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Weipeng Zhu

Publications and source records attributed to Weipeng Zhu.

At least 19 recordsLinked to original sources
arXiv2026

$L^2({\mathbb R}) $-Unconditional well-posedness for low dispersion fractional KdV equations

We show that the $ L^2({\mathbb R}) $-unconditional well-posedness, that is well-known for the KdV equation, is shared by KdV type equations with weaker dispersion. This is despite the difference in the nature of these equations, which are quasilinear while KdV is semilinear. More precisely we prove that the low dispersion fractional KdV equation $$ \partial_t u -D_x^α\partial_x u +\partial_x(u^2)=0 $$ is unconditionally globally well-posed in $L^2({\mathbb R}) $ for $α\in ]\frac{55}{38},2] $. Our method of proof combined refined bilinear estimates with the energy method enhanced with Bourgain's type estimates developed in Molinet-Vento (2015).

math.AP↗
arXiv2026

Nonexistence of Hölder continuous solution for the Camassa-Holm equation in Besov spaces

In the paper, we show that the continuity of the solution can not be improved to the Hölder continuity. Precisely speaking, the solution of the Camassa-Holm equation belongs to $\mathcal{C}([0,T];B^s_{p,r})$ but not to $\mathcal{C}^α([0,T];B^s_{p,r})$ with any $α\in(0,1)$. To the best of our knowledge, our work is the first one addressing the issue on the failure of Hölder continuous in time of solution to the classical Camassa-Holm equation. As a by-product, we establish the ill-posedness for the Camassa-Holm equation in $B^s_{p,\infty}(\mathbb{R})$ with $s>\max\big\{1+1/p, 3/2\big\}$ with $p\in[1,\infty]$ by proving the solution map to the Camassa-Holm equation starting from $u_0$ is discontinuous at $t = 0$ in $B^s_{p,\infty}(\mathbb{R})$.

math.AP↗
arXiv2025

Ill-posedness issue on the Oldroyd-B model in the critical Besov spaces

It is proved in \cite[J. Funct. Anal., 2020]{AP} that the Cauchy problem for some Oldroyd-B model is well-posed in $\B^{d/p-1}_{p,1}(\R^d) \times \B^{d/p}_{p,1}(\R^d)$ with $1\leq p<2d$. In this paper, we prove that the Cauchy problem for the same Oldroyd-B model is ill-posed in $\B^{d/p-1}_{p,r}(\R^d) \times \B^{d/p}_{p,r}(\R^d)$ with $1\leq p\leq \infty$ and $1< r\leq\infty$ due to the lack of continuous dependence of the solution.

math.AP↗
arXiv2024

Global existence and blow-up for the Euler-Poincaré equations with a class of initial data

In this paper we investigate the Cauchy problem of d-dimensional Euler-Poincaré equations. By choosing a class of new and special initial data, we can transform this d-dimensional Euler-Poincaré equations into the Camassa-Holm type equation in the real line. We first obtain some global existence results and then present a new blow-up result to the system under some different assumptions on this special class of initial data.

math.AP↗
arXiv2024

Nowhere-uniform continuity of the solution map of the Camassa-Holm equation in Besov spaces

In the paper, we gave a strengthening of our previous work in [32] (J. Differ. Equ. 269 (2020)) and proved that the data-to-solution map for the Camassa-Holm equation is nowhere uniformly continuous in $B^s_{p,r}(\R)$ with $s>\max\{1+1/{p},3/2\}$ and $(p,r)\in [1,\infty]\times[1,\infty)$. The method applies also to the b-family of equations which contain the Camassa-Holm and Degasperis-Procesi equations.

math.AP↗
arXiv2024

Non-uniform dependence on initial data for the Camassa--Holm equation in Besov spaces: Revisited

