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## Hyperbolic Systems (hyperbolic + system)
Kinds of Hyperbolic Systems
## Selected Abstracts## Data assimilation and inverse problem for fluid traffic flow models and algorithms INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 6 2008P. JaissonAbstract This article deals with traffic data assimilation and algorithms that are able to predict the traffic flow on a road section. The traffic flow is modellized by the Aw,Rascle hyperbolic system. We have to minimize a functional whose optimization variables are initial condition. We use the Roe method to compute the solution to the traffic flow modelling system. Then we compute the gradient of the functional by an adjoint method. This gradient will be used to optimize the functional. Copyright © 2008 John Wiley & Sons, Ltd. [source] ## PRICE: primitive centred schemes for hyperbolic systems INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 12 2003E. F. ToroAbstract We present first- and higher-order non-oscillatory primitive (PRI) centred (CE) numerical schemes for solving systems of hyperbolic partial differential equations written in primitive (or non-conservative) form. Non-conservative systems arise in a variety of fields of application and they are adopted in that form for numerical convenience, or more importantly, because they do not posses a known conservative form; in the latter case there is no option but to apply non-conservative methods. In addition we have chosen a centred, as distinct from upwind, philosophy. This is because the systems we are ultimately interested in (e.g. mud flows, multiphase flows) are exceedingly complicated and the eigenstructure is difficult, or very costly or simply impossible to obtain. We derive six new basic schemes and then we study two ways of extending the most successful of these to produce second-order non-oscillatory methods. We have used the MUSCL-Hancock and the ADER approaches. In the ADER approach we have used two ways of dealing with linear reconstructions so as to avoid spurious oscillations: the ADER TVD scheme and ADER with ENO reconstruction. Extensive numerical experiments suggest that all the schemes are very satisfactory, with the ADER/ENO scheme being perhaps the most promising, first for dealing with source terms and secondly, because higher-order extensions (greater than two) are possible. Work currently in progress includes the application of some of these ideas to solve the mud flow equations. The schemes presented are generic and can be applied to any hyperbolic system in non-conservative form and for which solutions include smooth parts, contact discontinuities and weak shocks. The advantage of the schemes presented over upwind-based methods is simplicity and efficiency, and will be fully realized for hyperbolic systems in which the provision of upwind information is very costly or is not available. Copyright © 2003 John Wiley & Sons, Ltd. [source] ## Chebyshev super spectral viscosity solution of a two-dimensional fluidized-bed model INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 3 2003Scott A. SarraArticle first published online: 13 MAY 200Abstract The numerical solution of a model describing a two-dimensional fluidized bed by a Chebyshev super spectral viscosity (SSV) method is considered. The model is in the form of a hyperbolic system of conservation laws with a source term, coupled with an elliptic equation for determining a stream function. The coupled elliptic equation is solved by a finite-difference method. The mixed SSV/finite-difference method produces physically shaped bubbles, on a very coarse grid. Fine scale details, which were not present in previous finite-difference solutions, are present in the solution. Copyright © 2003 John Wiley & Sons, Ltd. [source] ## Stability of travelling wave solutions to a semilinear hyperbolic system with relaxation MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 4 2009Yoshihiro UedaAbstract We study a semilinear hyperbolic system with relaxation and investigate the asymptotic stability of travelling wave solutions with shock profile. It is shown that the travelling wave solution is asymptotically stable, provided the initial disturbance is suitably small. Moreover, we show that the time convergence rate is polynomially (resp. exponentially) fast as t,, if the initial disturbance decays polynomially (resp. exponentially) for x,,. Our proofs are based on the space,time weighted energy method. Copyright © 2008 John Wiley & Sons, Ltd. [source] ## On stability of Alfvén discontinuities MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 3 2009Konstantin IlinAbstract We study Alfvén discontinuities for the equations of ideal compressible magnetohydrodynamics (MHD). The Alfvén discontinuity is a characteristic discontinuity for the hyperbolic system of the MHD equations but, as for shock waves, the gas crosses its front. By numerical testing of the Lopatinskii condition, we carry out spectral stability