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Properties of solutions of some dispersive equations

Abstract

Our project is divided into three parts: In the first part we want to prove a result of global well-posedness for the generalized Korteweg de Vries equation, g-KdV. Our method will be to refine a previous technique of Bourgain on high and low frequencies, along with some kind of a priori estimates and estimates refined of the group of KdV equation. The main difficulty of the iterative technique is that it is known in the literature for certain data within standard with energy greater than the norm of the solitary wave (in the same space of energy) there is blow-up and hence loss of the global existence, we try to use it to control the size of the initial data in each iteration. In the second part of the plan we will study a mixed KdV- nonlinear Schrödinger equation with variable coefficients. We intend here to improve a previous result of local existence without using a very strong tool of Harmonic Analysis: T1 Theorem. We also want to get some global existence result for this model in some Sobolev space, the difficulty here is that there are no conserved quantities for this model. Finally we try to prove some unique continuation result for this model. In the third part we will study a generalized equation of Benjamin Bona Mahony of fifth order (BBM5), this model is new in the literature and was obtained from a Boussinesq type system. We will follow the fixed point technique applied in the usual Sobolev spaces, we believe that the BBM5 equation can be locally well posed in Sobolev spaces with the usual index greater than or equal to one. We will also try to prove a result of global existence, obtaining conserved quantities and/or a priori estimates. Finally we will try to find a result of ill-posedness for this model. (AU)

Scientific publications (6)
(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)
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, X.; PANTHEE, M.; SCIALOM, M. Comparison between model equations for long waves and blow-up phenomena. Journal of Mathematical Analysis and Applications, v. 442, n. 1, p. 273-290, OCT 1 2016. Web of Science Citations: 2.
CARVAJAL, XAVIER; PANTHEE, MAHENDRA; SCIALOM, MARCIA. On well-posedness of the third-order nonlinear Schrodinger equation with time-dependent coefficients. COMMUNICATIONS IN CONTEMPORARY MATHEMATICS, v. 17, n. 4 AUG 2015. Web of Science Citations: 0.
CARVAJAL, XAVIER; PANTHEE, MAHENDRA. ON ILL-POSEDNESS FOR THE GENERALIZED BBM EQUATION. DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS, v. 34, n. 11, p. 4565-4576, NOV 2014. Web of Science Citations: 4.
CARVAJAL, XAVIER; PANTHEE, MAHENDRA. On the well-posedness of higher order viscous Burgers' equations. Journal of Mathematical Analysis and Applications, v. 417, n. 1, p. 1-22, SEP 1 2014. Web of Science Citations: 1.
CARVAJAL, XAVIER; GAMBOA, PEDRO. Global well-posedness for the critical Schrodinger-Debye system. DYNAMICS OF PARTIAL DIFFERENTIAL EQUATIONS, v. 11, n. 3, p. 251-268, SEP 2014. Web of Science Citations: 3.

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