1,721,011 research outputs found
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
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
Well-posedness for general 2 x 2 conservation laws
No full textWe consider the Cauchy problem for a strictly hyperbolic
system of conservation laws in one space dimension
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which is neither linearly degenerate nor genuinely non-linear.
We make the following assumption on the characteristic fields.
If denotes the -th right eigenvector of
and the corresponding eigenvalue, then
the set is
a smooth curve
in the -plane that is transversal to the vector field
Systems of conservation laws that fulfill such assumptions arise in studying
elastodynamics
or rigid heat conductors at low temperature.
For such systems we prove the existence
of a closed domain \ \Cal D
\subset L^1, \ containing all functions with sufficiently
small total variation, and of a uniformly Lipschitz continuous
semigroup S:{\Cal D} \times [0,+\infty)\rightarrow \Cal D
with the following properties.
Each trajectory \ \ of is a weak
solution of (1). Viceversa, if a piecewise Lipschitz,
entropic solution of (1) exists for
then it coincides with the trajectory of ,
i.e.
This result yields the uniqueness and continuous dependence of
weak, entropy-admissible solutions of the Cauchy problem (1)
with small initial data, for systems satysfying the above assumption
On the attainable set for scalar nonlinear conservation laws with boundary control
No full textSummary:
The paper treats the initial boundary value problem for a scalar conservation law with strictly convex flux function. The boundary data is a Lebesgue-measurable and bounded function regarded as a control and constrained to remain in a prescribed set of admissible controls. A time being fixed, the authors characterize the set consisting of the corresponding entropy solutions at the time . Under natural assumptions on , it is proven that is a compact subset of . Such a compactness property provides the key information in order to establish the existence of solutions for a class of optimisation problems. Finally the results are applied by the authors to an optimisation problem concerning a model of traffic flow on a highway
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
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