analysis, i.e. we find the parameter domains of stability and violent instability of planar Alfvén discontinuities. We also show that Alfvén discontinuities can be only weakly stable in the sense that the uniform Lopatinskii condition is never satisfied. Copyright © 2008 John Wiley & Sons, Ltd. [source] ## Artificial boundary conditions for viscoelastic flows MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 8 2008Sergueď A. NazarovAbstract The steady three-dimensional exterior flow of a viscoelastic non-Newtonian fluid is approximated by reducing the corresponding nonlinear elliptic,hyperbolic system to a bounded domain. On the truncation surface with a large radius R, nonlinear, local second-order artificial boundary conditions are constructed and a new concept of an artificial transport equation is introduced. Although the asymptotic structure of solutions at infinity is known, certain attributes cannot be found explicitly so that the artificial boundary conditions must be constructed with incomplete information on asymptotics. To show the existence of a solution to the approximation problem and to estimate the asymptotic precision, a general abstract scheme, adapted to the analysis of coupled systems of elliptic,hyperbolic type, is proposed. The error estimates, obtained in weighted Sobolev norms with arbitrarily large smoothness indices, prove an approximation of order O(R,2+,), with any ,>0. Our approach, in contrast to other papers on artificial boundary conditions, does not use the standard assumptions on compactly supported right-hand side f, leads, in particular, to pointwise estimates and provides error bounds with constants independent of both R and f. Copyright © 2007 John Wiley & Sons, Ltd. [source] ## BV-estimates of Lax,Friedrichs' scheme for hyperbolic conservation laws with relaxation MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 8 2008Jing ZhangAbstract In this paper, we will give BV-estimates of Lax,Friedrichs' scheme for a simple hyperbolic system of conservation laws with relaxation and get the global existence and uniqueness of BV-solution by the BV-estimates above. Furthermore, our results show that the solution converge towards the solution of an equilibrium model as the relaxation time ,>0 tends to zero provided sub-characteristic condition holds. Copyright © 2007 John Wiley & Sons, Ltd. [source] ## Asymmetric invariants for a class of strictly hyperbolic systems including the Timoshenko beam MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 11 2007Clelia MarchionnaAbstract We introduce a set of conserved quantities of energy-type for a strictly hyperbolic system of two coupled wave equations in one space dimension. The system is subject to mechanical boundary conditions. Some of these invariants are asymmetric in the sense that their defining quadratic form contains second order derivatives in only one of the unknowns. We study their independence with respect to the usual energies and characterize their sign. In many cases, our results provide sharp well-posedness and stability results. Finally, we apply some of our conservation laws to the study of a singular perturbation problem previously considered by J. Lagnese and J. L. Lions. Copyright © 2007 John Wiley & Sons, Ltd. [source] ## Well-posedness, smooth dependence and centre manifold reduction for a semilinear hyperbolic system from laser dynamics MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 8 2007Mark LichtnerAbstract We prove existence, uniqueness, regularity and smooth dependence of the weak solution on the initial data for a semilinear, first order, dissipative hyperbolic system with discontinuous coefficients. Such hyperbolic systems have successfully been used to model the dynamics of distributed feedback multisection semiconductor lasers. We show that in a function space of continuous functions the weak solutions generate a smooth skew product semiflow. Using slow fast structure and dissipativity we prove the existence of smooth exponentially attracting invariant centre manifolds for the non-autonomous model. Copyright © 2006 John Wiley & Sons, Ltd. [source] ## On the hyperbolic system of a mixture of Eulerian fluids: a comparison between single- and multi-temperature models MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 7 2007Tommaso RuggeriAbstract The first rational model of homogeneous mixtures of fluids was proposed by Truesdell in the context of rational thermodynamics. Afterwards, two different theories were developed: one with a single-temperature (ST) field of the mixture and the other one with several temperatures. The two systems are from the mathematical point of view completely different and the relationship between their solutions was not clarified. In this paper, the hyperbolic multi-temperature (MT) system of a mixture of Eulerian fluids will be explained and it will be shown that the corresponding single-temperature differential system is a principal subsystem of the MT one. As a consequence, the