文献[J. Funct. Anal., 2020]已证明，某类Oldroyd-B模型的柯西问题在空间$\boldsymbol{B}^{d/p-1}_{p,1}(\boldsymbol{R}^d) \times \boldsymbol{B}^{d/p}_{p,1}(\boldsymbol{R}^d)$（其中$1 \leq p < 2d$）中是适定的。本文证明，同一Oldroyd-B模型的柯西问题在空间$\boldsymbol{B}^{d/p-1}_{p,r}(\boldsymbol{R}^d) \times \boldsymbol{B}^{d/p}_{p,r}(\boldsymbol{R}^d)$（其中$1 \leq p \leq \infty$且$1 < r \leq \infty$）中是不适定的，原因是解缺乏连续依赖性。
英文摘要：
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.
We prove the global well-posedness in the critical Besov spaces for the incompressible Oldroyd-B model without damping mechanism on the stress tensor in $\mathbb{R}^d$ for the small initial data. Our proof is based on the observation that the behaviors of Green's matrix to the system of $\big(u,(-Δ)^{-\frac12}\mathbb{P}\nabla\cdotτ\big)$ as well as the effects of $τ$ change from the low frequencies to the high frequencies and the construction of the appropriate energies in different frequencies.
This paper is dedicated to the Cauchy problem of the incompressible Oldroyd-B model with general coupling constant $\om\in (0,1)$. It is shown that this set of equations admits a unique global solution in a certain hybrid Besov spaces for small initial data in $\dot{H}^s\cap\dot{B}^{\fr{d}{2}}_{2,1}$ with $-\fr{d}{2}<s<\fr{d}{2}-1$. In particular, if $d\ge3,$ and taking $s=0$, then $\dot{H}^0\cap\dot{B}^{\fr{d}{2}}_{2,1}\approx B^{\fr{d}{2}}_{2,1}$. Since $B^{s}_{2,\infty}\hookrightarrow B^{\fr{d}{2}}_{2,1}, s>\fr{d}{2}$, this result extends the work by Chen and Miao [Nonlinear Anal.,{68}(2008…
This paper is dedicated to the global well-posedness issue of the compressible Oldroyd-B model in the whole space $\R^d$ with $d\ge2$. It is shown that this set of equations admits a unique global solution in a certain critical Besov space provided the initial data, but not necessarily the coupling parameter, is small enough. This result extends the work by Fang and the author [{J. Differential Equations}, {256}(2014), 2559--2602] to the non-small coupling parameter case.
By a new energy approach involved in the high frequencies and low frequencies decomposition in the Besov spaces, we obtain the optimal decay for the incompressible Oldroyd-B model without damping mechanism in $\mathbb{R}^n$ ($n\ge 2$). More precisely, let $(u,τ)$ be the global small solutions constructed in [18], we prove for any $(u_0,τ_0)\in{\dot{B}_{2,1}^{-s}}(\mathbb{R}^n)$ that \begin{eqnarray*} \big\|Λ^α(u,Λ^{-1}\mathbb{P}\mathrm{div}τ)\big\|_{L^q} \le C (1+t)^{-\frac n4-\frac {(α+s)q-n}{2q}}, \quadΛ\stackrel{\mathrm{def}}{=}\sqrt{-Δ}, \end{eqnarray*} with $\frac n2-1<s<\frac np, $ $2\le…
The Cauchy problem of the compressible Oldroyd-B model without damping mechanism in R^n$ with $n\ge2$ is considered. The lack of dissipation in density and stress tensor in the model is compensated by exploiting an intrinsic structure and introducing new quantities between density, velocity and stress tensor. Therefore, global solutions to the system with small initial data in critical Besov spaces are obtained. As a byproduct, optimal time decay rates of the solutions are derived by using an energy estimation argument. The results remain valid for the compressible viscoelastic system without …
In this paper, we prove that the 2D viscous shallow water equations is ill-posed in the critical Besov spaces $\B^{\frac2p-1}_{p,1}(\R^2)$ with $p>4$. Our proof mainly depends on the method introduced by the paper \cite{C-M-Z4}.
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.
