arXiv subjects

Yanghai Yu

Publications and source records attributed to Yanghai Yu.

At least 19 recordsLinked to original sources

arXiv2026

Strong ill-posedness of the 2D Boussinesq equations in supercritical Besov spaces

In this paper, we prove that the 2D Boussinesq equations are strongly ill-posed in the supercritical Besov spaces $B^s_{p,q}$ and Sobolev spaces $W^{s,p}$ with $(p,q)\in(1,\infty)\times [1,\infty]$ and $s\in(0,1+2/p)$ by constructing an initial data with arbitrarily small norm for which the solution of the system exhibits norm inflation almost instantaneously. As a further application, we prove the instability of perturbations near the hydrostatic equilibrium for the 2D Boussinesq equations in the same $B^s_{p,q}$ and $W^{s,p}$.

math.AP↗

arXiv2026

Norm inflation and low-regularity ill-posedness for the rod equation

In this paper, we consider the Cauchy problem for the rod equation in the line. By constructing an explicit smooth initial data, we present a new method to prove that this problem is ill-posed in $H^s(\R)$ with $1< s<3/2$ in the sense of {\it norm inflation}, i.e., an initial data is smooth and arbitrarily small in $H^s(\R)$ with $1< s<3/2$, but the solution becomes arbitrarily large in the Sobolev space after an arbitrarily short time.

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arXiv2026

A new proof of unboundedness of Riesz operator in $L^\infty$ and applications to mild ill-posedness in $W^{1,\infty}$ of the Euler type equations

In this paper, we first present a new and simple proof of unboundedness of Riesz operator in $L^\infty$ and then establish the mild ill-posedness in $W^{1,\infty}$ of 3D rotating Euler equations and 2D Euler equations with partial damping. To the best of our knowledge, our work is the first one addressing the ill-posedness issue on the rotating Euler equations in $W^{1,\infty}$ without the vorticity formulation. As a further application, we prove the instability of perturbations for the 2D surface quasi-geostrophic equation and porous medium system in $W^{1,\infty}$.

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arXiv2025

Ill-posedness in $B^s_{p,\infty}$ of the Euler equations: Non-continuous dependence

In this paper, we solve an open problem left in the monographs \cite[Bahouri-Chemin-Danchin, (2011)]{BCD}. Precisely speaking, it was obtained in \cite[Theorem 7.1 on pp293, (2011)]{BCD} the existence and uniqueness of $B^s_{p,\infty}$ solution for the Euler equations. We furthermore prove that the solution map of the Euler equation is not continuous in the Besov spaces from $B^s_{p,\infty}$ to $L_T^\infty B^s_{p,\infty}$ for $s>1+d/p$ with $1\leq p\leq \infty$ and in the H\"{o}lder spaces from $C^{k,\alpha}$ to $L_T^\infty C^{k,\alpha}$ with $k\in \mathbb{N}^+$ and $\alpha\in(0,1)$, which later covers particularly the ill-posedness of $C^{1,\alpha}$ solution in \cite[Trans. Amer. Math. Soc., (2018)]{MYtams}. Beyond purely technical aspects on the choice of initial data, a remarkable novelty of the proof is the construction of an approximate solution to the Burgers equation.

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arXiv2025

Ill-posedness and inviscid limit of basic equations of fluid dynamics in Besov spaces

In this paper, we consider the Cauchy problem to the basic equations of fluid dynamics on the torus. Firstly, we construct a new initial data and provide a simple proof on the ill-posedness of $B^s_{p,\infty}$ solution of the Euler equations and the surface quasi-geostrophic equation, which covers the results obtained by Cheskidov-Shvydkoy \cite{CS} and Misio{\l}ek-Yoneda \cite{MY}. Secondly, we prove the failure of the $B^s_{p,\infty}$-convergence in the inviscid limit for both the Navier-Stokes equations and the surface quasi-geostrophic equation.

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arXiv2025

Non-convergence of the Navier-Stokes equations toward the Euler equations in the endpoint Besov spaces

In this paper, we consider the inviscid limit problem to the higher dimensional incompressible Navier-Stokes equations in the whole space. It was proved in \cite[J. Funct. Anal., 276 (2019)]{GZ} that given initial data $u_0\in B^{s}_{p,r}$ with $1\leq r<\infty$, the solution of the Navier-Stokes equations converges strongly in $B^{s}_{p,r}$ to the solution of the Euler equations as the viscosity parameter tends to zero. In the case when $r=\infty$, we prove the failure of the $B^{s}_{p,\infty}$-convergence of the Navier-Stokes equations toward the Euler equations in the inviscid limit.

