Problem For Quasilinear Hyperbolic Systems Research Articles

Local Solution of Three-Dimensional Axisymmetric Supersonic Flow in a Nozzle

In this paper, we construct a local supersonic flow in a 3-dimensional axis-symmetry nozzle when a uniform supersonic flow inserts the throat. We apply the local existence theory of boundary value problem for quasilinear hyperbolic system to solve this problem. The boundary value condition is set in particular to guarantee the character number condition. By this trick, the theory in quasilinear hyperbolic system can be employed to a large range of the boundary value problem.

Saturated boundary feedback control of quasi-linear hyperbolic balance laws with application to LWR traffic flow stabilization

The saturated boundary stabilization problem for quasi-linear hyperbolic systems of balance laws is considered under H2-norm in this paper, where the boundary conditions of the system are subject to actuator saturations. The resulting closed-loop system is proven to be locally exponentially stable with respect to the steady states in the presence of saturations. To this end, the sector nonlinearity model is introduced to deal with the saturation term, and then sufficient conditions for ensuring the locally exponential stability are established in terms of a set of matrix inequalities by employing the Lyapunov function method along with a sector condition. Furthermore, these results are applied to the stabilization of the two-lane traffic flow dynamic represented by Lighthill–Whitham–Richards (LWR) model. By utilizing variable speed limit (VSL) devices, a saturated boundary feedback controller is designed to stabilize the two-lane traffic flow, and the exponential convergence of the quasi-linear traffic flow system in H2 sense is validated by numerical simulations.

Global classical solutions of a kind of boundary value problem for quasilinear hyperbolic systems

In this paper, we consider a stability problem of a kind of boundary value problem for quasilinear hyperbolic systems. For small boundary data, we prove that the solution exists globally in time when the system is weakly linearly degenerate. In the special case of linear degeneracy, the smallness assumption on the boundary data is weakened. The error estimate in space between two different solutions with different boundary data is also obtained. In our proof, an important estimate which describes the interaction of different waves is established by constructing a continuous Glimm function. This estimate together with uniform a prior estimates with respect to the time leads to the global‐in‐time existence of smooth solutions. Finally, we apply the stability results to the isentropic Euler equations and the system of the motion of an elastic string.

Mixed problem for quasilinear hyperbolic system with coefficients functionally dependent on solution

Abstract The mixed problem for quasilinear hyperbolic system with coefficients functionally dependent on the solution is studied. We assume that the coefficients are continuous nonlinear operators in the Banach space C1(ℝ) satisfying some additional assumptions. Under these assumptions we prove the uniqueness and existence of local in time C1 solution, provided that the initial data are also of class C1.

Asymptotic stability of equilibrium state to the mixed initial-boundary value problem for quasilinear hyperbolic systems

Under the internal dissipative condition, the Cauchy problem for inhomogeneous quasilinear hyperbolic systems with small initial data admits a unique global C1 solution, which exponentially decays to zero as t → +∞, while if the coefficient matrix Θ of boundary conditions satisfies the boundary dissipative condition, the mixed initial-boundary value problem with small initial data for quasilinear hyperbolic systems with nonlinear terms of at least second order admits a unique global C1 solution, which also exponentially decays to zero as t → +∞. In this paper, under more general conditions, the authors investigate the combined effect of the internal dissipative condition and the boundary dissipative condition, and prove the global existence and exponential decay of the C1 solution to the mixed initial-boundary value problem for quasilinear hyperbolic systems with small initial data. This stability result is applied to a kind of models, and an example is given to show the possible exponential instability if the corresponding conditions are not satisfied.

Riemann solution for one dimensional non-ideal isentropic magnetogasdynamics

In the present paper, we study the Riemann problem for quasilinear hyperbolic system of partial differential equations governing the one dimensional non-ideal isentropic magnetogasdynamics with transverse magnetic field. We discuss the properties of rarefaction waves, shocks and contact discontinuities. Differently from single equation methods rooted in the ideal gasdynamics, the new approach is based on the system of two nonlinear algebraic equations imposing the equality of total pressure and velocity, assuming as unknowns the two values of densities, on both sides of the contact discontinuity. Newton iterative method is used to obtain densities. The resulting exact solver is implemented with the examples of general applicability of the proposed approach.

