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Journal article
Well-posedness of fully nonlinear KdV-type evolution equationsDedicated with admiration to the memory of our friend Walter Craig
Nonlinearity, v 32(8), pp 2914-2954
17 Jul 2019
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Abstract
We study the well-posedness of the initial value problem for fully nonlinear evolution equations, where f may depend on up to the first three spatial derivatives of u. We make three primary assumptions about the form of a regularity assumption, a dispersivity assumption, and an assumption related to the strength of backwards diffusion. Because the third derivative of u is present in the right-hand side and we effectively assume that the equation is dispersive, we say that these fully nonlinear evolution equations are of KdV-type. We prove the well-posedness of the initial value problem in the Sobolev space , which is close to optimal for the energy estimates we make. The proof relies on gauged energy estimates which follow after making two regularizations, a parabolic regularization and mollification of the initial data. There is evidence that the backward diffusion condition we express is optimal.
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Details
- Title
- Well-posedness of fully nonlinear KdV-type evolution equationsDedicated with admiration to the memory of our friend Walter Craig
- Creators
- Timur Akhunov - College of New JerseyDavid M Ambrose - Drexel UniversityJ Douglas Wright - Drexel University Department of Mathematics, Philadelphia, PA, United States of America
- Publication Details
- Nonlinearity, v 32(8), pp 2914-2954
- Publisher
- IOP Publishing
- Number of pages
- 41
- Grant note
- 1511488; 1515849 / Division of Mathematical Sciences (https://doi.org/10.13039/100000121)
- Resource Type
- Journal article
- Language
- English
- Academic Unit
- Mathematics
- Web of Science ID
- WOS:000476588900005
- Scopus ID
- 2-s2.0-85072292173
- Other Identifier
- 991019168150104721
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- Collaboration types
- Domestic collaboration
- Web of Science research areas
- Mathematics, Applied
- Physics, Mathematical