We prove Glimm interaction estimates for a 3 3 hyperbolic system of conservation laws arising in ... more We prove Glimm interaction estimates for a 3 3 hyperbolic system of conservation laws arising in the modeling of multi-phase ows. No smallness of the interacting waves is assumed. Our proof simplies and improves a previous result by Y.-J. Peng [9].
In this chapter we illustrate our approach to the error estimate analysis, for a scalar, space-de... more In this chapter we illustrate our approach to the error estimate analysis, for a scalar, space-dependent, non-resonant balance law. A wave-front tracking scheme is analyzed, leading to a generic linear dependence in time of the error. Numerical illustrations are given for accretive case and for the case of periodic forcing.
We study a 2×2 system of balance laws that describes the evolution of a granular material (avalan... more We study a 2×2 system of balance laws that describes the evolution of a granular material (avalanche) flowing downhill. The original model was proposed by Hadeler and Kuttler [13]. The Cauchy problem for this system is studied by the authors in the recent papers [2,16]. In this paper, we first consider an initial-boundary value problem. The bound-ary condition is given by the flow of the incoming material. For this problem we prove the global existence of BV solutions for a suitable class of data, with bounded but possibly large total variations. We then study the “slow erosion (or deposition) limit”. We show that, if the thickness of the moving layer remains small, then the profile of the standing layer depends only on the total mass of the avalanche flowing downhill, not on the time-law describing at which rate the material slides down. More precisely, in the limit as the thickness of the moving layer tends to zero, the slope of the mountain is provided by an entropy solution to a...
We consider a hydrodynamic model of flocking-type with all-to-all interaction kernel in one-space... more We consider a hydrodynamic model of flocking-type with all-to-all interaction kernel in one-space dimension and establish global existence of entropy weak solutions to the Cauchy problem for any BV initial data that has finite total mass confined in a bounded interval and initial density uniformly positive therein. In addition, under a suitable condition on the initial data, we show that entropy weak solutions admit time-asymptotic flocking.
In this paper, we present a modified wave-front tracking algorithm which is suitable for the anal... more In this paper, we present a modified wave-front tracking algorithm which is suitable for the analysis of scalar conservation laws with nonlocal terms. This method has been first employed in Shen and Zhang (Arch Ration Mech Anal 204:837–879, 2012) to analyze a nonlocal Hamilton-Jacobi equation related to a granular flow and later used in other works. Such an approach leads to a possibly simpler analysis in obtaining rigorous quantitative estimates on approximate solutions, compared to a classical iteration procedure based on the recomputation of the nonlocal term at each time step. Here, we delineate this method for a nonlocal equation namely “the Kuramoto-Sakaguchi equation” arising from the kinetic modeling of collective motion of large ensemble of Kuramoto oscillators, for which BV-weak solutions and their large time behavior are investigated in Amadori et al. (J Differ Equ 262, 978–1022, 2017).
In this paper we study a \begin{document}$ 2\times2 $\end{document} semilinear hyperbolic system ... more In this paper we study a \begin{document}$ 2\times2 $\end{document} semilinear hyperbolic system of partial differential equations, which is related to a semilinear wave equation with nonlinear, time-dependent damping in one space dimension. For this problem, we prove a well-posedness result in \begin{document}$ L^\infty $\end{document} in the space-time domain \begin{document}$ (0,1)\times [0,+\infty) $\end{document}. Then we address the problem of the time-asymptotic stability of the zero solution and show that, under appropriate conditions, the solution decays to zero at an exponential rate in the space \begin{document}$ L^{\infty} $\end{document}. The proofs are based on the analysis of the invariant domain of the unknowns, for which we show a contractive property. These results can yield a decay property in \begin{document}$ W^{1,\infty} $\end{document} for the corresponding solution to the semilinear wave equation.
We prove Glimm interaction estimates for a 3 3 hyperbolic system of conservation laws arising in ... more We prove Glimm interaction estimates for a 3 3 hyperbolic system of conservation laws arising in the modeling of multi-phase ows. No smallness of the interacting waves is assumed. Our proof simplies and improves a previous result by Y.-J. Peng [9].
In this chapter we illustrate our approach to the error estimate analysis, for a scalar, space-de... more In this chapter we illustrate our approach to the error estimate analysis, for a scalar, space-dependent, non-resonant balance law. A wave-front tracking scheme is analyzed, leading to a generic linear dependence in time of the error. Numerical illustrations are given for accretive case and for the case of periodic forcing.
We study a 2×2 system of balance laws that describes the evolution of a granular material (avalan... more We study a 2×2 system of balance laws that describes the evolution of a granular material (avalanche) flowing downhill. The original model was proposed by Hadeler and Kuttler [13]. The Cauchy problem for this system is studied by the authors in the recent papers [2,16]. In this paper, we first consider an initial-boundary value problem. The bound-ary condition is given by the flow of the incoming material. For this problem we prove the global existence of BV solutions for a suitable class of data, with bounded but possibly large total variations. We then study the “slow erosion (or deposition) limit”. We show that, if the thickness of the moving layer remains small, then the profile of the standing layer depends only on the total mass of the avalanche flowing downhill, not on the time-law describing at which rate the material slides down. More precisely, in the limit as the thickness of the moving layer tends to zero, the slope of the mountain is provided by an entropy solution to a...
We consider a hydrodynamic model of flocking-type with all-to-all interaction kernel in one-space... more We consider a hydrodynamic model of flocking-type with all-to-all interaction kernel in one-space dimension and establish global existence of entropy weak solutions to the Cauchy problem for any BV initial data that has finite total mass confined in a bounded interval and initial density uniformly positive therein. In addition, under a suitable condition on the initial data, we show that entropy weak solutions admit time-asymptotic flocking.
In this paper, we present a modified wave-front tracking algorithm which is suitable for the anal... more In this paper, we present a modified wave-front tracking algorithm which is suitable for the analysis of scalar conservation laws with nonlocal terms. This method has been first employed in Shen and Zhang (Arch Ration Mech Anal 204:837–879, 2012) to analyze a nonlocal Hamilton-Jacobi equation related to a granular flow and later used in other works. Such an approach leads to a possibly simpler analysis in obtaining rigorous quantitative estimates on approximate solutions, compared to a classical iteration procedure based on the recomputation of the nonlocal term at each time step. Here, we delineate this method for a nonlocal equation namely “the Kuramoto-Sakaguchi equation” arising from the kinetic modeling of collective motion of large ensemble of Kuramoto oscillators, for which BV-weak solutions and their large time behavior are investigated in Amadori et al. (J Differ Equ 262, 978–1022, 2017).
In this paper we study a \begin{document}$ 2\times2 $\end{document} semilinear hyperbolic system ... more In this paper we study a \begin{document}$ 2\times2 $\end{document} semilinear hyperbolic system of partial differential equations, which is related to a semilinear wave equation with nonlinear, time-dependent damping in one space dimension. For this problem, we prove a well-posedness result in \begin{document}$ L^\infty $\end{document} in the space-time domain \begin{document}$ (0,1)\times [0,+\infty) $\end{document}. Then we address the problem of the time-asymptotic stability of the zero solution and show that, under appropriate conditions, the solution decays to zero at an exponential rate in the space \begin{document}$ L^{\infty} $\end{document}. The proofs are based on the analysis of the invariant domain of the unknowns, for which we show a contractive property. These results can yield a decay property in \begin{document}$ W^{1,\infty} $\end{document} for the corresponding solution to the semilinear wave equation.
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Papers by Debora Amadori