Analisi delle soluzioni di entropia alla legge di conservazione con flusso discontinuo nello spazio e nel tempo

In this thesis, we consider a first order partial differential equation with coefficient that contains a jump in the space and time variables. The equation is deeply inspired by the one-dimensional model of pedestrian flow introduced by Hughes, commonly referred to as \textit{the Hughes' model} \cite{HUGHES2002507}. To be precise, our equation consists in a scalar conservation law whose flux coefficient is a time dependent function that switches sign according to the location of the 'turning curve', $\xi(t)$, given \emph{a priori} in the space domain. The equation writes \[ \pt\rho + \px\big( \sign(x-\xi(t))f(\rho)\big) = 0. %~~(t, x)\in\Omega= (0, +\infty)\times (-\infty, +\infty), \] The main object of this thesis is to analyse the well-posedness of the above equation via entropy solutions taken in an appropriate sense. In this text, the basic notions of scalar conservation law with continuous and discontinuous flux functions are revisited with a detailed discussion and comparison of existing results regarding entropy admissibility criteria for both classes of equations. Furthermore, the theory of $ L^1$-dissipative germ introduced by Andreianov, Karlsen and Risebro \cite{Andreianov2011} is reviewed and is then extended to our equation with the goal of analysing the entropy solutions for the case of non-classical flux interface coupling at the turning curve. \medskip In a series of steps leading to the existence of solutions to the above equation, we properly define the entropy solutions and construct the exact solution of the Riemann problem that arises at the turning curve. Using the Riemann solver at $\xi(t),$ the total variation in the solution as the slope of $\xi$ changes in time and as $\xi$ interactions with classical waves (i.e. shock and rarefaction) are studied. \medskip Finally, a numerical scheme with a moving mesh adaptation and a modified numerical flux near the turning curve $x=\xi(t)$ is also proposed. In order to prove that the scheme converges to the weak entropy solution, we first establish that it is well-balanced and stable in $\Lp\infty$ space. The approximate solution is further analysed in the sense of entropy process solutions. We also present some explicit examples and their approximate solutions to numerically check convergence of the approximate solution provided by the scheme to the weak entropy solutions.