357 research outputs found
A wavefront tracking algorithm for N×N nongenuinely nonlinear conservation laws
No full textAbstractWe introduce a wavefront tracking algorithm for N×N hyperbolic systems of conservation lawsut+F(u)x=0, that admits characteristic fields that are neither genuinely nonlinear nor linearly degenerate in the sense of Lax. Instead we assume that, for any nongenuinely nonlinear ith characteristic family, the derivative of the ith eigenvalue λi(u) of DF(u) in the direction of the ith right eigenvector ri(u), vanishes on a single (N−1)-dimensional hypersurface in the u-space, transversal to the field ri(u). Systems that fulfill this type of assumptions are of particular interest in studying elastodynamic or rigid heat conductors at low temperature. The first proof of the existence of weak solutions for nongenuinely nonlinear systems was given by T. P. Liu (Mem. Amer. Math. Soc.30 (1981), no. 240), based on a Glimm scheme. Our construction here provides an alternative method for establishing the global existence of weak solutions for such systems. Moreover, relying on the stability analysis developed in Ancona and Marson, preprint S.I.S.S.A.-I.S.A.S. 27/99/11, 1999, and preprint, 2000, we show that these solutions are entropy admissible in the sense of Lax
SBV-like Regularity of Entropy Solutions for a Scalar Balance Law
No full textWe demonstrate that solutions to scalar balance laws,in one space dimension, which exhibit bounded variation, must be functions of special bounded variation (SBV). This case study illustrates the strategy ap- plied in [Ancona-Caravenna-Marson, preprint, University of Padova, 2024] to systems of balance laws, extending the methodologies developed in pioneering previous works by several authors. While for a single balance law a more general work is already available, generalizing the first breakthrough related to a conserva- tion law, the case of 1D systems presents new behaviors that require a different strategy. This is why in this note we make the effort to introduce the notation and tools that are required for the case of more equations. When the flux presents linear degeneracies, it is known that entropy solutions can really present nasty fractal Cantor-like behaviours, although f′(u) is still SBV: We thus discuss SBV-like regularity, as SBV-regularity fails
Nonconvex conservation laws and Ordinary Differential Equations
No full textThis paper deals with the well posedness of a class of ordinary
differential equations. The vector field depends on the solution to a
scalar conservation law, whose flux function is assumed to have a
single inflection point (whence ``nonconvex''). We consider Filippov
solutions to the o.d.e. and prove H\"older continuous dependence on
the initial data. The problem is motivated by a model of traffic flow
Scalar non-linear conservation laws with integrable boundary data
No full textSummary: The paper deals with the initial-boundary value problem for a 1-D scalar conservation law in the case of initial and boundary data. The generalized solutions are constructed via semigroup theory, and a comparison principle as well as explicit (variational) representations for the solutions are obtained. Also the attainable set as is described for integrable boundary data considered as control. These results are applied to the concrete problem of optimal traffic fl
On the convergence rate for the Glimm scheme
No full textWe consider a Cauchy problem for a strictly hyperbolic,
quasilinear system in one space dimension , where the
matrix valued map is smooth and with non genuinely nonlinear
characteristic fields.
We introduce a Glimm type functional, quadratic in
the sizes of waves whose strengths are smaller than some
fixed threshold parameter. Next, we investigate the rate of
convergence of approximate solutions constructed via
the Glimm scheme. Moreover, we give a conjecture on the rate
of convergence without any additional assumption on beyond
the strict hyperbolicity and regularity
A note on the Riemann problem for general n×n conservation laws
No full textAbstractWe analyze the structure of the general solution of the Riemann problem for a strictly hyperbolic system of conservation laws whose characteristic fields are neither genuinely non-linear nor linearly degenerate in the sense of Lax
Basic estimates for a front tracking algorithm for general 2×2 conservation laws
No full textThe authors analyze a front-tracking algorithm for 2×2 systems of conservation laws with non-genuinely nonlinear characteristic fields. The convergence of the corresponding approximate Riemann solvers is established and basic interaction estimates for the front-tracking approximate solutions are provided
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