Well-posedness and Gevrey Analyticity of the Generalized Keller-Segel System in Critical Besov Spaces

In this paper, we study the Cauchy problem for the generalized Keller-Segel system with the cell diffusion being ruled by fractional diffusion: \begin{equation*} \begin{cases} \partial_{t}u+\Lambda^{\alpha}u-\nabla\cdot(u\nabla \psi)=0\quad &\mbox{in}\ \ \mathbb{R}^n\times(0,\infty), -\Delta \psi=u\quad &\mbox{in}\ \ \mathbb{R}^n\times(0,\infty), u(x,0)=u_0(x), \ \ &\mbox{in}\ \ \mathbb{R}^n. \end{cases} \end{equation*} In the case that $1<\alpha\leq 2$, we prove local well-posedness for any initial data and global well-posedness for small initial data in critical Besov spaces $\dot{B}^{-\alpha+\frac{n}{p}}_{p,q}(\mathbb{R}^{n})$ with $1\leq p<\infty$, $1\leq q\leq \infty$, and analyticity of solutions for initial data $u_{0}\in \dot{B}^{-\alpha+\frac{n}{p}}_{p,q}(\mathbb{R}^{n})$ with $1< p<\infty$, $1\leq q\leq \infty$. Moreover, the global existence and analyticity of solutions with small initial data in critical Besov spaces $\dot{B}^{-\alpha}_{\infty,1}(\mathbb{R}^{n})$ is also established. In the limit case that $\alpha=1$, we prove global well-posedness for small initial data in critical Besov spaces $\dot{B}^{-1+\frac{n}{p}}_{p,1}(\mathbb{R}^{n})$ with $1\leq p<\infty$ and $\dot{B}^{-1}_{\infty,1}(\mathbb{R}^{n})$, and show analyticity of solutions for small initial data in $\dot{B}^{-1+\frac{n}{p}}_{p,1}(\mathbb{R}^{n})$ with $1<p<\infty$ and $\dot{B}^{-1}_{\infty,1}(\mathbb{R}^{n})$, respectively.Comment: 24 page

The Optimal Temporal Decay Estimates for the Fractional Power Dissipative Equation in Negative Besov Spaces

In this paper, we first generalize a new energy approach, developed by Y. Guo and Y. Wang \cite{GW12}, in the framework of homogeneous Besov spaces for proving the optimal temporal decay rates of solutions to the fractional power dissipative equation, then we apply this approach to the supercritical and critical quasi-geostrophic equation and the critical Keller-Segel system. We show that the negative Besov norm of solutions is preserved along time evolution, and obtain the optimal temporal decay rates of the spatial derivatives of solutions by the Fourier splitting approach and the interpolation techniques.Comment: 23 pages. arXiv admin note: text overlap with arXiv:1508.0202

A Beale-Kato-Majda Blow-up Criterion for a Hydrodynamic System Modeling Vesicle and Fluid Interactions

In this paper, we establish an analog of the Beale-Kato-Majda type criterion for singularities of smooth solutions of a hydrodynamic system modeling vesicle and fluid interactions. The result shows that the maximum norm of the vorticity alone controls the breakdown of smooth solutions.Comment: 16 page

Logarithmical Blow-up Criteria for the Nematic Liquid Crystal Flows

We investigate the blow-up criterion for the local in time classical solution of the nematic liquid crystal flows in dimension two and three. More precisely, $0<T_{*}<+\infty$ is the maximal time interval if and only if (i) for $n=3$, {align*} \int_{0}^{T_{*}}\frac{\|\omega\|_{\dot{B}^{0}_{\infty,\infty}}+\|\nabla d\|_{\dot{B}^{0}_{\infty,\infty}}^{2}}{\sqrt{1+\text{ln}(e+\|\omega\|_{\dot{B}^{0}_{\infty,\infty}} +\|\nabla d\|_{\dot{B}^{0}_{\infty,\infty}})}}\text{d}t=\infty, {align*} or {align*} \int_{0}^{T_{*}}\frac{\|\nabla u\|_{\dot{B}^{-1}_{\infty,\infty}}^{2}+\|\nabla d\|_{\dot{B}^{0}_{\infty,\infty}}^{2}}{\sqrt{1+\text{ln}(e+\|\nabla u\|_{\dot{B}^{-1}_{\infty,\infty}} +\|\nabla d\|_{\dot{B}^{0}_{\infty,\infty}})}}\text{d}t=\infty; {align*} and (ii) for $n=2$, {align*} \int_{0}^{T_{*}}\frac{\|\nabla d\|_{\dot{B}^{0}_{\infty,\infty}}^{2}}{\sqrt{1+\text{ln}(e +\|\nabla d\|_{\dot{B}^{0}_{\infty,\infty}})}}\text{d}t=\infty. {align*}Comment: 17 page

