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Long Time Existence via Modified Energy Method
Student-Run Analysis & PDE| Speaker: | Kyle Chickering, UC Davis |
| Location: | 1147 MSB |
| Start time: | Thu, Feb 6 2020, 1:10PM |
We discuss Hunter, Ifram, Tataru, and Wong's recent work \cite{hunter_2015} showing cubically nonlinear existence for the quadratically nonlinear Burger Hilbert equation. They present a straightforward proof which relies on insight about the structure of a normal form transformation to obtain long time existence. This method is significantly more straightforward than the previous proof of the same result by Hunter and Ifram \cite{hunter_2012}.
We explore the difficulties obtaining cubically nonlinear existence for this equation, including difficulties in estimating the transformed equation. We show how the modified energy method overcomes these difficulties by comparing the estimates of the normal form transformation to the original equation's energy. Finally, we discuss applications to other nonlinear problems.