In the paper, we revisit the uniform continuity properties of the data-to-solution map of the Camassa--Holm equation on the real-line case. We show that the data-to-solution map of the Camassa--Holm equation is not uniformly continuous on the initial data in Besov spaces $B_{p, r}^s(\mathbb{R})$ with $s>\frac{1}{2}$ and $1\leq p, r< \infty$, which improves the previous works [Himonas et al., Asian J. Math., 11 (2007)], [Li et al., J. Differ. Equ., 269 (2020)] and [Li et al., J. Math. Fluid Mech., 23 (2021)]. Furthermore, we present a strengthening of our previous work in [Li et al., J. Differ. Equ., 269 (2020)] and prove that the data-to-solution map for the Camassa--Holm equation is nowhere uniformly continuous in $B^s_{p,r}(\mathbb{R})$ with $s>\max\{1+1/{p},3/2\}$ and $(p,r)\in [1,\infty]\times[1,\infty)$. The method applies also to the b-family of equations which contain the Camassa--Holm and Degasperis--Procesi equations.

math.AP↗
arXiv2023

Non-uniform convergence of solution for the Camassa-Holm equation in the zero-filter limit

In the short note, we prove that given initial data $\mathcal{u}_0 \in \pmb{H}^s(\mathbb{R})$ with $s>\frac32$ and for some $T>0$, the solution of the Camassa-Holm equation does not converges uniformly with respect to the initial data in $\pmb{L}^\infty$ $(0,T;H^s(\mathbb{R}))$ to the inviscid Burgers equation as the filter parameter $α$ tends to zero. This is a supplement to our recent result on the zero-filter limit.

math.AP↗
arXiv2023

Anomalous Dissipation for the d-dimensional Navier-Stokes Equations

The purpose of this paper is to study the vanishing viscosity limit for the d-dimensional Navier--Stokes equations in the whole space: \begin{equation*} \begin{cases} \partial_tu^\varepsilon+u^\varepsilon\cdot \nabla u^\varepsilon-\varepsilonΔu^\varepsilon+\nabla p^\varepsilon=0,\\ \mathrm{div}\ u^\varepsilon=0. \end{cases} \end{equation*} We aim to presenting a simple rigorous examples of initial data which generates the corresponding solutions of the Navier--Stokes equations do exhibit anomalous dissipation. Precisely speaking, we show that there are (classical) solutions for which the dissipation rate of the kinetic energy is bounded away from zero.

math.AP↗
arXiv2023

Loss of Uniform Convergence for Solutions of the Navier--Stokes Equations in the Inviscid Limit

In this paper, we consider the inviscid limit problem to the higher dimensional incompressible Navier--Stokes equations in the whole space. It is shown in [Guo, Li, Yin: J. Funct. Anal., 276 (2019)] that given initial data $u_0\in B^{s}_{p,r}$ and for some $T>0$, the solutions of the Navier--Stokes equations converge strongly in $L^\infty_TB^{s}_{p,r}$ to the Euler equations as the viscosity parameter tends to zero. We furthermore prove the failure of the uniform (with respect to the initial data) $B^{s}_{p,r}$ convergence in the inviscid limit of a family of solutions of the Navier-Stokes equations towards a solution of the Euler equations.

math.AP↗
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↗
arXiv2023

Ill-posedness for a generalized Camassa-Holm equation with higher-order nonlinearity in the critical Besov space

In this paper, we prove that the Cauchy problem for a generalized Camassa-Holm equation with higher-order nonlinearity is ill-posed in the critical Besov space $B^1_{\infty,1}(\R)$. It is shown in (J. Differ. Equ., 327:127-144,2022) that the Camassa-Holm equation is ill-posed in $B^1_{\infty,1}(\R)$, here we turn our attention to a higher-order nonlinear generalization of Camassa-Holm equation proposed by Hakkaev and Kirchev (Commun Partial Differ Equ 30:761-781,2005). With newly constructed initial data, we get the norm inflation in the critical space $B^1_{\infty,1}(\R)$ which leads to ill-posedness.

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 two component Degasperis-Procesi equation in critical Besov space

In this paper, we study the Cauchy problem for the two component Degasperis-Procesi equation in critical Besov space $B^1_{\infty,1}(\mathbb R)$. By presenting a new construction of initial data, we proved the norm inflation of the corresponding solutions in $B^1_{\infty,1}(\mathbb R)$ and hence ill-posedness. This is quite different from the local well-posedness result for the Degasperis-Procesi equation in critical Besov space $B^1_{\infty,1}(\mathbb R)$ due to the coupled structure of density function.

math.AP↗