Searcharxiv⌕ Search

arXiv subjects

Yanghai Yu

Publications and source records attributed to Yanghai Yu.

At least 37 records · Page 2Linked to original sources
arXiv2023

On the well-posedness and non-uniform continuous dependence for the Novikov equation in the Triebel-Lizorkin spaces

In this paper we study the Cauchy problem of the Novikov equation in $\mathbb{R}$ for initial data belonging to the Triebel-Lizorkin spaces, i.e, $u_0\in F^{s}_{p,r}$ with $1< p, r<\infty$ and $s>\max\{\frac32,1+\frac1p\}$. We prove local-in-time unique existence of solution to the Novikov equation in $F^{s}_{p,r}$. Furthermore, we obtain that the data-to-solution of this equation is continuous but not uniformly continuous in the same space.

math.AP↗
arXiv2023

Ill-posedness for the periodic Camassa--Holm type equations in the end-point critical Besov space $B^{1}_{\infty,1}$

For the real-line case, it is shown that both the Camassa--Holm \cite{Guo} and Novikov equations \cite{Li-arx} are ill-posed in $B_{\infty,1}^{1}$. In this paper, by presenting a new construction of initial data which leads to the norm inflation phenomena, we prove that both the periodic Camassa--Holm and Novikov equations are also ill-posed in $B_{\infty,1}^{1}$.

math.AP↗
arXiv2022

Ill-posedness of the hyperbolic Keller-Segel model in Besov spaces

In this paper, we give a new construction of $u_0\in B^σ_{p,\infty}$ such that the corresponding solution to the hyperbolic Keller-Segel model starting from $u_0$ is discontinuous at $t = 0$ in the metric of $B^σ_{p,\infty}(\R^d)$ with $d\geq1$ and $1\leq p\leq\infty$, which implies the ill-posedness for this equation in $B^σ_{p,\infty}$. Our result generalizes the recent work in \cite{Zhang01} (J. Differ. Equ. 334 (2022)) where the case $d=1$ and $p=2$ was considered.

math.AP↗
arXiv2022

Ill-posedness issue on a multidimensional chemotaxis equations in the critical Besov spaces

In this paper, we aim to solving the open question left in [Nie, Yuan: Nonlinear Anal 196 (2020); J. Math. Anal. Appl 505 (2022)) and Xiao, Fei: J. Math. Anal. Appl 514 (2022)]. We prove that a multidimensional chemotaxis system is ill-posedness in $\dot{B}_{2d, r}^{-\frac{3}{2}} \times\big(\dot{B}_{2d, r}^{-\frac{1}{2}}\big)^{d}$ when $1\leq r<d$ due to the lack of continuity of the solution.

math.AP↗
arXiv2022

Ill-posedness of the Novikov equation in the critical Besov space $B^{1}_{\infty,1}(\mathbb{R})$

It is shown that both the Camassa-Holm and Novikov equations are ill-posed in $B_{p,r}^{1+1/p}(\mathbb{R})$ with $(p,r)\in[1,\infty]\times(1,\infty]$ in \cite{Guo2019} and well-posed in $B_{p,1}^{1+1/p}(\mathbb{R})$ with $p\in[1,\infty)$ in \cite{Ye}. Recently, the ill-posedness for the Camassa-Holm equation in $B^{1}_{\infty,1}(\mathbb{R})$ has been proved in \cite{Guo}. In this paper, we shall solve the only left an endpoint case $r=1$ for the Novikov equation. More precisely, we prove the ill-posedness for the Novikov equation in $B^{1}_{\infty,1}(\mathbb{R})$ by exhibiting the norm inflation phenomena.

math.AP↗
arXiv2022

Norm inflation and ill-posedness for the Fornberg-Whitham equation

In this paper, we prove that the Cauchy problem for the Fornberg-Whitham equation is not locally well-posed in $B^s_{p,r}(\R)$ with $(s,p,r)\in (1,1+\frac1p)\times[2,\infty)\times [1,\infty]$ or $(s,p,r)\in \{1\}\times[2,\infty)\times [1,2]$ by showing norm inflation phenomena of the solution for some special initial data.

math.AP↗
arXiv2022

Ill-posedness for the stationary Navier-Stokes equations in critical Besov spaces

This paper presents some progress toward an open question which proposed by Tsurumi (Arch. Ration. Mech. Anal. 234:2, 2019): whether or not the stationary Navier-Stokes equations in $\R^d$ is well-posed from $\dot{B}_{p, q}^{-2}$ to $\mathbb{P} \dot{B}_{p, q}^{0}$ with $p=d$ and $1 \leq q \leq 2$. In this paper, we prove that for the case $1\leq q<\frac d2$ with $d\geq4$ the stationary Navier-Stokes equations is ill-posed from $\dot{B}_{d, q}^{-2}(\R^d)$ to $\mathbb{P} \dot{B}_{d, q}^{0}(\R^d)$ by showing that a sequence of external forces is constructed to show discontinuity of the solution map at zero. Indeed in such case of $q$, there exists a sequence of external forces which converges to zero in $\dot{B}_{d, q}^{-2}$ and yields a sequence of solutions which does not converge to zero in $\dot{B}_{d, q}^{0}$. In particular, we also prove that the stationary Navier-Stokes equations is well-posed from $\dot{B}_{d, 2}^{-2}(\R^d)$ to $\mathbb{P} \dot{B}_{d, 2}^{0}(\R^d)$ with $d=3,4$. Based on these two cases, we demonstrate that the above open question for the dimension $d\geq4$ has been solved completely.

