Well-posedness of the Cauchy problem for 𝑛×𝑛 systems of conservation laws

Bressan, Alberto; Crasta, Graziano; Piccoli, Benedetto

doi:10.1090/memo/0694

Cited by 155 publications

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“…This result was first proved in [15] for 2 × 2 systems, then in [16] for general n × n systems, using a (lengthy and technical) homotopy method. Here the idea is to consider a path of initial data γ 0 : θ → u θ (0) connecting u(0) with v(0).…”

Section: Nonlinear Systemsmentioning

confidence: 92%

Hyperbolic Conservation Laws: An Illustrated Tutorial

Bressan

2012

Lecture Notes in Mathematics

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These notes provide an introduction to the theory of hyperbolic systems of conservation laws in one space dimension. The various chapters cover the following topics: 1. Meaning of a conservation equation and definition of weak solutions. 2. Hyperbolic systems. Explicit solutions in the linear, constant coefficients case. Nonlinear effects: loss of regularity and wave interactions. 3. Shock waves: Rankine-Hugoniot equations and admissibility conditions. 4. Genuinely nonlinear and linearly degenerate characteristic fields. Centered rarefaction waves. The general solution of the Riemann problem. Wave interaction estimates. 5. Weak solutions to the Cauchy problem, with initial data having small total variation. Approximations generated by the front-tracking method and by the Glimm scheme. 6. Continuous dependence of solutions w.r.t. the initial data, in the L 1 distance. 7. Characterization of solutions which are limits of front tracking approximations. Uniqueness of entropy-admissible weak solutions. 8. Vanishing viscosity approximations. 9.Extensions and open problems. The survey is concluded with an Appendix, reviewing some basic analytical tools used in the previous sections.Throughout the exposition, technical details are mostly left out. The main goal of these notes is to convey basic ideas, also with the aid of a large number of figures.

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Section: Nonlinear Systemsmentioning

confidence: 92%

Hyperbolic Conservation Laws: An Illustrated Tutorial

Bressan

2012

Lecture Notes in Mathematics

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“…which satisfies the following stability estimates in L 1 (0, l): 13) for some constant L 1 , L 2 depending only on A and on δ 1 .…”

Section: Introductionmentioning

confidence: 94%

“…An existence result for hyperbolic boundary value problems was proved in [25,31] using an adaptation of the Glimm scheme introduced in [24]. Improvements of the results in [25,31] have been obtained by a wave-front tracking technique introduced in [9] and later used in a series of papers ( [10,12,13,17,15,14,16]) to establish the well posedness of the Cauchy problem. Such a wave-front tracking technique was adapted to the initial-boundary value problem in [1], where a substantial improvement of the results in [25,31] was achieved.…”

Section: Introductionmentioning

confidence: 99%

Vanishing viscosity solutions of a 2 \times 2 triangular hyperbolic system with Dirichlet conditions on two boundaries

Spinolo¹

2007

Indiana Univ. Math. J.

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We consider the 2 × 2 parabolic systemswith Dirichlet boundary conditions imposed at x = 0 and at x = l. The matrix A is assumed to be in triangular form and strictly hyperbolic, and the boundary is not characteristic, i.e. the eigenvalues of A are different from 0. We show that, if the initial and boundary data have sufficiently small total variation, then the solution u ε exists for all t ≥ 0 and depends Lipschitz continuously in L 1 on the initial and boundary data.Moreover, as ε → 0 + , the solutions u ε (t) converge in L 1 to a unique limit u(t), which can be seen as the vanishing viscosity solution of the quasilinear hyperbolic systemThis solution u(t) depends Lipschitz continuously in L 1 w.r.t the initial and boundary data. We also characterize precisely in which sense the boundary data are assumed by the solution of the hyperbolic system.2000 Mathematics Subject Classification: 35L65.

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“…Desirable stability properties and well-posedness of systems having the form of (2.19) (neglecting the source term ϕ) are established by hyperbolicity in the eigenstructure of A x (q) and A y (q) (e.g. [20,21]). The system is hyperbolic if, for all unit vectors (n x , n y ) T and admissible solutions q, the matrix…”

Section: Summary Of the Mathematical Model (A) Governing Equationsmentioning

confidence: 99%

A depth-averaged debris-flow model that includes the effects of evolving dilatancy. II. Numerical predictions and experimental tests

2014

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We evaluate a new depth-averaged mathematical model that is designed to simulate all stages of debris-flow motion, from initiation to deposition. A companion paper shows how the model's five governing equations describe simultaneous evolution of flow thickness, solid volume fraction, basal pore-fluid pressure and two components of flow momentum. Each equation contains a source term that represents the influence of state-dependent granular dilatancy. Here, we recapitulate the equations and analyse their eigenstructure to show that they form a hyperbolic system with desirable stability properties.To solve the equations, we use a shock-capturing numerical scheme with adaptive mesh refinement, implemented in an open-source software package we call D-Claw. As tests of D-Claw, we compare model output with results from two sets of largescale debris-flow experiments. One set focuses on flow initiation from landslides triggered by rising pore-water pressures, and the other focuses on downstream flow dynamics, runout and deposition. D-Claw performs well in predicting evolution of flow speeds, thicknesses and basal pore-fluid pressures measured in each type of experiment. Computational results illustrate the critical role of dilatancy in linking coevolution of the solid volume fraction and porefluid pressure, which mediates basal Coulomb friction and thereby regulates debris-flow dynamics.

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Well-posedness of the Cauchy problem for 𝑛×𝑛 systems of conservation laws

Abstract: Pag 2 d i c h i a r a z i o n e s o s t i t u t i v a d i c e r t i f i c a z i o n e(art. 46 D.P.R. 445 del 28.12.2000)La/il sottoscritta/o……………………………………………………………………………………… nata/o a .

Cited by 155 publications

References 0 publications

Hyperbolic Conservation Laws: An Illustrated Tutorial

Hyperbolic Conservation Laws: An Illustrated Tutorial

Vanishing viscosity solutions of a 2 \times 2 triangular hyperbolic system with Dirichlet conditions on two boundaries

A depth-averaged debris-flow model that includes the effects of evolving dilatancy. II. Numerical predictions and experimental tests

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