Balance laws : non local mixed systems and IBVPs

Abstract

Scalar hyperbolic balance laws in several space dimensions play a central role in this thesis. First, we deal with a new class of mixed parabolic–hyperbolic systems on all Rn: we obtain the basic well–posedness theorems, devise an ad hoc numerical algorithm, prove its convergence and investigate the qualitative properties of the solutions. The extension of these results to bounded domains requires a deep understanding of the initial boundary value problem (IBVP) for hyperbolic balance laws. The last part of the thesis provides rigorous estimates on the solution to this IBVP, under precise regularity assumptions. In Chapter 1 we introduce a predator–prey model. A non local and non linear balance law is coupled with a parabolic equation: the former describes the evolution of the predator density, the latter that of prey. The two equations are coupled both through the convective part of the balance law and the source terms. The drift term is a non local function of the prey density. This allows the movement of predators to be directed towards the regions where the concentration of prey is higher. We prove the well–posedness of the system, hence the existence and uniqueness of solution, the continuous dependence on the initial data and various stability estimates. In Chapter 2 we devise an algorithm to compute approximate solutions to the mixed system introduced above. The balance law is solved numerically by a Lax–Friedrichs type method via dimensional splitting, while the parabolic equation is approximated through explicit finite–differences. Both source terms are integrated by means of a second order Runge–Kutta scheme. The key result in Chapter 2 is the convergence of this algorithm. The proof relies on a careful tuning between the parabolic and the hyperbolic methods and exploits the non local nature of the convective part in the balance law. This algorithm has been implemented in a series of Python scripts. Using them, we obtain information about the possible order of convergence and we investigate the qualitative properties of the solutions. Moreover, we observe the formation of a striking pattern: while prey diffuse, predators accumulate on the vertices of a regular lattice. The analytic study of the system above is on all Rn. However, both possible biological applications and numerical integrations suggest that the boundary plays a relevant role. With the aim of studying the mixed hyperbolic–parabolic system in a bounded domain, we noticed that for balance laws known results lack some of the estimates necessary to deal with the coupling. In Chapter 3 we then focus on the IBVP for a general balance law in a bounded domain. We prove the well–posedness of this problem, first with homogeneous boundary condition, exploiting the vanishing viscosity technique and the doubling of variables method, then for the non homogeneous case, mainly thanks to elliptic techniques. We pay particular attention to the regularity assumptions and provide rigorous estimates on the solution.