FINER ANALYSIS OF CHARACTERISTIC CURVE AND ITS APPLICATION TO EXACT, OPTIMAL CONTROLLABILITY, STRUCTURE OF THE ENTROPY SOLUTION OF A SCALAR CONSERVATION LAW WITH CONVEX FLUX

Goal of this thesis is to study four problems. In chapters 3-5, we consider scalar conser- vation law in one space dimension with strictly convex flux. First problem is to know the profile of the entropy solution. In spite of the fact that, this was studied extensively in last several decades, the complete profile of the entropy solution is not well understood. Second problem is the exact controllability. This was studied for Burgers equation and some partial results are obtained for large time. It was a challenging problem to know the controllability for all time and also for general convex flux. In a seminal paper [25], Dafermos introduces the characteristic curves and obtain some qualitative properties of a solution of a convex conservation law. In this thesis, we further study the finer properties of these characteristic curves. Here we solve these two problems in complete generality. In view of the explicit formulas of Lax - Oleinik [31], Joseph - Gowda [40], target func- tions must satisfy some necessary conditions. In this thesis we prove that these are also sufficient. Method of the proof depends highly on the characteristic methods and explicit formula given by Lax - Oleinik and the proof is constructive. Third problem is to solve the optimal controllability problem. In chapter 5 we derive a method to obtain a solution of an optimal control problem for the scalar conservation laws with convex flux. By using the method of descent, this type of problem was considered by Castro-Palacios-Zuazua in [23] for the Burgers equation. Our approach is simple and based on the explicit formulas of Hopf and Lax-Olenik. Last but not the least is about the problem of total variation bound for solution of scalar conservation laws with discontinuous flux. For the scalar con- servation laws with discontinuous flux, an infinite family (A, B)-interface entropies are introduced and each one of them has been shown to form an L1 -contraction semigroup (see, [8]). One of the main unsettled questions concerning conservation law with discon- tinuous flux is boundedness of total variation of the solution. Away from the interface, boundedness of total variation of the solution has been proved in a recent paper [16]. In the chapter 6, we discuss this particular issue in detail and produce a counter example to show that the solution, in general, has unbounded total variation near the interface. In fact this example illustrates that smallness of BV norm of the initial data is immaterial. We hereby settled the question of determining for which of the aforementioned (A, B) pairs, the solution will have bounded total variation in case of strictly convex fluxes