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

When: 

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

Where: 

Rettaliata Engineering Center, Room 106

Speaker: 

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

Description: 

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