Nonlinear behavior of model equations which are linearly ill-posed
- 1 January 1988
- journal article
- research article
- Published by Taylor & Francis in Communications in Partial Differential Equations
- Vol. 13 (4) , 423-467
- https://doi.org/10.1080/03605308808820548
Abstract
We discuss two model equations of nonlinear evolution which demonstrate that linearly ill-posed problems may be well-posed in a mild sense. For the nonlocal equation (1.4), smooth solutions exist for all time, are unique, and depend continuously on the initial data in low norms. For the partial differential equation (1.1), solutions always exist; we do not know whether they are unique, but if they are, they also have continuous dependence on data. The large-time behavior of solutions and other qualitative properties are discussedKeywords
This publication has 7 references indexed in Scilit:
- Desingularization of periodic vortex sheet roll-upJournal of Computational Physics, 1986
- A study of singularity formation in a vortex sheet by the point-vortex approximationJournal of Fluid Mechanics, 1986
- Nonlinear Kelvin–Helmholtz instability of a finite vortex layerJournal of Fluid Mechanics, 1985
- Existence of Infinitely Many Solutions for a Forward Backward Heat EquationTransactions of the American Mathematical Society, 1983
- A semi-analytic approach to the self-induced motion of vortex sheetsJournal of Fluid Mechanics, 1981
- Vortex simulations of the Rayleigh–Taylor instabilityPhysics of Fluids, 1980
- The spontaneous appearance of a singularity in the shape of an evolving vortex sheetProceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 1979