsadze (1988). Some Classes of Partial Differential Equations . Gordon and Breach, New York; A.V. Bitsadze and A.N. Nakhushev (1972). Theory of degenerating hyperbolic equations. Dokl. Akad. Nauk, SSSR , 204 , 1289-1291 (Russian); M.H. Protter (1954). The Cauchy problem for a hyperbolic second order equation. Can. J. Math ., 6 , 542-553], the authors discussed some boundary value problems for hyperbolic equations of second order.

Global classical discontinuous solutions to a class of generalized riemann problem for general quasilinear hyperbolic systems of conservation laws

In this paper, the authors prove the global existence and uniqueness of piecewise C1 solution u = u(t, x) containing only n contact discontinuities with small amplitude to the generalized Riemann problem for general linearly degenerate quasilinear hyperbolic systems of conservation laws with small decay initial data. This solution has a global structure similar to the similarity solution u=U(x/t) to the corresponding Riemann problem. The result shows that the similarity solution u=U(x/t) possesses a global nonlinear structural stability.

Asymptotic behavior of solutions for a degenerate hyperbolic system of viscous conservation laws with boundary effect

This paper studies the asymptotic behavior of solutions to the initial boundary value problem on the half space R+ a 2 x 2 degenerate system of viscous conservation laws. We show that the solution of the IBVP exists globally in time and tends toward a shifted critical viscous shock wave as time goes to infinity. We also prove that the system does not exhibit "phase transition" and remains in the degenerate hyperbolic state, when the initial perturbations are small. The proof is given by a weighted energy method.

Well-posedness of the shallow water equations in the presence of a front

We consider the initial value problem for the inviscid shallow-water equations in the case where a "front" is present, i.e., a boundary where the fluid depth tends to zero. Since the wave speed in shallow water behaves like the square root of the depth, this results in a degenerate hyperbolic system "on the edge" of change of type. It is shown that smooth solutions exist for smooth initial data.

Removing small divisors in the method of characteristic

The traditional method of characteristics applied to linear systems ofconservation laws with “clore” speeds of propagation could lead to expressions of solutions which contain small divisors. These could cause dificulty in compuations We propose to eliminate thew small divisors by developing a uniform representation for solutions of a linear degenerate systems of conservation laws. The uniform representation is achieved by global triangularization of matrix fnctions and the calculus of finite differences. We also obtain the asymptotic behaviour for large times t of solutions of degenerate hyperbolic systems among other by products.

Low regularity solutions for linearly degenerate hyperbolic systems

Low regularity solutions for linearly degenerate hyperbolic systems