In this paper, we first establish the local well-posedness (existence, uniqueness and continuous dependence) for the Fornberg-Whitham equation in both supercritical Besov spaces $B^s_{p,r},\ s>1+\frac{1}{p},\ 1\leq p,r\leq+\infty$ and critical Besov spaces $B^{1+\frac{1}{p}}_{p,1},\ 1\leq p<+\infty$, which improves the previous work \cite{y2,ho,ht}. Then, we prove the solution is not uniformly continuous dependence on the initial data in supercritical Besov spaces $B^s_{p,r},\ s>1+\frac{1}{p},\ 1\leq p\leq+\infty,\ 1\leq r<+\infty$ and critical Besov spaces $B^{1+\frac{1}{p}}_{p,1},\ 1\leq p<+…
In this paper, we present a construction of $u_{0}\in B^{1}_{\infty,1}$ and get the local ill-posedness for the rotation Camassa-Holm equation modelling the equatorial water waves with the weak Coriolis effect by proving the norm inflation.
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.
We give a construction of a divergence-free vector field $u_0 \in H^s \cap B^{-1}_{\infty,\infty}$, for all $s<1/2$, such that any Leray-Hopf solution to the Navier-Stokes equation starting from $u_0$ is discontinuous at $t=0$ in the metric of $B^{-1}_{\infty,\infty}$. For the Euler equation a similar result is proved in all Besov spaces $B^s_{r,\infty}$ where $s>0$ if $r>2$, and $s>n(2/r-1)$ if $1 \leq r \leq 2$.
We prove the ill-posedness of the 3-D baratropic Navier-Stokes equation for the initial density and velocity belonging to the critical Besov space $(\dot{B}^{\f 3p}_{p,1}+\barρ,\,\dot{B}^{\f 3p-1}_{p,1})$ for $p>6$ in the sense that a ``norm inflation" happens in finite time, here $\barρ$ is a positive constant. Our argument also shows that the compressible viscous heat-conductive flows is ill-posed for the initial density, velocity and temperature belonging to the critical Besov space $(\dot{B}^{\f 3p}_{p,1}+\barρ,\,\dot{B}^{\f 3p-1}_{p,1},\,\dot{B}^{\f 3p-2}_{p,1})$ for $p>3$. These results …
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-p…
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…
We prove the ill-posedness for the 3D incompressible inhomogeneous Navier-stokes equations in critical Besov space. In particular, a norm inflation happens in finite time with the initial data satisfying $$\|a_0\|_{\dot{B}_{p,1}^\frac{3}{p}}+\|u_0\|_{\dot{B}_{6,1}^{-\frac{1}{2}}}\le δ,\ p>6$$ or $$\|a_0\|_{\dot{B}_{6,1}^\frac{1}{2}}+\|u_0\|_{\dot{B}_{p,1}^{\frac{3}{p}-1}}\le δ,\ p>6.$$ To obtain the norm inflation, we construct a special class of initial data and introduce a modified pressure. Comparing with the classical Navier-Stokes equations in $L^\infty$ framework, we can obtain the ill-p…
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.
In the paper, by constructing a initial data $u_{0}\in B^σ_{p,\infty}$ with $σ-2>\max\{1+\frac 1 p, \frac 3 2\}$, we prove that the corresponding solution to the higher dimensional Camassa-Holm equations starting from $u_{0}$ is discontinuous at $t=0$ in the norm of $B^σ_{p,\infty}$, which implies that the ill-posedness for the higher dimensional Camassa-Holm equations in $B^σ_{p,\infty}$.
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}.
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 $B_{\infty,1}^{1}$.
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.
We study well-posedness and ill-posedness for Cauchy problem of the three-dimensional viscous primitive equations describing the large scale ocean and atmosphere dynamics. By using the Littlewood-Paley analysis technique, in particular Chemin-Lerner's localization method, we prove that the Cauchy problem with Prandtl number $P=1$ is locally well-posed in the Fourier-Besov spaces $[\dot{FB}^{2-\frac{3}{p}}_{p,r}(\mathbb{R}^3)]^4$ for $1<p\leq\infty,1\leq r<\infty$ and $[\dot{FB}^{-1}_{1,r}(\mathbb{R}^3)]^4$ for $1\leq r\leq 2$, and globally well-posed in these spaces when the initial data $(u_0…
We prove that the Navier-Stokes initial value problem is well-posed in the logrithmically refined Besov spaces when the second index is not less than certain critical value, and ill-posed in such spaces when the second index is less than this critical value. The well-posedness result is proved by using some sharp bilinear estimates obtained from some Hardy-Littlewood type inequalities. The ill-posedness assertion is proved by refining the arguments of Wang [18] and Yoneda [20].
