Abstract:We introduce a new approach to obtain sharp

pointwiseerror estimates for viscosity approximation (and in fact --- more general approximations) to scalar conservation laws with piecewise smooth solutions. To this end, we derive a transport inequality for an appropriatelyweightederror function. The key ingredient in our approach is a one-sided interpolation inequality between classical $L^1$ error estimates and $Lip^+$ stability bounds. The one-sided interpolation, interesting for its own sake, enables us to convert a global $L^1$ result into a (non-optimal) local estimate. This, in turn, provides the necessary bounds on the coefficients of the above mentioned transport inequality. Estimates on the weighted error then follow from the maximum principal, and a bootstrap argument yields optimal pointwise error bound for the viscosity approximation.Unlike previous works in this direction, our method can deal with finitely many waves with possible collisions. Moreover, in our approach one does not follow the characteristics but instead makes use of the energy method, and hence this approach could be extended to other types of approximate solutions.

**Paper:**- Available as PostScript
**Title:**- Pointwise error estimates for scalar conservation laws with piecewise smooth solutions
**Author(s):**- Eitan Tadmor , <tadmor@math.ucla.edu>
- Tao Tang, <ttang@math.ucla.edu>
**Publishing information:****Comments:****Submitted by:**- <tadmor@math.ucla.edu> February 4 1998.

[ 1996 Preprints | 1997 Preprints | 1998 Preprints | All Preprints | Preprint Server Homepage ]

Conservation Laws Preprint Server <conservation@math.ntnu.no> Last modified: Thu Feb 5 11:31:45 1998