Well-posedness of the Cauchy problem and stability theory for nonlinear dispersive equations

Abstract

In this project we plan to study nonlinear evolution partial differential equations of dispersive type. This has become a very active subject of research in the field of mathematical analysis, particularly in the last two decades, with several of the most important results and techniques having been developed by very notable mathematicians like Jean Bourgain, Carlos Kenig, Gustavo Ponce, Luis Vega or Terence Tao,among others ([2]-[12]).For sufficiently smooth data, local well-posedness issues are studied using combinations of energy methods, contraccion principle, and estimates of Strichartz, smoothing and maximal type ([7]). But to deal with low regularity data these might not suffice. The recently introduced Fourier restriction norm method by J. Bourgain [2] is adequate to deal with very low regularity data. Although this method is very technical and depends highly on the structure of the associated Fourier symbol of the linear part of the equations, it has become very successful in obtaining sharp results [6]. For global well-posedness, one usually tries to find conservation laws associated to the equations, which naturally yield a priori estimates in certain function spaces. These can then be used to iterate known local solutions, extending them 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. The absence of adequate a priori estimates therefore prevents, in these cases, the implementation of the successive iterations. To deal with this adverse situation, a new concept, called I-method and almost conserved quantities, has been introduced [5, 6], in order to prove global well-posedness in function spaces of lower regularity than the ones provided by the conservation laws.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. With this project we plan to study these issues for some models that fall within the dispersive class of evolution equations, in particular the higher order modified Korteweg-de Vries (mKdV), Fifth order Benjamin-Bona-Mahony (BBM) and Zakharov-Kuznetsov (ZK) equations, as well as the Schrödinger-Debye (SD), Boussinesq type and Davey-Stewartson (DS) systems.Besides well-posedness issues, we also plan to develop research on singularity formation of solutions to these equations, as this is the phenomenon preventing global existence. The work developed in the last few years by Frank Merle and his collaborators (see [10, 11]) has solved long standing open questions on existence of H^1 blow-up solutions for L^2 critical Schrödinger and mKdV equations. The methods thus far used are intimately connected with stability of ground state, or soliton, solutions for these equations. We are interested in exploring the possibility of adapting these methods to the Schrodinger-Debye and Davey-Stewartson systems.Our next interest is to study the assymptotic behavior of the solution like unique continuation principle (UCP) which we studied extensively during our doctoral and post-doctoral research. Some of our recent results in this direction can be found in [3, 12]. (AU)