2017 Fiscal Year Research-status Report
Long time behavior of solutions of nonlinear dispersive equations
| Project/Area Number |
15K17570
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| Research Institution | Nagoya University |
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Principal Investigator |
Roy Tristan 名古屋大学, 高等研究院(多元), 特任助教 (80751073)
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| Project Period (FY) |
2015-04-01 – 2019-03-31
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| Keywords | dispersive equations / global well-posedness / loglog supercritical / local well-posedness / third-order nonlinearity |
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| Outline of Annual Research Achievements |
I managed to prove no formation of singularities and linear asymptotic behavior of nonsmooth solutions of a loglog supercritical Schrodinger equation with a repulsive nonlinearity and with radial data. My strategy relies upon a long-time estimate (namely a Morawetz-type estimate), concentration techniques, and Jensen-type inequalities to control the nonlinearity when I use the concentration techniques. This paper will appear in International Mathematics Research Notices.
I managed to do the same for a loglog supercritical Schrodinger equation with an attractive nonlinearity and with radial data. My strategy relies upon a long-time estimate (namely a virial-type estimate), concentration techniques, and Jensen-type inequalities to control the nonlinearity when I prove the virial-type estimate. That paper is currently under submission.
I am finishing my common project with Prof. Kotaro Tsugawa regarding the construction of local solutions of nonlinear dispersive equations with a third-order nonlinearity. Our strategy relies upon a more general approach that combines standard energy estimates with the gauge transform introduced by Hayashi and Ozawa. It is applied to fully nonlinear third-order nonlienarities.
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| Current Status of Research Progress |
Current Status of Research Progress
1: Research has progressed more than it was originally planned.
Reason
I have a good chance to believe that my common project with Prof Kotaro Tsugawa will succeed. The approach we have developed allows us to extend the results to more general third-order nonlinearities. I do believe that the success of this approach is based upon a fruitful collaboration that led us to ask the right questions and find the right answers.
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| Strategy for Future Research Activity |
I am planning to finish my common project with Prof Kotaro Tsugawa. I am also trying to engage in collaborative work with other japananese mathematicians who work in my field of research, i.e the field of nonlinear dispersive equations. In particular I am interested in studying the asymptotic behavior of solutions of more complicated models (such as damped wave equations or equations with a large power or a small power). I am also trying to extend some results regarding the non-formation of singularities of solutions of nonlinear dispersive equations with oscillatory data.
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| Causes of Carryover |
K. Tsugawa and I have constructed local solutions of dispersive equations with a third-order polynomial nonlinearity. We believe that we can extend our result to a general third-order nonlinearity.
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