arXiv subjects

Jianzhong Lu

Publications and source records attributed to Jianzhong Lu.

3 recordsLinked to original sources

arXiv2025

A remark on the zero-filter limit for the Camassa-Holm equation in $B^s_{2,\infty}(\R)$

This paper investigates the zero-filter limit problem associated with the Camassa-Holm equation. In the work cited as \cite{C.L.L.W.L}, it was established that, under the hypothesis of initial data $u_0\in B^s_{2,r}(\R)$ with $s>\frac32$ and $1\leq r<\infty$, the solutions $\mathbf{S}_{t}^{\mathbfα}(u_0)$ of the Camassa-Holm equation exhibit convergence in the $L^\infty_T(B^s_{2,r})$ norm to the unique solution of the Burgers equation as $α\rightarrow 0$. Contrary to this result, the present study demonstrates that for initial data $u_0\in B^s_{2,\infty}(\R)$ the solutions of the Camassa-Holm equation fail to converge strongly in the $L^\infty_T(B^s_{2,\infty})$ norm to the Burgers equation as $α\rightarrow 0$.

math.AP↗

arXiv2024

Well-posedness and no-uniform dependence for the Euler-Poincaré equations in Triebel-Lizorkin spaces

In this paper, we study the Cauchy problem of the Euler-Poincaré equations in $\R^d$ with initial data belonging to the Triebel-Lizorkin spaces. We prove the local-in-time unique existence of solutions to the Euler-Poincaré equations in $F^s_{p,r}(\R^d)$. Furthermore, we obtain that the data-to-solution of this equation is continuous but not uniformly continuous in these spaces.

math.AP↗

arXiv2023

Zero-filter limit issue for the Camassa-Holm equation in Besov spaces

In this paper, we focus on zero-filter limit problem for the Camassa-Holm equation in the more general Besov spaces. We prove that the solution of the Camassa-Holm equation converges strongly in $L^\infty(0,T;B^s_{2,r}(\R))$ to the inviscid Burgers equation as the filter parameter $α$ tends to zero with the given initial data $u_0\in B^s_{2,r}(\R)$. Moreover, we also show that the zero-filter limit for the Camassa-Holm equation does not converges uniformly with respect to the initial data in $B^s_{2,r}(\R)$.

math.AP↗