238 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
Interoception and its role in shaping words meaning: behavioral and electrophysiological explorative evidence on the concreteness of Abstract concepts
No full textAccording to Embodied theories of language, sensorimotor systems are actively involved in the processing of words meaning. Different studies support this theoretical proposition, showing that modality-specific sensorimotor representations built on actual experiences are reenacted during words processing. While Concrete concepts are thought to rely on reactivation of experiences in exteroceptive modalities; Abstract concepts are thought to reactivate bodily sensations through the interoceptive modality. The main aim of the current thesis is to provide evidence advocating for the prominent role of interoception in embodiment of Abstract concepts, providing additional insight on the body-language coupling adopting behavioral and electrophysiological techniques. In Experiment 1, we compared perceptual strength ratings across languages, explored the importance of interoception in defining Abstract concepts, and addressed its implication for trait measures of empathy and interoceptive sensitivity. Moreover, cluster analysis was conducted with the aim of examining whether perceptual strength ratings could provide a reliable way of distinguishing conceptual categories. In detail, we explored perceptual strength ratings of 40 Abstract and 40 Concrete concepts in Italian, Hebrew, and English native speakers. As far as we are aware of, no previous studies directly compared different languages combining perceptual strength ratings, trait measurements, and clustering analysis. We observed high consistencies between languages for what concerns: 1) pattern of perceptual ratings for Abstract and Concrete concepts for each modality; 2) the importance of interoception in embodiment of Abstract concepts; and 3) the perceptual-based distinction in conceptual categories. In addition, some cross-linguistic differences emerged in relation to the levels in which interoceptive components of concepts influence empathy. In addition, to understand better the contribution of bodily states and interoception to word processing, we further examined the effects of being immersed in a perceptual deprivation chamber aimed at increasing saliency of bodily and internal states on electrophysiological correlates of word processing. To the best of our knowledge no previous studies explored the effect of this kind of manipulation on word processing. Consequently, in Experiment 2, we collected ERPs related to the processing of Abstract and Concrete words, focusing on the N400 and a late N700-like negativity. We hypothesized that the immersion in the perceptual deprivation chamber would reduce the differences between ERPs’ amplitudes for Abstract and Concrete words as an effect of increased interoceptive and decreased exteroceptive saliency. In line with our hypothesis, we observed that: 1) the N400 difference between Abstract and Concrete concepts was reduced in the perceptual deprivation chamber; and 2) Abstract concepts elicited stronger late N700-like negativity in the perceptual deprivation chamber. We suggest that increased attentional focus on internal states would reduce the retrieval of sensorimotor components during Concrete concepts processing and facilitate multimodal imagery processes during Abstract words’ processing. Taken together, the current thesis supports the bidirectional relationship between bodily states and language processing in which words act as cue to the retrieval of perceptual experiences (language-to-body) and the bodily state facilitate the retrieval of multimodal situated representations (body-to-language). Furthermore, the importance of directing attention towards bodily and internal states and the implications of our results on the Abstract-Concrete distinction are discussed. In conclusion, bodily states appear to be involved in word processing even more than previously thought. Future studies aimed at providing deeper understanding of the body-language coupling including additional physiological measures are recommended
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
Sharp Convergence Rate of the Glimm Scheme for General Nonlinear Hyperbolic Systems
No full textConsider a general strictly hyperbolic,
quasilinear system, in one space dimension
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u_t+A(u) u_x=0,
\eqno (1)
%
where , , is a smooth
matrix-valued map. Given an initial datum with small
total variation, let be the corresponding (unique)
vanishing viscosity solution of (1) obtained as limit of solutions to
the viscous parabolic approximation , as
. For every , we prove the a-priori bound
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\big\|u^\eps(T,\cdot)-u(T,\cdot)\big\|_{\elleuno}=o(1)\cdot\sqrt\eps\,|\log\eps|
\eqno (2)
%
for an approximate solution u^\eps of (1) constructed by the Glimm
scheme, with mesh size \Delta x = \Delta t = \eps, and with a
suitable choice of the sampling sequence. This result provides for
general hyperbolic systems the same type of error estimates valid for
Glimm approximate solutions of hyperbolic systems of conservation laws
satisfying the classical Lax or Liu assumptions on the
eigenvalues and on the eigenvectors of the
Jacobian matrix .
The estimate (2) is obtained introducing a new wave interaction
functional with a cubic term that controls the nonlinear coupling of
waves of the same family and at the same time decreases at
interactions by a quantity that is of the same order of the product of
the wave strength times the change in the wave speeds. This is
precisely the type of errors arising in a wave tracing analysis of the
Glimm scheme, which is crucial to control in order to achieve an
accurate estimate of the convergence rate as~(2)
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
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