math.AP↗
arXiv2022

Ill-posedness for the Euler equations in Besov spaces

In the paper, we consider the Cauchy problem to the Euler equations in $\mathbb{R}^d$ with $d\geq2$. We construct an initial data $u_0\in B^σ_{p,\infty}$ showing that the corresponding solution map of the Euler equations starting from $u_0$ is discontinuous at $t = 0$ in the metric of $B^σ_{p,\infty}$, which implies the ill-posedness for this equation in $B^σ_{p,\infty}$. We generalize the periodic result of Cheskidov and Shvydkoy \cite{Cheskidov}.

math.AP↗
arXiv2022

Ill-posedness for the Camassa-Holm equation in $B_{p,1}^{1}\cap C^{0,1}$

In this paper, we study the Cauchy problem for the Camassa-Holm equation on the real line. By presenting a new construction of initial data, we show that the solution map in the smaller space $B_{p,1}^{1}\cap C^{0,1}$ with $p\in(2,\infty]$ is discontinuous at origin. More precisely, $u_0\in B_{p,1}^{1}\cap C^{0,1}$ can guarantee that the Camassa-Holm equation has a unique local solution in $W^{1,p}\cap C^{0,1}$, however, this solution is instable and can have an inflation in $B_{p,1}^{1}\cap C^{0,1}$ for certain initial data.

math.AP↗
arXiv2021

Sharp ill-posedness for the generalized Camassa-Holm equation in Besov spaces

In this paper, we consider the Cauchy problem for the generalized Camassa-Holm equation that includes the Camassa-Holm as well as the Novikov equation on the line. We present a new and unified method to prove the sharp ill-posedness for the generalized Camassa-Holm equation in $B^s_{p,\infty}$ with $s>\max\{1+1/p, 3/2\}$ and $1\leq p\leq\infty$ in the sense that the solution map to this equation starting from $u_0$ is discontinuous at $t = 0$ in the metric of $B^s_{p,\infty}$. Our results cover and improve the previous work given in [J. Li, Y. Yu, W. Zhu, Ill-posedness for the Camassa-Holm and related equations in Besov spaces, J. Differential Equations, 306 (2022), 403--417], solving an open problem left in [J. Li, Y. Yu, W. Zhu, Ill-posedness for the Camassa-Holm and related equations in Besov spaces, J. Differential Equations, 306 (2022), 403--417].

math.AP↗
arXiv2021

Non-uniform dependence on initial data for the generalized Camassa-Holm-Novikov equation in Besov space

Considered in this paper is the generalized Camassa-Holm-Novikov equation with high order nonlinearity, which unifies the Camassa-Holm and Novikov equations as special cases. We show that the solution map of generalized Camassa-Holm-Novikov equation is not uniformly continuous on the initial data in Besov spaces $B_{p, r}^s(\mathbb{R})$ with $s>\max\{1+\frac{1}{p}, \frac{3}{2}\}$, $1\leq p, r< \infty$ as well as in critical space $B_{2, 1}^{\frac{3}{2}}(\mathbb{R}).$ Our result covers and improves the previous work given by Li et al. \cite{Li 2020, 1Li 2020, Li 2021}(J. Differ. Equ. 269 (2020) 8686-8700; J. Math. Fluid Mech. 22 (2020) 4:50; J. Math. Fluid Mech., (2021) 23:36).

math.AP↗
arXiv2021

Ill-posedness for the Camassa-Holm and related equations in Besov spaces

In this paper, we give a construction of $u_0\in B^σ_{p,\infty}$ such that corresponding solution to the Camassa-Holm equation starting from $u_0$ is discontinuous at $t = 0$ in the metric of $B^σ_{p,\infty}$, which implies the ill-posedness for this equation in $B^σ_{p,\infty}$. We also apply our method to the b-equation and Novikov equation.

math.AP↗
arXiv2020

Non-uniform dependence on initial data for the 2D viscous shallow water equations

The failure of uniform dependence on the data is an interesting property of classical solution for a hyperbolic system. In this paper, we consider the solution map of the Cauchy problem to the 2D viscous shallow water equations which is a hyperbolic-parabolic system. We prove that the solution map of this problem is not uniformly continuous in Sobolev spaces $H^s\times H^{s}$ for $s>2$.

math.AP↗