subcharacteristic conditions for characteristic speeds hold and this gives an upper-bound esteem for pulse speeds in an ST model. Global behaviour of smooth solutions for large time for both systems will also be discussed through the application of the Shizuta,Kawashima condition. Finally, as an application, the particular case of a binary mixture is considered. Copyright © 2006 John Wiley & Sons, Ltd. [source] ## Asymptotic and spectral properties of operator-valued functions generated by aircraft wing model MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 3 2004A. V. BalakrishnanAbstract The present paper is devoted to the asymptotic and spectral analysis of an aircraft wing model in a subsonic air flow. The model is governed by a system of two coupled integro-differential equations and a two parameter family of boundary conditions modelling the action of the self-straining actuators. The differential parts of the above equations form a coupled linear hyperbolic system; the integral parts are of the convolution type. The system of equations of motion is equivalent to a single operator evolution,convolution equation in the energy space. The Laplace transform of the solution of this equation can be represented in terms of the so-called generalized resolvent operator, which is an operator-valued function of the spectral parameter. More precisely, the generalized resolvent is a finite-meromorphic function on the complex plane having a branch-cut along the negative real semi-axis. Its poles are precisely the aeroelastic modes and the residues at these poles are the projectors on the generalized eigenspaces. The dynamics generator of the differential part of the system has been systematically studied in a series of works by the second author. This generator is a non-selfadjoint operator in the energy space with a purely discrete spectrum. In the aforementioned series of papers, it has been shown that the set of aeroelastic modes is asymptotically close to the spectrum of the dynamics generator, that this spectrum consists of two branches, and a precise spectral asymptotics with respect to the eigenvalue number has been derived. The asymptotical approximations for the mode shapes have also been obtained. It has also been proven that the set of the generalized eigenvectors of the dynamics generator forms a Riesz basis in the energy space. In the present paper, we consider the entire integro-differential system which governs the model. Namely, we investigate the properties of the integral convolution-type part of the original system. We show, in particular, that the set of poles of the adjoint generalized resolvent is asymptotically close to the discrete spectrum of the operator that is adjoint to the dynamics generator corresponding to the differential part. The results of this paper will be important for the reconstruction of the solution of the original initial boundary-value problem from its Laplace transform and for the analysis of the flutter phenomenon in the forthcoming work. Copyright © 2004 John Wiley & Sons, Ltd. [source] ## Pseudo-reflection phenomena for singularities in thin elastic shells MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 17 2003P. Karamian-SurvilleWe consider problems of statics of thin elastic shells with hyperbolic middle surface subjected to boundary conditions ensuring the geometric rigidity of the surface. The asymptotic behaviour of the solutions when the relative thickness tends to zero is then given by the membrane approximation. It is a hyperbolic problem propagating singularities along the characteristics. We address here the reflection phenomena when the propagated singularities arrive to a boundary. As the boundary conditions are not the classical ones for a hyperbolic system, there are various cases of reflection. Roughly speaking, singularities provoked elsewhere are not reflected at all at a free boundary, whereas at a fixed (or clamped) boundary the reflected singularity is less singular than the incident one. Reflection of singularities provoked along a non-characteristic curve C are also considered. Copyright © 2003 John Wiley & Sons, Ltd. [source] ## Homogenizing the acoustic properties of a porous matrix containing an incompressible inviscid fluid MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 10 2003J. L. FerrinWe undertake a rigorous derivation of the Biot's law for a porous elastic solid containing an inviscid fluid. We consider small displacements of a linear elastic solid being itself a connected periodic skeleton containing a pore structure of the characteristic size ,. It is completely saturated by an incompressible inviscid fluid. The model is described by the equations of the linear elasticity coupled with the linearized incompressible Euler system. We study the homogenization limit when the pore size ,tends to zero. The main difficulty is obtaining an a priori estimate for the gradient of the fluid velocity in the pore structure. Under the assumption that the solid