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The Cauchy Problem for the Shallow Water Type Equations in Low Regularity Spaces on the Circle


Sep 14, 2017 - 12:45pm to 1:45pm


Rettaliata Engineering Center, Room 106


Wei Yan
School of Mathematics and Information Science, Henan Normal University


In this paper, we investigate the Cauchy problem for the shallow water type equation $$u_{t} + u_{xxx} + \frac{1}{2}(u^{2})_{x}+\partial_{x}(1−\partial_{x} ^{2}) ^{-1}\left(u^{2}+\frac{1}{2}(u^{2})_{x}\right)=0,\quad x\in T=R/2πλ$$ with low regularity data in the periodic settings and \(\lambda≥\). We prove that the bilinear estimate in \(X_{s,b}\) with \(s<1/2\) is invalid. We also prove that the problem is locally wellposed in \(H^{s}(T)\) with \(1/6<s<1/2\) for small initial data. The result of this paper improves the result of case \(j=1\) of Himonas and Misiolek (Communications in Partial Differential Equations, 23 (1998), 123-139.). The new ingredients are some new function spaces and some new Strichartz estimates.

Event Type: 

Department of Applied Mathematics - Stochastic & Multiscale Modeling and Computation Seminar