| Of concern is the Cauchy problem ut+divf u+ guH x=0,x∈Rn ,t>0, 1 ux,0=u0 x,∈Rn , where the flux function &phis; satisfies some mild regularity at the origin, g is globally bounded and Lipschitzian, H is positive and bounded away from zero, and finally the initial data u0, is a bounded and integrable function.; Well-posedness of the problem has been established by Goldstein and Park, even for the case of singular perturbation. Our concern is the regularity of such solution. In this dissertation we propose a new notion of regularity, and establish a regularity theory based on this notion. Finally we aim to show that this new notion of regularity is actually equivalent to the standard one for this type of problem.; In the context of semigroups of operators theory, showing some regularity of a Cauchy problem amounts to finding subsets, or subspaces which are invariant under the action of the semigroup. In more practical terms, our first objective is to construct a Banach space which is invariant under the semigroup which governs the Cauchy problem (1), and then to show the Banach space that we construct is actually the Banach space of bounded variation functions on Rn . |
