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Nonlinear Evolution Equations of Dispersive Type

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)

Scientific publications (8)
(References retrieved automatically from Web of Science and SciELO through information on FAPESP grants and their corresponding numbers as mentioned in the publications by the authors)
CARVAJAL, X.; PANTHEE, M.; PASTRAN, R. On the well-posedness, ill-posedness and norm-inflation for a higher order water wave model on a periodic domain. NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS, v. 192, MAR 2020. Web of Science Citations: 0.
NOGUEIRA, MARCELO; PANTHEE, MAHENDRA. ON THE SCHRODINGER-DEBYE SYSTEM IN COMPACT RIEMANNIAN MANIFOLDS. COMMUNICATIONS ON PURE AND APPLIED ANALYSIS, v. 19, n. 1, p. 425-453, JAN 2020. Web of Science Citations: 0.
CARVAJAL, XAVIER; PANTHEE, MAHENDRA. Sharp well-posedness for a coupled system of mKdV-type equations. JOURNAL OF EVOLUTION EQUATIONS, v. 19, n. 4, p. 1167-1197, DEC 2019. Web of Science Citations: 0.
CARVAJAL, X.; PANTHEE, M. On sharp global well-posedness and ill-posedness for a fifth-order KdV-BBM type equation. Journal of Mathematical Analysis and Applications, v. 479, n. 1, p. 688-702, NOV 1 2019. Web of Science Citations: 1.
LINARES, FELIPE; PANTHEE, MAHENDRA; ROBERT, TRISTAN; TZVETKOV, NIKOLAY. ON THE PERIODIC ZAKHAROV-KUZNETSOV EQUATION. DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS, v. 39, n. 6, p. 3521-3533, JUN 2019. Web of Science Citations: 0.
BHATTARAI, SANTOSH; CORCHO, ADAN J.; PANTHEE, MAHENDRA. Well-Posedness for Multicomponent Schrodinger-gKdV Systems and Stability of Solitary Waves with Prescribed Mass. Journal of Dynamics and Differential Equations, v. 30, n. 2, p. 845-881, JUN 2018. Web of Science Citations: 2.
BONA, J. L.; CARVAJAL, X.; PANTHEE, M.; SCIALOM, M. Higher-Order Hamiltonian Model for Unidirectional Water Waves. JOURNAL OF NONLINEAR SCIENCE, v. 28, n. 2, p. 543-577, APR 2018. Web of Science Citations: 2.
CARVAJAL, XAVIER; ESFAHANI, AMIN; PANTHEE, MAHENDRA. Well-Posedness Results and Dissipative Limit of High Dimensional KdV-Type Equations. BULLETIN OF THE BRAZILIAN MATHEMATICAL SOCIETY, v. 48, n. 4, p. 505-550, DEC 2017. Web of Science Citations: 1.

Please report errors in scientific publications list by writing to: cdi@fapesp.br.