생체의 모델링과 Simulation

This paper develops a new approach in the analysis of the Camassa–Holm equation. By introducing a new set of independent and dependent variables, the equation is transformed into a semilinear system, whose solutions are obtained as fixed points of a contractive transformation. These new variables resolve all singularities due to possible wave breaking. Returning to the original variables, we obtain a semigroup of global solutions, depending continuously on the initial data. Our solutions are conservative, in the sense that the total energy equals a constant, for almost every time.

/pdf/global-conservative-solutions-of-the-camassa-holm-equation-2gi6h3ifov.pdf

Global conservative solutions of the Camassa-Holm equation

Backstepping boundary control for first-order hyperbolic PDEs and application to systems with actuator and sensor delays

The goal of this final chapter is to show how the boundary value problems of mathematical physics can be solved by the methods of the preceding chapters. This will be done by solving a variety of specific problems that illustrate the principal types of problems that were formulated in Chapter 7. Additional applications are developed in the Exercises. The primary solution method is Fourier’s method of separation of variables and the associated Sturm-Liouville theory of Chapter 8.

Boundary Value Problems of Mathematical Physics

This paper is devoted to the continuation of solutions to the Camassa–Holm equation after wave breaking. By introducing a new set of independent and dependent variables, the evolution problem is rewritten as a semilinear hyperbolic system in an L∞ space, containing a non-local source term which is discontinuous but has bounded directional variation. For a given initial condition, the Cauchy problem has a unique solution obtained as fixed point of a contractive integral transformation. Returning to the original variables, we obtain a semigroup of global dissipative solutions, defined for every initial data $\bar u\in H^1 ({\mathbb R})$, and continuously depending on the initial data. The new variables resolve all singularities due to possible wave breaking and ensure that energy loss occurs only through wave breaking.

/pdf/global-dissipative-solutions-of-the-camassa-holm-equation-58ml9ceryw.pdf

Global dissipative solutions of the camassa–holm equation

This paper is concerned with a fluidodynamic model for traffic flow. More precisely, we consider a single conservation law, deduced from the conservation of the number of cars, defined on a road network that is a collection of roads with junctions. The evolution problem is underdetermined at junctions; hence we choose to have some fixed rules for the distribution of traffic plus optimization criteria for the flux. We prove existence of solutions to the Cauchy problem and we show that the Lipschitz continuous dependence by initial data does not hold in general, but it does hold under special assumptions.Our method is based on a wave front tracking approach [A. Bressan, Hyperbolic Systems of Conservation Laws. The One-dimensional Cauchy Problem, Oxford University Press, Oxford, UK, 2000] and works also for boundary data and time-dependent coefficients of traffic distribution at junctions, including traffic lights.

Traffic Flow on a Road Network

This paper is concerned with a fluidodynamic model for traffic flow. More precisely, we consider a single conservation law, deduced from conservation of the number of cars, defined on a road network that is a collection of roads with junctions. The evolution problem is underdetermined at junctions, hence we choose to have some fixed rules for the distribution of traffic plus an optimization criteria for the flux. We prove existence, uniqueness and stability of solutions to the Cauchy problem. 
Our method is based on wave front tracking approach, (B), and works also for boundary data and time dependent coefficients of traffic distribution at junctions, so including traffic lights.

https://www.math.ntnu.no/conservation/2005/028.pdf

On the well-posedness of the Degasperis-Procesi equation

We consider a generalized hyperelastic-rod wave equation (or generalized Camassa--Holm equation) describing nonlinear dispersive waves in compressible hyperelastic rods. We establish existence of a strongly continuous semigroup of global weak solutions for any initial data from $H^1(\R)$. We also present a "weak equals strong" uniqueness result.

/pdf/global-weak-solutions-to-a-generalized-hyperelastic-rod-wave-52dyzl5bzp.pdf

Global weak solutions to a generalized hyperelastic-rod wave equation

where α, γ, ω are given real constants. Equation (1) was first introduced as a model describing propagation of unidirectional gravitational waves in a shallow water approximation over a flat bottom, with u representing the fluid velocity [DGH01]. For α = 0 and for α = 1, γ = 0 we obtain the Korteweg–de Vries and the Camassa–Holm [CH93, J02] equations, respectively. Both of them describe unidirectional shallow water waves. Moreover, all these three equations have a bi-Hamiltonian structure, they are completely integrable, they have infintely many conserved quantities. From a mathematical point of view the Camassa–Holm equation is well studied, see [CHK05.1] for an extensive list of references. In particular, we recall that existence and uniqueness results for global weak solutions have been proved by Coclite, Holden, and Karlsen [CHK05.1], Constantin and Escher [CE98], Constantin and Molinet [CM00], and Xin and Zhang [XZ00, XZ02], see also Danchin [D01, D03] as well as others.

Giuseppe Maria Coclite

Papers

Traffic Flow on a Road Network

Traffic Flow on a Road Network

On the well-posedness of the Degasperis-Procesi equation

Global weak solutions to a generalized hyperelastic-rod wave equation

Global Weak Solutions for a Shallow Water Equation