| Grant number: | 16/25864-6 |
| Support type: | Regular Research Grants |
| Duration: | March 01, 2017 - August 31, 2019 |
| Field of knowledge: | Physical Sciences and Mathematics - Mathematics - Analysis |
| Principal researcher: | Mahendra Prasad Panthee |
| Grantee: | Mahendra Prasad Panthee |
| Home Institution: | Instituto de Matemática, Estatística e Computação Científica (IMECC). Universidade Estadual de Campinas (UNICAMP). Campinas , SP, Brazil |
Abstract
In this project we plan to study nonlinear evolution equations of dispersive type. We are interested in studying the associated Cauchy problem and certain properties of the solution like local and global existence, controllability, stabilization and stability of solitary waves, unique continuation property (UCP) and its generalization among others.The nonlinear Schrodinger (NLS), Boussinesq and Korteweg-de Vries (KdV) equations that appear to describe several physical phenomena are the typical examples that fall in this class. In recent time, a very notable mathematicians like, J. Bona, J. Bourgain, C. Kenig, G. Ponce, L. Vega or T. Tao, among others (see [7]-[41]) are devoted in developing new techniques to respond long standing problems in this area. For sufficiently regular data, local well-posedness issues for the associated Cauchy problem are studied using combinations of energy methods, contraction principle, and estimates of Strichartz, smoothing and maximal type ([25, 26]). However in the case of data with low Sobolev regularity the technique explained above might not suce. To deal with this situaton, the recently introduced Fourier restriction norm method in [9] is adequate and has proved to be very successful in obtaining sharp results in several situations [17]. To prove global well-posedness, one usually tries to use conservation laws associated to the equations, to obtain a priori estimates in certain function spaces which can be used to iterate the known local solutions into a global ones. But there are several cases where a gap exists between the function spaces for which local existence is known and the spaces associated to the conservation laws. To deal with this adverse situation, a new concept, called I-method and almost conserved quantities, has been introduced [16, 17]. These quantities are used to prove global well-posedness in function spaces of lower regularity than the ones given by the conservation laws.It is worth mentioning that the methods described above depend heavily on the structure of the model in question and the domain in which the model is posed. Despite the extensive research carried out in this subject, mostly in the last two decades, the question of well-posedness (both local and global) is still open for most models, particularly for the low regularity data. In this project we plan to study these issues for some models that fall within the dispersive class of evolution equations. Also, we also plan to develop research on singularity formation of solutions to these equations, controllability and the asymptotic behavior like UCP and stability.NOTE: For references, see the attached research project. (AU)