Search

The mapping properties of the time evolution operator E(t) of nonlinear hyperbolic scalar conservation laws is investigated. It is shown that this operator is Lipschitz in the Hausdorff metric in one space dimension whenever the flux is convex and one of the initial conditions satisfies a one-sided Lipschitz condition. The Hausdorff distance between the graphs of the solutions measures the closeness in L∞ in the regions where the solutions are smooth, as well as the closeness between the locations of shocks. A similar result on Hausdorff stability is proved with respect to a perturbation of the flux function. These results complement the well known L1 contractivity of the solution operator. They are used in a subsequent paper to prove new smoothness results for solutions to such conservation laws. Negative results are proved in the case of non-convex and genuinely multidimensional fluxes. [ABSTRACT FROM AUTHOR]

The smoothness of the solutions of 1D scalar conservation laws is investigated and it is shown that if the initial value has smoothness of order α in Lq with α > 1 and q = 1/α, this smoothness is preserved at any time t > 0 for the graph of the solution viewed as a function in a suitably rotated coordinate system. The precise notion of smoothness is expressed in terms of a scale of Besov spaces which also characterizes the functions that are approximated at rate N-α in the uniform norm by piecewise polynomials on N adaptive intervals. An important implication of this result is that a properly designed adaptive strategy should approximate the solution at the same rate N-α in the Hausdorff distance between the graphs. [ABSTRACT FROM AUTHOR]