Riemann solution for ideal isentropic magnetogasdynamics

In the present paper, we study the Riemann problem for quasilinear hyperbolic system of partial differential equations governing the one dimensional ideal isentropic magnetogasdynamics with transverse magnetic field. We discuss the properties of rarefaction waves, shocks and contact discontinuities. Differently from single equation methods rooted in the ideal gasdynamics, the new approach is based on the system of two nonlinear equations imposing the equality of total pressure and velocity, assuming as unknowns the two values of densities, on both sides of the contact discontinuity. Newton iterative method is used to obtain densities. The resulting exact solver is implemented with the examples of general applicability of the proposed approach. For comparisons with exact solution we also shown numerical results obtained by the total variation diminishing slope limiter centre scheme. It is shown that both analytical and numerical results demonstrate the broad applicability and robustness of the new Riemann solver.

Optimal control problem for quasilinear hyperbolic system: the Slutsky equation (in Ukrainian)

Optimal control problem for quasilinear hyperbolic system: the Slutsky equation (in Ukrainian)

Global existence of weak discontinuous solutions to the Cauchy problem with small BV initial data for quasilinear hyperbolic systems

In this paper, we study the Cauchy problem for quasilinear hyperbolic system with a kind of non‐smooth initial data. Under the assumption that the initial data possess a suitably small bounded variation norm, a necessary and sufficient condition is obtained to guarantee the existence and uniqueness of global weak discontinuous solution. Copyright © 2014 John Wiley & Sons, Ltd.

The lifespan of classical solution to the Cauchy problem for the hyperbolic mean curvature flow with a linear forcing term

We investigate the hyperbolic mean curvature flow with a linear forcing term. By means of support function, a hyperbolic Monge-Ampere equation is derived and this equation could be reduced to a system in Riemann invariants. Using the theory of the local solution to Cauchy problem for quasilinear hyperbolic system, we discuss the lifespan (the maximal existence time of unique local classical solutions) of classical solution to the Cauchy problem for the hyperbolic mean curvature flow with a linear forcing term.

Cauchy problem for quasilinear hyperbolic systems of shallow water equations

This article investigates the Cauchy problem for two different models (modified and classical), governed by quasilinear hyperbolic systems that arise in shallow water theory. Under certain reasonable hypotheses on the initial data, we obtain the global smooth solutions for both the systems. The bounds on simple wave solutions of the modified system are shown to depend on the parameter H characterizing the advective transport of impulse. Similarly the bounds on simple wave solutions of the classical system describing the flow over a sloping bottom with profile b(x) are shown to depend on the bottom topography. On the other hand, if the initial data are specified differently, then it is shown that solutions for both the systems exhibit finite time blow-up from specific smooth initial data. Moreover, we show that an increase in H and convexity of b would reduce the time taken for the solutions to blow up.

Global existence of classical solutions of Goursat problem for quasilinear hyperbolic systems with BV data

We investigate the existence of a global classical solution to the Goursat problem for linearly degenerate quasilinear hyperbolic systems. As the result in [A. Bressan, Contractive metrics for nonlinear hyperbolic systems, Indiana Univ. Math. J. 37 (1988) 409–421] suggests that one may achieve global smoothness even if the C1 norm of the initial data is large, we prove that, if the C1 norm of the boundary data is bounded but possibly large, and the BV norm of the boundary data is sufficiently small, then the solution remains C1 globally in time. Applications include the equation of time‐like extremal surfaces in Minkowski space R1 + (1 + n) and the one‐dimensional Chaplygin gas equations. Copyright © 2013 John Wiley & Sons, Ltd.

Asymptotic behavior of global classical solutions to Goursat problem for quasilinear hyperbolic systems with small BV data

This paper is concerned with the asymptotic behavior of global C 1 solutions of the Goursat problem for quasilinear hyperbolic systems. Based on the existence result on the global classical solution, we prove that when t tends to the infinity, the solution approaches a combination of Lipschitz continuous and piecewise C 1 traveling wave solutions, provided that the C 1 norm of the boundary data is bounded but possibly large, and the BV norm of the boundary data is sufficiently small. Applications include the 1D compressible Euler equations for Chaplygin gases.

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.

Asymptotic behavior of global classical solutions to Goursat problem of quasilinear hyperbolic systems

This work is concerned with the asymptotic behavior of global C1 solutions of the Goursat problem for quasilinear hyperbolic systems. We prove that when t tends to the infinity, the solutions approach a combination of C1 traveling wave solutions at algebraic rate (1+t)−μ, provided that the boundary data decay at the rate (1+t)−(1+μ), where μ is a positive constant. Finally, we give an application to the equation for motion of an elastic string.

