WAVE SOLUTIONS FOR HYPERBOLIC SYSTEMS

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.

Estimate of the time of occurrence of discontinuities in the solution of a boundary value problem for a second order quasilinear hyperbolic system: PMM vol. 36, n≗3, 1972, pp. 528–532

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...

Using a result on the existence and uniqueness of the semiglobal C 1 solution to the mixed initial-boundary value problem for first order quasi-linear hyperbolic systems with general nonlinear boundary conditions, we establish the exact boundary controllability for quasi-linear hyperbolic systems if the C 1 norm of initial and final states is small enough.

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.

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.

Exact internal controllability of nodal profile for first order quasilinear 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.

In this paper, the mixed initial-boundary value problem for general first order quasilinear hyperbolic systems with nonlinear boundary conditions in the domain D = {(t, x) | t ≥ 0, x ≥ 0} is considered. A sufficient condition to guarantee the existence and uniqueness of global weakly discontinuous solution is given.

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.

Numerical study of multiphase hyperbolic models

Exact time of blow-up for classical solutions to quasilinear hyperbolic systems

By introducing the concept of the weak linear degeneracy the authors give a complete result for the global existence and for the life span of C1 solutions to the Cauchy problem for general first order quasilinear hyperbolic systems with initial data small in C1 norm and with compact support.

\n \nBy means of a direct and constructive method based on the theory of\n semi-global C1 solution, the local exact boundary\n observability is established for one-dimensional first order\n quasilinear hyperbolic systems with general nonlinear boundary conditions. An implicit duality between the\n exact boundary controllability and the exact boundary observability is then shown in the quasilinear case.\n\n

In this Chapter we will generalize the exact boundary controllability of nodal profile for 1-D first order quasilinear hyperbolic systems on a single spatial interval , discussed in Chap. 4, to that on a tree-like network. A general framework can be established for general 1-D first order quasilinear hyperbolic systems with general nonlinear boundary conditions and general nonlinear interface conditions , provided that there are full of boundary controls in both boundary conditions and interface conditions (see Gu and Li [8]).