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Jinlu Li

Publications and source records attributed to Jinlu Li.

At least 73 records · Page 4Linked to original sources
arXiv2022

Generalized projections on general Banach spaces

In general Banach spaces, the metric projection map lacks the powerful properties it enjoys in Hilbert spaces. There are a few generalized projections that have been proposed in order to resolve many of the deficiencies of the metric projection. However, such notions are predominantly studied in Banach spaces with rich topological structures, such as uniformly convex Banach spaces. In this paper, we investigate two notions of generalized projection in general Banach spaces. Various examples are provided to demonstrate the proposed notions and the loss of structure in the generalized projections after migrating from specially structured Banach spaces to general Banach spaces. Connections between the generalized projection and the metric projection are thoroughly explored.

math.FA↗
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↗
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↗
arXiv2022

The Fixed Point Property of Quasi-Point-Separable Topological Vector Spaces

In this paper, we introduce the concept of quasi-point-separable topological vector spaces, which has the following important properties: 1.In general, the conditions for a topological vector space to be quasi-point-separable is not very difficult to check; 2.The class of quasi-point-separable topological vector spaces is very large that includes locally convex topological vector spaces and pseudonorm adjoint topological vector spaces as special cases; 3.Every quasi-point-separable Housdorrf topological vector space has the fixed point property (that is, every continuous self-mapping on any given nonempty closed and convex subset has a fixed point), which is the result of the main theorem of this paper (Theorem 4.1). Furthermore, we provide some concrete examples of quasi-point-separable topological vector spaces, which are not locally convex. It follows that the main theorem of this paper is a proper extension of Tychonoffs fixed point theorem on locally convex topological vector spaces.

math.FA↗
arXiv2021

Embedding of partially ordered topological spaces in Fell topological hyperspaces

In this paper, we prove a theorem about embedding of some partially ordered topological spaces in topological hyperspaces equipped with Fell topology. Then we give some examples to show that the map defining the embedding may not be continuous with respect to Vietoris topology or Hausdorff topology equipped on the hyperspaces.

math.GN↗
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

Global large, smooth solutions of the 2D surface quasi-geostrophic equations

In this paper, we prove the global regularity of smooth solutions to 2D surface quasi-geostrophic (SQG) equations with super-critical dissipation for a class of large initial data, where the velocity and temperature can be arbitrarily large in spaces $L^\infty(\mathbb{R}^2)$ and $H^3(\mathbb{R}^2)$. This result could be seen as an improvement work of Liu-Pan-Wu \cite{Liu}, for it's without any smallness hypothesis of the $L^\infty(\mathbb{R}^2)$ norm of the initial data.

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

An Extension of Tychonoffs Fixed Point Theorem to Quasi-point Separable Topological Vector Spaces

In this paper, we introduce the concepts of m-quasiconvex, originally m-quasiconvex,and generalized m-quasiconvex functionals on topological vector spaces. Then we extend the concept of point separable topological vector spaces (by the topological dual spaces) to quasi-point separable topological vector spaces by families of generalized m-quasiconvex functionals. We show that every pseudonorm adjoint topological vector space, which includes locally convex topological vector spaces as special cases, is quasi-point separable. By the Fan-KKM theorem, we prove a fixed point theorem on quasi-point separable topological vector spaces. It is an extension of the fixed point theorem on pseudonorm adjoint topological vector spaces proved in [8]. Therefore, it is a properextension of Tychonoffs fixed point theorem on locally convex topological vector spaces, which are demonstrated by some examples.

math.FA↗
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↗
arXiv2020

Non-uniform continuity of the generalized Camassa-Holm equation in Besov spaces

In this paper, we consider the Cauchy problem for the generalized Camassa-Holm equation proposed by Hakkaev and Kirchev (2005) \cite{Hakkaev 2005}. We prove that the solution map of the generalized Camassa-Holm equation is not uniformly continuous on the initial data in Besov spaces. Our result include the present work (2020) \cite{Li 2020,Li 2020-1} on Camassa-Holm equation with $Q=1$ and extends the previous non-uniform continuity in Sobolev spaces (2015) \cite{Mi 2015} to Besov spaces. In addition, the non-uniform continuity in critical space $B_{2, 1}^{\frac{3}{2}}(\mathbb{R})$ is the first to be considered in our paper.

math.AP↗
arXiv2020

A class large solution of 3D incompressible micropolar fluid system

In this paper, we construct a class global large solution to the 3D incompressible micropolar fluid system. Precisely speaking, by choosing a special initial data which can be arbitrarily large in $\dot{B}^{-1}_{\infty,\infty}$, the system has a unique global solution.

math.AP↗