part is connected and using results on the first order elliptic systems, we obtain the required estimate. It allows us to apply appropriate results from the 2-scale convergence. Then it is proved that the microscopic displacements and the fluid pressure converge in 2-scales towards a linear hyperbolic system for an effective displacement and an effective pressure field. Using correctors, we also give a strong convergence result. The obtained system is then compared with the Biot's law. It is found that there is a constitutive relation linking the effective pressure with the divergences of the effective fluid and solid displacements. Then we prove that the homogenized model coincides with the Biot's equations but with the added mass ,a being a matrix, which is calculated through an auxiliary problem in the periodic cell for the tortuosity. Furthermore, we get formulas for the matricial coefficients in the Biot's effective stress,strain relations. Finally, we consider the degenerate case when the fluid part is not connected and obtain Biot's model with the relative fluid displacement equal to zero. Copyright © 2003 John Wiley & Sons, Ltd. [source] ## An instability of the Godunov scheme COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 11 2006Alberto BressanWe construct a solution to a 2 × 2 strictly hyperbolic system of conservation laws, showing that the Godunov scheme [13] can produce an arbitrarily large amount of oscillations. This happens when the speed of a shock is close to rational, inducing a resonance with the grid. Differently from the Glimm scheme or the vanishing-viscosity method, for systems of conservation laws our counterexample indicates that no a priori BV bounds or L1 -stability estimates can in general be valid for finite difference schemes. © 2006 Wiley Periodicals, Inc. [source] ## An approximate-state Riemann solver for the two-dimensional shallow water equations with porosity INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 12 2010P. Finaud-GuyotAbstract PorAS, a new approximate-state Riemann solver, is proposed for hyperbolic systems of conservation laws with source terms and porosity. The use of porosity enables a simple representation of urban floodplains by taking into account the global reduction in the exchange sections and storage. The introduction of the porosity coefficient induces modified expressions for the fluxes and source terms in the continuity and momentum equations. The solution is considered to be made of rarefaction waves and is determined using the Riemann invariants. To allow a direct computation of the flux through the computational cells interfaces, the Riemann invariants are expressed as functions of the flux vector. The application of the PorAS solver to the shallow water equations is presented and several computational examples are given for a comparison with the HLLC solver. Copyright © 2009 John Wiley & Sons, Ltd. [source] ## A finite-volume particle method for conservation laws on moving domains INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 9 2008D. TeleagaAbstract The paper deals with the finite-volume particle method (FVPM), a relatively new method for solving hyperbolic systems of conservation laws. A general formulation of the method for bounded and moving domains is presented. Furthermore, an approximation property of the reconstruction formula is proved. Then, based on a two-dimensional test problem posed on a moving domain, a special Ansatz for the movement of the particles is proposed. The obtained numerical results indicate that this method is well suited for such problems, and thus a first step to apply the FVPM to real industrial problems involving free boundaries or fluid,structure interaction is taken. Finally, we perform a numerical convergence study for a shock tube problem and a simple linear advection equation. Copyright © 2008 John Wiley & Sons, Ltd. [source] ## MUSTA schemes for multi-dimensional hyperbolic systems: analysis and improvements INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 2 2005V. A. TitarevAbstract We develop and analyse an improved version of the multi-stage (MUSTA) approach to the construction of upwind Godunov-type fluxes whereby the solution of the Riemann problem, approximate or exact, is not required. The new MUSTA schemes improve upon the original schemes in terms of monotonicity properties, accuracy and stability in multiple space dimensions. We incorporate the MUSTA technology into the framework of finite-volume weighted essentially nonoscillatory schemes as applied to the Euler equations of compressible gas dynamics. The results demonstrate that our new schemes are good alternatives to current centred methods and to conventional upwind methods as applied to complicated hyperbolic systems for which the solution of the Riemann problem is costly or unknown. Copyright © 2005 John Wiley & Sons, Ltd. [source] ## A method of coupling non-linear hyperbolic systems: examples in CFD and plasma physics INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 10-11 2005E. GodlewskiAbstract This paper analyses a method of coupling systems of conservation laws with examples in two fluid flows. Copyright © 2005 John Wiley & Sons, Ltd. [source] ## PRICE: primitive centred schemes for hyperbolic systems INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 12 2003E. F. ToroAbstract We present first- and higher-order non-oscillatory primitive (PRI) centred (CE) numerical schemes for solving systems of hyperbolic partial differential equations written in primitive (or non-conservative) form. Non-conservative systems arise in a variety of fields of application and they are adopted in that form for numerical convenience, or more importantly, because they do not posses a known conservative form; in the latter case there is no option but to apply non-conservative methods. In addition we have chosen a centred, as distinct from upwind, philosophy. This is because the systems we are ultimately interested in (e.g. mud flows, multiphase flows) are exceedingly complicated and the eigenstructure is difficult, or very costly or simply impossible to obtain. We derive six new basic schemes and then we study two ways of extending the most successful of these to produce second-order non-oscillatory methods. We have used the MUSCL-Hancock and the ADER approaches. In the ADER approach we have used two ways of dealing with linear reconstructions so as to avoid spurious oscillations: the ADER TVD scheme and ADER with ENO reconstruction. Extensive numerical experiments suggest that all the schemes are very satisfactory, with the ADER/ENO scheme being perhaps the most promising, first for dealing with source terms and secondly, because higher-order extensions (greater than two) are possible. Work currently in progress includes the application of some of these ideas to solve the mud flow equations. The schemes presented are generic and can be applied to any hyperbolic system in non-conservative form and for which solutions include smooth parts, contact discontinuities and weak shocks. The advantage of the schemes presented over upwind-based methods is simplicity and efficiency, and will be fully realized for hyperbolic systems in which the provision of upwind information is very costly or is not available. Copyright © 2003 John Wiley & Sons, Ltd. [source] ## Multidimensional FEM-FCT schemes for arbitrary time stepping INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 3 2003D. KuzminAbstract The flux-corrected-transport paradigm is generalized to finite-element schemes based on arbitrary time stepping. A conservative flux decomposition procedure is proposed for both convective and diffusive terms. Mathematical properties of positivity-preserving schemes are reviewed. A nonoscillatory low-order method is constructed by elimination of negative off-diagonal entries of the discrete transport operator. The linearization of source terms and extension to hyperbolic systems are discussed. Zalesak's multidimensional limiter is employed to switch between linear discretizations of high and low order. A rigorous proof of positivity is provided. The treatment of non-linearities and iterative solution of linear systems are addressed. The performance of the new algorithm is illustrated by numerical examples for the shock tube problem in one dimension and scalar transport equations in two dimensions. Copyright © 2003 John Wiley & Sons, Ltd. [source] ## Exact boundary controllability for a kind of second-order quasilinear hyperbolic systems and its applications MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 3 2010Lixin YuAbstract In this paper, by means of a constructive method based on the existence and uniqueness of the semi-global C2 solution, we establish the local exact boundary controllability for a kind of second-order quasilinear hyperbolic systems. As an application, we obtain the one-sided local exact boundary controllability for the 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. Copyright © 2009 John Wiley & Sons, Ltd. [source] ## Point-wise decay estimate for the global classical solutions to quasilinear hyperbolic systems MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 13 2009Yi ZhouAbstract In 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. [source] ## Mechanism of the formation of singularities for quasilinear hyperbolic systems with linearly degenerate characteristic fields MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 2 2008Ta-Tsien LiAbstract One often believes that there is no shock formation for the Cauchy problem of quasilinear hyperbolic systems (of conservation laws) with linearly degenerate characteristic fields. It has been a conjecture for a long time (see Arch. Rational Mech. Anal. 2004; 172:65,91; Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables. Springer: New York, 1984) and it is still an open problem in the general situation up to now. In this paper, a framework to justify this conjecture is proposed, and, by means