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 t…
In this paper, we consider the Cauchy problem for a two-component Novikov system on the line. By specially constructed initial data $(ρ_0, u_0)$ in $B_{p, \infty}^{s-1}(\mathbb{R})\times B_{p, \infty}^s(\mathbb{R})$ with $s>\max\{2+\frac{1}{p}, \frac{5}{2}\}$ and $1\leq p \leq \infty$, we show that any energy bounded solution starting from $(ρ_0, u_0)$ does not converge back to $(ρ_0, u_0)$ in the metric of $B_{p, \infty}^{s-1}(\mathbb{R})\times B_{p, \infty}^s(\mathbb{R})$ as time goes to zero, thus results in discontinuity of the data-to-solution map and ill-posedness.
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 inflati…
We study the Cauchy problem of the 2D viscous shallow water equations in some critical Besov spaces $\dot B^{\frac{2}{p}}_{p,1}(\mathbb{R}^2)\times \dot B^{\frac{2}{p}-1}_{p,q}(\mathbb{R}^2)$. As is known, this system is locally well-posed for large initial data as well as globally well-posed for small initial data in $\dot B^{\frac{2}{p}}_{p,1}(\mathbb{R}^2)\times \dot B^{\frac{2}{p}-1}_{p,1}(\mathbb{R}^2)$ for $p<4$ and ill-posed in $\dot B^{\frac{2}{p}}_{p,1}(\mathbb{R}^2)\times \dot B^{\frac{2}{p}-1}_{p,1}(\mathbb{R}^2)$ for $p>4$. In this paper, we prove that this system is ill-posed for …
Strong ill-posedness of the 2D Boussinesq equations in supercritical Besov spaces
School of Mathematics and Computer Sciences, Gannan Normal University（赣南师范大学数学与计算机科学学院）；School of Mathematics and Statistics, Anhui Normal University（安徽师范大学数学与统计学院）
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 \…
We study the Cauchy problem for the incompressible Navier-Stokes equation \begin{align} u_t -Δu+u\cdot \nabla u +\nabla p=0, \ \ {\rm div} u=0, \ \ u(0,x)= δu_0. \label{NS} \end{align} For arbitrarily small $δ>0$, we show that the solution map $δu_0 \to u$ in critical Besov spaces $\dot B^{-1}_{\infty,q}$ ($\forall \ q\in [1,2]$) is discontinuous at origin. It is known that the Navier-Stokes equation is globally well-posed for small data in $BMO^{-1}$. Taking notice of the embedding $\dot B^{-1}_{\infty,q} \subset BMO^{-1}$ ($q\le 2$), we see that for sufficiently small $δ>0$, $u_0\in \dot B^{…
Ill-posedness and inviscid limit of the basic equations of fluid dynamics in Besov spaces
School of Mathematics and Statistics, Anhui Normal University（安徽师范大学数学与统计学院）；School of Mathematics and Computer Sciences, Gannan Normal University（赣南师范大学数学与计算机科学学院）；College of Artificial Intelligence, Henan Agricultural University（河南农业大学人工智能学院）
In this paper, we mainly consider the Cauchy problem of a weakly dissipative Camassa-Holm equation. We first establish the local well-posedness of equation in Besov spaces $B^{s}_{p,r}$ with $s>1+\frac 1 p$ and $s=1+\frac 1 p , r=1,p\in [1,\infty).$ Then, we prove the global existence for small data, and present two blow-up criteria. Finally, we get two blow-up results, which can be used in the proof of the ill-posedness in critical Besov spaces.
We introduce a notion of global weak solution to the Navier-Stokes equations in three dimensions with initial values in the critical homogeneous Besov spaces $\dot{B}^{-1+\frac{3}{p}}_{p,\infty}$, $p > 3$. These solutions satisfy a certain stability property with respect to the weak-$\ast$ convergence of initial conditions. To illustrate this property, we provide applications to blow-up criteria, minimal blow-up initial data, and forward self-similar solutions. Our proof relies on a new splitting result in homogeneous Besov spaces that may be of independent interest.
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 m…
We construct an example showing that the solution map of the Euler equations is not continuous in the Hölder space from $C^{1,α}$ to $L^\infty_tC^{1,α}_x$ for any $0<α<1$. On the other hand we show that it is continuous when restricted to the little Hölder subspace $c^{1,α}$. We apply the latter to prove an ill-posedness result for solutions of the vorticity equations in Besov spaces near the critical space $B^1_{2,1}$. As a consequence we show that a sequence of best constants of the Sobolev embedding theorem near the critical function space is not continuous.
