Errata to "Symmetric hyperbolic systems with boundary conditions that do not satisfy the Kreiss-Sakamoto condition" (Appl. Math. (Warsaw) 35 (2008), 323-333)
Exact boundary observability for quasilinear hyperbolic systems
By means of a direct and constructive method based on the theory of semi-global C1 solution, the local exact boundary observability is established for one-dimensional first order quasilinear hyperbolic systems with general nonlinear boundary conditions. An implicit duality between the exact boundary controllability and the exact boundary observability is then shown in the quasilinear case.
Existence of classical solutions and feedback stabilization for the flow in gas networks
We consider the flow of gas through pipelines controlled by a compressor station. Under a subsonic flow assumption we prove the existence of classical solutions for a given finite time interval. The existence result is used to construct Riemannian feedback laws and to prove a stabilization result for a coupled system of gas pipes with a compressor station. We introduce a Lyapunov function and prove exponential decay with respect to the L2-norm.
Existence of classical solutions and feedback stabilization for the flow in gas networks
We consider the flow of gas through pipelines controlled by a compressor station. Under a subsonic flow assumption we prove the existence of classical solutions for a given finite time interval. The existence result is used to construct Riemannian feedback laws and to prove a stabilization result for a coupled system of gas pipes with a compressor station. We introduce a Lyapunov function and prove exponential decay with respect to the L2-norm.
Explicit solutions for a system of first-order partial differential equations. II.
Explicit solutions for a system of first-order partial differential equations.
Exponential stability and transfer functions of processes governed by symmetric hyperbolic systems
In this paper we study the frequency and time domain behaviour of a heat exchanger network system. The system is governed by hyperbolic partial differential equations. Both the control operator and the observation operator are unbounded but admissible. Using the theory of symmetric hyperbolic systems, we prove exponential stability of the underlying semigroup for the heat exchanger network. Applying the recent theory of well-posed infinite-dimensional linear systems, we prove that the system is...
Exponential Stability and Transfer Functions of Processes Governed by Symmetric Hyperbolic Systems
In this paper we study the frequency and time domain behaviour of a heat exchanger network system. The system is governed by hyperbolic partial differential equations. Both the control operator and the observation operator are unbounded but admissible. Using the theory of symmetric hyperbolic systems, we prove exponential stability of the underlying semigroup for the heat exchanger network. Applying the recent theory of well-posed infinite-dimensional linear systems, we prove that the system...
Generalized solutions to boundary value problems for quasilinear hyperbolic systems of partial differential-functional equations
Generalized solutions to quasilinear hyperbolic systems in the second canonical form are investigated. A theorem on existence, uniqueness and continuous dependence upon the boundary data is given. The proof is based on the methods due to L. Cesari and P. Bassanini for systems which are not functional.
Generalized solutions to singular initial-boundary hyperbolic problems with non-lipshitz nonlinearities
Geometric optics expansions with amplification for hyperbolic boundary value problems: Linear problems
We compute and justify rigorous geometric optics expansions for linear hyperbolic boundary value problems that do not satisfy the uniform Lopatinskii condition. We exhibit an amplification phenomenon for the reflection of small high frequency oscillations at the boundary. Our analysis has two important consequences for such hyperbolic boundary value problems. Firstly, we make precise the optimal energy estimate in Sobolev spaces showing that losses of derivatives must occur from the source terms...
Global classical solutions to a kind of mixed initial-boundary value problem for inhomogeneous quasilinear hyperbolic systems
In this paper, the mixed initial-boundary value problem for inhomogeneous quasilinear strictly hyperbolic systems with nonlinear boundary conditions in the first quadrant is investigated. Under the assumption that the right-hand side satisfies a matching condition and the system is strictly hyperbolic and weakly linearly degenerate, we obtain the global existence and uniqueness of a solution and its stability with certain small initial and boundary data.
Increasing smoothness of solutions to a hyperbolic system on the plane with delay in the boundary conditions.
Initial-boundary value problem for the discrete Boltzmann equation
Interface model coupling via prescribed local flux balance
This paper deals with the non-conservative coupling of two one-dimensional barotropic Euler systems at an interface at x = 0. The closure pressure laws differ in the domains x < 0 and x > 0, and a Dirac source term concentrated at x = 0 models singular pressure losses. We propose two numerical methods. The first one relies on ghost state reconstructions at the interface while the second is based on a suitable relaxation framework. Both methods satisfy a well-balanced property for stationary...
Linear hyperbolic problems in the whole scale of Sobolev-type spaces of periodic functions
We study one-dimensional linear hyperbolic systems with -coefficients subjected to periodic conditions in time and reflection boundary conditions in space. We derive a priori estimates and give an operator representation of solutions in the whole scale of Sobolev-type spaces of periodic functions. These spaces give an optimal regularity trade-off for our problem.
Local generalized solutions of mixed problems for quasilinear hyperbolic systems of functional partial differential equations in two independent variables
Mathematical and numerical analysis of a stratigraphic model
In this paper, we consider a multi-lithology diffusion model used in stratigraphic modelling to simulate large scale transport processes of sediments described as a mixture of lithologies. This model is a simplified one for which the surficial fluxes are proportional to the slope of the topography and to a lithology fraction with unitary diffusion coefficients. The main unknowns of the system are the sediment thickness , the surface concentrations in lithology of the sediments at the top...
Mathematical and numerical analysis of a stratigraphic model
In this paper, we consider a multi-lithology diffusion model used in stratigraphic modelling to simulate large scale transport processes of sediments described as a mixture of L lithologies. This model is a simplified one for which the surficial fluxes are proportional to the slope of the topography and to a lithology fraction with unitary diffusion coefficients. The main unknowns of the system are the sediment thickness h, the L surface concentrations in lithology i of the sediments at the...
Measure-valued solutions and well-posedness of multi-dimensional conservation laws in a bounded domain.