# 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