of the concept such as the strict block hyperbolicity, the part richness and the successively block-closed system, some general kinds of quasilinear hyperbolic systems, which verify the conjecture, are given. Copyright © 2007 John Wiley & Sons, Ltd. [source] ## Asymmetric invariants for a class of strictly hyperbolic systems including the Timoshenko beam MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 11 2007Clelia MarchionnaAbstract We introduce a set of conserved quantities of energy-type for a strictly hyperbolic system of two coupled wave equations in one space dimension. The system is subject to mechanical boundary conditions. Some of these invariants are asymmetric in the sense that their defining quadratic form contains second order derivatives in only one of the unknowns. We study their independence with respect to the usual energies and characterize their sign. In many cases, our results provide sharp well-posedness and stability results. Finally, we apply some of our conservation laws to the study of a singular perturbation problem previously considered by J. Lagnese and J. L. Lions. Copyright © 2007 John Wiley & Sons, Ltd. [source] ## Well-posedness, smooth dependence and centre manifold reduction for a semilinear hyperbolic system from laser dynamics MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 8 2007Mark LichtnerAbstract We prove existence, uniqueness, regularity and smooth dependence of the weak solution on the initial data for a semilinear, first order, dissipative hyperbolic system with discontinuous coefficients. Such hyperbolic systems have successfully been used to model the dynamics of distributed feedback multisection semiconductor lasers. We show that in a function space of continuous functions the weak solutions generate a smooth skew product semiflow. Using slow fast structure and dissipativity we prove the existence of smooth exponentially attracting invariant centre manifolds for the non-autonomous model. Copyright © 2006 John Wiley & Sons, Ltd. [source] ## Low-gain adaptive stabilization of semilinear second-order hyperbolic systems MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 18 2004Toshihiro KobayashiAbstract In this paper low-gain adaptive stabilization of undamped semilinear second-order hyperbolic systems is considered in the case where the input and output operators are collocated. The linearized systems have an infinite number of poles and zeros on the imaginary axis. The adaptive stabilizer is constructed by a low-gain adaptive velocity feedback. The closed-loop system is governed by a non-linear evolution equation. First, the well-posedness of the closed-loop system is shown. Next, an energy-like function and a multiplier function are introduced and the exponential stability of the closed-loop system is analysed. Some examples are given to illustrate the theory. Copyright © 2004 John Wiley & Sons, Ltd. [source] ## The Cauchy problem for quasilinear SG-hyperbolic systems MATHEMATISCHE NACHRICHTEN, Issue 7 2007Marco CappielloAbstract We study the Cauchy problem for a class of quasilinear hyperbolic systems with coefficients depending on (t, x) , [0, T ] × ,n and presenting a linear growth for |x | , ,. We prove well-posedness in the Schwartz space ,, (,n). The result is obtained by deriving an energy estimate for the solution of the linearized problem in some weighted Sobolev spaces and applying a fixed point argument. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] ## Exact boundary controllability of unsteady flows in a network of open canals MATHEMATISCHE NACHRICHTEN, Issue 3 2005Tatsien LiAbstract By means of the general results on the exact boundary controllability for quasilinear hyperbolic systems, the author establishes the exact boundary controllability of unsteady flows in both a single open canal and a network of open canals with star configuration respectively. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] ## On Hamiltonian perturbations of hyperbolic systems of conservation laws I: Quasi-Triviality of bi-Hamiltonian perturbations COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 4 2006Boris DubrovinWe study the general structure of formal perturbative solutions to the Hamiltonian perturbations of spatially one-dimensional systems of hyperbolic PDEs vt + [,(v)]x = 0. Under certain genericity assumptions it is proved that any bi-Hamiltonian perturbation can be eliminated in all orders of the perturbative expansion by a change of coordinates on the infinite jet space depending rationally on the derivatives. The main tool is in constructing the so-called quasi-Miura transformation of jet coordinates, eliminating an arbitrary deformation of a semisimple bi-Hamiltonian structure of hydrodynamic type (the quasi-triviality theorem). We also describe, following [35], the invariants of such bi-Hamiltonian structures with respect to the group of Miura-type transformations depending polynomially on the derivatives. © 2005 Wiley Periodicals, Inc. [source] |