The Characteristic Initial Value Problem for the Einstein-Yang-Mills-Higgs System in Weighted Sobolev Spaces

We revisit and complete existence and uniqueness results stated and partially established by Muller zum Hagen in 1990 for the characteristic initial value problem for quasilinear hyperbolic systems of second order with data prescribed on two intersecting smooth null hypersurfaces. The new ingredient of this investigation consists of some Moser estimates expressed in the same weighted Sobolev spaces as those used by Muller zum Hagen. These estimates, combined with energy inequalities obtained by Muller zum Hagen for the linearized Goursat problem, permit us to develop a fixed point method which leads clearly to an existence and uniqueness result for the quasilinear Goursat problem. As an application we locally solve, under finite differentiability conditions, the characteristic initial value problem for the Einstein-Yang-Mills-Higgs system using harmonic gauge for the gravitational potentials and Lorentz gauge for the Yang-Mills potentials.

Global classical solutions of mixed initial-boundary value problem for quasilinear hyperbolic systems

In this paper, we consider the mixed initial-boundary value problem for quasilinear hyperbolic systems with nonlinear boundary conditions in the first quadrant { ( t , x ) | t ≥ 0 , x ≥ 0 } . Under the assumptions that the system is strictly hyperbolic and linearly degenerate or weakly linearly degenerate, the global existence and uniqueness of C 1 solutions are obtained for small initial and boundary data. We also present two applications for physical models.

Global existence of classical solutions to the mixed initial–boundary value problem for quasilinear hyperbolic systems of diagonal form with large BV data

In this paper, we investigate the mixed initial–boundary value problem with large BV data for linearly degenerate quasilinear hyperbolic systems of diagonal form with general nonlinear boundary conditions in the half space { ( t , x ) | t ⩾ 0 , x ⩾ 0 } . As the result in [A. Bressan, Contractive metrics for nonlinear hyperbolic systems, Indiana Univ. Math. J. 37 (1988) 409–421] suggests that one may achieve global smoothness even if the C 1 norm of the initial data is large, we prove that, if the C 1 norm and the BV norm of the initial and boundary data are bounded but possibly large, then the solution remains C 1 globally in time and possesses uniformly bounded total variation in x for all t ⩾ 0 . As an application, we apply the result to the system describing the motion of relativistic closed strings in the Minkowski space R 1 + n .

The mixed initial–boundary value problem for quasilinear hyperbolic systems with linearly degenerate characteristics

In this paper, we investigate the mixed initial–boundary value problem with small BV data for linearly degenerate quasilinear hyperbolic systems with general nonlinear boundary conditions in the half space { ( t , x ) | t ≥ 0 , x ≥ 0 } . As the result in [A. Bressan, Contractive metrics for nonlinear hyperbolic systems, Indiana Univ. Math. J. 37 (1988) 409–421] suggests that one may achieve global smoothness even if the C 1 norm of the initial data is large, we prove that, if the C 1 norm of the initial and boundary data is bounded but possibly large, and the BV norm of the initial and boundary data is sufficiently small, then the solution remains C 1 globally in time. Applications to quasilinear hyperbolic systems arising in physics and mechanics, particularly to the system describing the motion of the relativistic string in the Minkowski space–time R 1 + n , are also given.

Point‐wise decay estimate for the global classical solutions to quasilinear hyperbolic systems

AbstractIn this paper, we first consider the Cauchy problem for quasilinear strictly hyperbolic systems with weak linear degeneracy. The existence of global classical solutions for small and decay initial data was established in (Commun. Partial Differential Equations 1994; 19:1263–1317; Nonlinear Anal. 1997; 28:1299–1322; Chin. Ann. Math. 2004; 25B:37–56). We give a new, very simple proof of this result and also give a sharp point‐wise decay estimate of the solution. Then, we consider the mixed initial‐boundary‐value problem for quasilinear hyperbolic systems with nonlinear boundary conditions in the first quadrant. Under the assumption that the positive eigenvalues are weakly linearly degenerate, the global existence of classical solution with small and decay initial and boundary data was established in (Discrete Continuous Dynamical Systems 2005; 12(1):59–78; Zhou and Yang, in press). We also give a simple proof of this result as well as a sharp point‐wise decay estimate of the solution. Copyright © 2008 John Wiley & Sons, Ltd.