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Logarithmic Convexity and the 'Slow Evolution' Constraint in Ill-Posed Initial Value Problems

Published

Author(s)

Alfred S. Carasso

Abstract

This paper examines a wide class of ill-posed initial value problems for partial differential equations, and surveys logarithmic convexity results leading to Hoelder-continuous dependence on data for solutions satisfying prescribed bounds. The discussion includes analytic continuation in the unit disc, time-reversed parabolic equations in Lp spaces, the time-reversed Navier-Stokes equations, as well as a large class of non-local evolution equations that can be obtained by randomizing the time variable in well-posed Cauchy problems. It is shown that in many cases, the resulting Hoelder-continuity is much too weak to permit useful continuation from imperfect data. However, considerable reduction in the growth of errors occurs, and continuation becomes feasible, for solutions satisfying the slow evolution from the continuation boundary (SECB) constraint, previously introduced by the author.
Citation
Siam Journal on Mathematical Analysis
Volume
30
Issue
3

Keywords

growing diffusion coefficients, ill-posed problems, parabolic equations, SECB constraint

Citation

Carasso, A. (1999), Logarithmic Convexity and the 'Slow Evolution' Constraint in Ill-Posed Initial Value Problems, Siam Journal on Mathematical Analysis, [online], https://tsapps.nist.gov/publication/get_pdf.cfm?pub_id=150727 (Accessed June 20, 2024)

Issues

If you have any questions about this publication or are having problems accessing it, please contact reflib@nist.gov.

Created January 15, 1999, Updated June 2, 2021