The blow up phenomenon in the first step of the classical Picard's scheme was proved in this paper. For certain initial spaces, Bourgain-Pavlović and Yoneda proved the ill-posedness of the Navier-Stokes equations by showing the norm inflation in certain solution spaces. But Chemin and Gallagher said the space $\dot{B}^{-1,\infty}_{\infty}$ seems to be optimal for some solution spaces best chosen. In this paper, we consider more general initial spaces than Bourgain-Pavlović and Yoneda did and establish ill-posedness result independent of the choice of solution space. Our result is a complement …
Many questions related to well-posedness/ill-posedness in critical spaces for hydrodynamic equations have been open for many years. In this article we give a new approach to studying norm inflation (in some critical spaces) for a wide class of equations arising in hydrodynamics. As an application, we prove strong ill-posedness of the $n$-dimensional Euler equations in the class $C^1\cap L^2 (Ω)$ and also in $C^k \cap L^2(Ω)$ where $Ω$ can be the whole space, a smooth bounded domain, or the torus. We also apply our method to the Oldroyd B, surface quasi-geostrophic, and Boussinesq systems.
We show that the wave map equation in $\R^{1+1}$ is in general ill-posed in the critical space $\dot H^{1/2}$, and the Besov space $\dot B^{1/2,1}_2$. The problem is attributed to the bad behaviour of the one-dimensional bilinear expression $D^{-1}(f Dg)$ in these spaces.
For the famous Camassa-Holm equation, the well-posedness in $B^{1+\frac{1}{p}}_{p,1}(\mathbb{R})$ with $1\leq p\leq2$ and the ill-posedness in $B^{1+\frac{1}{p}}_{p,r}(\mathbb{R})$ with $1\leq p\leq+\infty,\ 1<r\leq+\infty$ had been studied in \cite{d1,d2,glmy}. That is to say, it left an open problem in the critical case $B^{1+\frac{1}{p}}_{p,1}(\mathbb{R})$ with $2<p\leq+\infty$ proposed by Danchin in \cite{d1,d2}. In this paper, we solve this problem. The main difficulty is to prove the uniqueness, which usually needs to use the Moser-type inequality, resulting in the index $p$ belongs to $…
Nonexistence of Hölder continuous solution for the Camassa-Holm equation in Besov spaces
School of Mathematics and Statistics, Anhui Normal University（安徽师范大学数学与统计学院）；School of Mathematics and Computer Sciences, Gannan Normal University（赣南师范大学数学与计算机科学学院）；School of Mathematics, Foshan University（佛山大学数学学院）
We show that the incompressible Euler equations on $\mathbb{R}^2$ are not locally well-posed in the sense of Hadamard in the Besov space $B^1_{\infty,1}$. Our approach relies on the technique of Lagrangian deformations of Bourgain and Li. We show that the assumption that the data-to-solution map is continuous in $B^1_{\infty,1}$ leads to a contradiction with a well-posedness result in $W^{1,p}$ of Kato and Ponce.
We prove norm inflation and hence ill-posedness for a class of shallow water wave equations, such as the Camassa-Holm equation, Degasperis-Procesi equation and Novikov equation etc., in the critical Sobolev space $H^{3/2}$ and even in the Besov space $B^{1+1/p}_{p,r}$ for $p\in [1,\infty], r\in (1,\infty]$. Our results cover both real-line and torus cases (only real-line case for Novikov), solving an open problem left in the previous works (\cite{Danchin2,Byers,HHK}).
We study a class of higher-order KdV equations. We show that the associated initial value problem is well posed in weighted Besov and Sobolev spaces for small initial data. We also prove ill-posedness results when in H^s(\R), for any real s.
This work is concerned about the Cauchy problem for the following generalized KdV- Burgers equation \begin{equation*} \left\{\begin{array}{l} \partial_tu+\partial_x^3u+L_pu+u\partial_xu=0, u(0,\,x)=u_0(x). \end{array} \right. \end{equation*} where $L_p$ is a dissipative multiplicator operator. Using Besov-Bourgain Spaces, we establish a bilinear estimate and following the framework developed in Molinet, L. & Vento, S. (2011) we prove sharp global well-posedness in the Sobolev spaces $H^{-p/2}(I\!\!R)$ and sharp ill-posedness in $H^s(I\!\!R)$ when $s<-p/2$ with $p \geq 2$.
