arXiv subjects
Publications and source records attributed to Weipeng Zhu.
In this paper, we investigate the continuous dependence on initial data of solutions to the Euler-Poincaré system. By constructing a sequence approximate solutions and calculating the error terms, we show that the data-to-solution map is not uniformly continuous in Sobolev space $H^s(\mathbb{R}^d)$ for $s>1+\frac d2$.
In this paper, we establish the continuous dependence for the non-resistive MHD equations in Sobolev spaces. Our obtained result fills considerably the recent result [C. Fefferman, D. McCormick, J. Robinson and J. Rodrigo, Higher order commutator estimates and local existence for the non-resistive MHD equations and related models, J. Funct. Anal. 267 (2014), 1035--1056] for the continuous dependence on initial data. We also show that this result is optimal. More precisely, we proved that the data-to-solution map is not uniform continuous dependence in Sobolev spaces.
In this paper we consider the Cauchy problem to the Korteweg system with the general pressure in dimension $d\geq2$, and establish the global well-posedness of strong solution for the small initial data in $L^p$ type critical Besov spaces by using the Friedrich method and compactness arguments. Furthermore, we also obtain the optimal decay rate for the Korteweg system in $L^2$ type Besov spaces.
We study the Cauchy problem of the incompressible micropolar fluid system in $\mathbb{R}^{3}$. In a recent work of the first author and Jihong Zhao \cite{ZhuZ18}, it is proved that the Cauchy problem of the incompressible micropolar fluid system is locally well-posed in the Fourier--Besov spaces $\F^{2-\frac{3}{p}}_{p,r}$ for $1<p\leq\infty$ and $1\leq r<\infty$, and globally well-posed in these spaces with small initial data. In this work we consider the critical case $p=1$. We show that this problem is locally well-posed in $\F^{-1}_{1,r}$ for $1\leq r\leq 2$, and is globally well-posed in these spaces with small initial data. Furthermore, we prove that such problem is ill-posed in $\F^{-1}_{1,r}$ for $2< r\leq \infty$, which implies that the function space $\F^{-1}_{1,2}$ is sharp for well-posedness. In addition, using a similar argument we also prove that this problem is ill-posed in the Besov space $\B^{-1}_{\infty,r}$ with $2<r\leq\infty$.
In this paper we consider the global well-posedness of compressible magnetohydrodynamic system in $\R^d$ with $d\geq2$, in the framework of the critical Besov spaces. We can show that if the initial data, the shear viscosity and the magnetic diffusion coefficient are small comparing with the volume viscosity, then compressible magnetohydrodynamic system has a unique global solution.
In the paper, we consider the Cauchy problem for a generalized Degasperis-Procesi equation. We prove that the data-to-solution map is not uniformly continuous.
In this paper, we study the Cauchy problem for the following Hamilton-Jacobi equation \bbal\bca \pa_tu-\De u=|\na u|^2,\quad t>0, \ x\in \R^d,\\ u(0,x)=u_0, \quad \quad x\in \R^d. \eca\end{align*} We show that the solution map in Besov spaces $B^0_{\infty,q}(\R^d),1\leq q\leq \infty$ is discontinuous at origin. That is, we can construct a sequence initial data $\{u^N_0\}$ satisfying $||u^N_0||_{B^0_{\infty,q}(\R^d)}\rightarrow 0, \ N\rightarrow \infty$ such that the corresponding solution $\{u^N\}$ with $u^N(0)=u^N_0$ satisfies \bbal ||u^N||_{L^\infty_T(B^0_{\infty,q}(\R^d))}\geq c_0, \qquad \forall \ T>0, \quad N\gg 1, \end{align*} with a constant $c_0>0$ independent of $N$.