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arXiv2025

On the continuous properties for the 3D incompressible rotating Euler equations

In this paper, we consider the Cauchy problem for the 3D Euler equations with the Coriolis force in the whole space. We first establish the local-in-time existence and uniqueness of solution to this system in $B^s_{p,r}(\R^3)$. Then we prove that the Cauchy problem is ill-posed in two different sense: (1) the solution of this system is not uniformly continuous dependence on the initial data in the same Besov spaces, which extends the recent work of Himonas-Misio{\l}ek \cite[Comm. Math. Phys., 296, 2010]{HM1} to the more general framework of Besov spaces; (2) the solution of this system cannot be H\"{o}lder continuous in time variable in the same Besov spaces. In particular, the solution of the system is discontinuous in the weaker Besov spaces at time zero. To the best of our knowledge, our work is the first one addressing the issue on the failure of H\"{o}lder continuous in time of solution to the classical Euler equations with(out) the Coriolis force.

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arXiv2024

Non-uniform dependence on initial data for the generalized Camassa-Holm equation in $C^1$

It is shown in \cite[Adv. Differ. Equ(2017)]{HT} that the Cauchy problem for the generalized Camassa-Holm equation is well-posed in $C^1$ and the data-to-solution map is H\"{o}lder continuous from $C^\alpha$ to $\mathcal{C}([0,T];C^\alpha)$ with $\alpha\in[0,1)$. In this paper, we further show that the data-to-solution map of the generalized Camassa-Holm equation is not uniformly continuous on the initial data in $C^1$. In particular, our result also can be a complement of previous work on the classical Camassa-Holm equation in \cite[Geom. Funct. Anal(2002)]{G02}.

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arXiv2024

Global existence and blow-up for the Euler-Poincar\'{e} equations with a class of initial data

In this paper we investigate the Cauchy problem of d-dimensional Euler-Poincar\'{e} equations. By choosing a class of new and special initial data, we can transform this d-dimensional Euler-Poincar\'{e} 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.

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arXiv2024

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.

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arXiv2024

Nonexistence of H\"{o}lder 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 H\"{o}lder continuity. Precisely speaking, the solution of the Camassa-Holm equation belongs to $\mathcal{C}([0,T];B^s_{p,r})$ but not to $\mathcal{C}^\alpha([0,T];B^s_{p,r})$ with any $\alpha\in(0,1)$. To the best of our knowledge, our work is the first one addressing the issue on the failure of H\"{o}lder 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 $B^s_{p,\infty}(\mathbb{R})$.

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arXiv2024

On the ill-posedness for the Navier--Stokes equations in the weakest Besov spaces

It is proved in \cite{IO21} that the Cauchy problem for the full compressible Navier--Stokes equations of the ideal gas is ill-posed in $\dot{B}_{p, q}^{2 / p}(\mathbb{R}^2) \times \dot{B}_{p, q}^{2 / p-1}(\mathbb{R}^2) \times \dot{B}_{p, q}^{2 / p-2}(\mathbb{R}^2) $ with $1\leq p\leq \infty$ and $1\leq q<\infty$. In this paper, we aim to solve the end-point case left in \cite{IO21} and prove that the Cauchy problem is ill-posed in $\dot{B}_{p, \infty}^{d / p}(\mathbb{R}^d) \times \dot{B}_{p, \infty}^{d / p-1}(\mathbb{R}^d) \times \dot{B}_{p, \infty}^{d / p-2}(\mathbb{R}^d)$ with $1\leq p\leq\infty$ by constructing a sequence of initial data which shows that the solution map is discontinuous at zero. As a by-product, we demonstrate that the incompressible Navier--Stokes equations is also ill-posed in $\dot{B}_{p,\infty}^{d/p-1}(\mathbb{R}^d)$, which is an interesting open problem in itself.

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arXiv2023

A remark on the vanishing diffusivity limit of the Keller-Segel equations in Besov spaces

It is shown in \cite[J. Differ. Equ., (2022)]{22jde} that given initial data $u_0\in B^{s}_{p,r}$ and for some $T>0$, the solutions of the parabolic-type Keller-Segel equations converge strongly in $L^\infty_TB^{s}_{p,r}$ to the hyperbolic Keller-Segel equations as the diffusivity parameter $\epsilon$ tends to zero. In this paper, we furthermore prove this solution maps do not converge uniformly with respect to the initial data $u_0$ as $\epsilon\to0$ in the same topology of Besov spaces.

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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 $\alpha$ tends to zero. This is a supplement to our recent result on the zero-filter limit.

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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-\varepsilon\Delta u^\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.

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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 $B_{\infty,1}^{1}$.

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