Logarithmically Improved Blow-up Criteria for a Phase Field Navier-Stokes Vesicle-Fluid Interaction Model

In this paper, we study a hydrodynamical system modeling the deformation of vesicle membrane under external incompressible viscous flow fields. The system is in the Eulerian formulation and is governed by the coupling of the incompressible Navier-Stokes equations with a phase field equation. In the three dimensional case, we establish two logarithmically improved blow-up criteria for local smooth solutions of this system in terms of the vorticity field only in the homogeneous Besov spaces.Comment: 22 page

Blow-up Criteria for the Three Dimensional Nonlinear Dissipative System Modeling Electro-hydrodynamics

In this paper, we investigate some sufficient conditions for the breakdown of local smooth solutions to the three dimensional nonlinear nonlocal dissipative system modeling electro-hydrodynamics. This model is a strongly coupled system by the well-known incompressible Navier-Stokes equations and the classical Poisson-Nernst-Planck equations. We show that the maximum of the vorticity field alone controls the breakdown of smooth solutions, which reveals that the velocity field plays a more dominant role than the density functions of charged particles in the blow-up theory of the system. Moreover, some Prodi-Serrin type blow-up criteria are also established.Comment: 21 page

A regularity criterion for the solution of the nematic liquid crystal flows in terms of B˙∞,∞−1\dot{B}^{-1}_{\infty,\infty}-norm

In this paper, we investigate regularity criterion for the solution of the nematic liquid crystal flows in dimension three and two. We prove the solution $(u,d)$ is smooth up to time $T$ provided that there exists a positive constant $\varepsilon_{0}>0$ such that (i) for n=3, |(u,\nabla d)|_{L^{\infty}(0,T;\dot{B}^{-1}_{\infty,\infty})}\leq \varepsilon_{0}, and (ii) for $n=2$, |\nabla d|_{L^{\infty}(0,T;\dot{B}^{-1}_{\infty,\infty})}\leq \varepsilon_{0}.Comment: 15. arXiv admin note: text overlap with arXiv:1209.562

Existence of Solutions for the Debye-H\"{u}ckel System with Low Regularity Initial Data

In this paper we study existence of solutions for the Cauchy problem of the Debye-H\"{u}ckel system with low regularity initial data. By using the Chemin-Lerner time-space estimate for the heat equation, we prove that there exists a unique local solution if the initial data belongs to the Besov space $\dot{B}^{s}_{p,q}(\mathbb{R}^{n})$ for $-3/2<s\leq-2+\frac{n}{2}$, $p=\frac{n}{s+2}$ and $1\leq q\leq \infty$, and furthermore, if the initial data is sufficiently small then the solution is global. This result improves the regularity index of the initial data space in previous results on this model.Comment: 9 page

Global Existence and Stability for a Hydrodynamic System in the Nematic Liquid Crystal Flows

In this paper we consider a coupled hydrodynamical system which involves the Navier-Stokes equations for the velocity field and kinematic transport equations for the molecular orientation field. By applying the Chemin-Lerner's time-space estimates for the heat equation and the Fourier localization technique, we prove that when initial data belongs to the critical Besov spaces with negative-order, there exists a unique local solution, and this solution is global when initial data is small enough. As a corollary, we obtain existence of global self-similar solutions. In order to figure out the relation between the solution obtained here and weak solution of standard sense, we establish a stability result, which yields in a direct way that all global weak solutions associated with the same initial data must coincide with the solution obtained here, namely, weak-strong uniqueness holds.Comment: 18 page

Sparsity Aware Normalized Least Mean p-power Algorithms with Correntropy Induced Metric Penalty

For identifying the non-Gaussian impulsive noise systems, normalized LMP (NLMP) has been proposed to combat impulsive-inducing instability. However, the standard algorithm is without considering the inherent sparse structure distribution of unknown system. To exploit sparsity as well as to mitigate the impulsive noise, this paper proposes a sparse NLMP algorithm, i.e., Correntropy Induced Metric (CIM) constraint based NLMP (CIMNLMP). Based on the first proposed algorithm, moreover, we propose an improved CIM constraint variable regularized NLMP(CIMVRNLMP) algorithm by utilizing variable regularized parameter(VRP) selection method which can further adjust convergence speed and steady-state error. Numerical simulations are given to confirm the proposed algorithms.Comment: 5 pages, 4 figures, submitted for DSP201