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In this paper, we are mainly concerned with the well-posed problem of the fractional Keller--Segel system in the framework of variable Lebesgue spaces. Based on carefully examining the algebraical structure of the system, we reduced the fractional Keller--Segel system into the generalized nonlinear heat equation to overcome the difficulties caused by the boundedness of the Riesz potential in a variable Lebesgue spaces, then by mixing some structural properties of the variable Lebesgue spaces with the optimal decay estimates of the fractional heat kernel, we were able to establish two well-posedness results of the fractional Keller--Segel system in this functional setting., Comment: arXiv admin note: substantial text overlap with arXiv:2404.05127

In this paper, we are mainly concerned with the well-posedness of the dissipative surface quasi-geostrophic equation in the framework of variable Lebesgue spaces. Based on some analytical results developed in the variable Lebesgue spaces and the $L^{p}$-$L^{q}$ decay estimates of the fractional heat kernel, we establish the local existence and regularity of solutions to the 2D dissipative surface quasi-geostrophic equation in the variable Lebesgue space., Comment: 11 pages

In this paper, we investigate global existence of large solutions for the parabolic-elliptic Keller-Segel system in the homogeneous Besov type spaces. A class of initial data was presented, generating a global smooth solution although the $\dot{B}^{-2}_{\infty,\infty}$-norm of the initial data may be chosen arbitrarily large., Comment: 7 pages

This work is concerned with the global existence of large solutions to the three-dimensional dissipative fluid-dynamical model, which is a strongly coupled nonlinear nonlocal system characterized by the incompressible Navier--Stokes--Poisson--Nernst--Planck equations. Making full use of the algebraic structure of the system, we obtain the global existence of solutions without smallness assumptions imposed on the third component of the initial velocity field and the summation of initial densities of charged species. More precisely, we prove that there exist two positive constants $c_{0}, C_{0}$ such that if the initial data satisfies \begin{align*} \big(\|u_{0}^{h}\|_{\dot{B}^{-1+\frac{3}{p}}_{p,1}}+\|N_{0}-P_{0}\|_{\dot{B}^{-2+\frac{3}{q}}_{q,1}} \big) \exp\Big\{C_{0}\big(\|u_{0}^{3}\|_{\dot{B}^{-1+\frac{3}{p}}_{p,1}}^{2}+(\|N_{0}+P_{0}\|_{\dot{B}^{-2+\frac{3}{r}}_{r,1}}+1)\exp\big\{C_{0}\|u_{0}^{3}\|_{\dot{B}^{-1+\frac{3}{p}}_{p,1}}\big\}+1\big)\Big\} \leq c_{0}, \end{align*} then the incompressible Navier--Stokes--Poisson--Nernst--Planck equations admits a unique global solution., Comment: 18 pages

In this paper, logarithmically improved regularity criteria for the Navier--Stokes/Poisson--Nernst--Planck system are established in terms of both the pressure and the gradient of pressure in the homogeneous Besov space., Comment: 18pages

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 pages

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:1508.02023

In \cite{GW12} (Y. Guo, Y. Wang, Decay of dissipative equations and negative Sobolev spaces, Commun. Partial Differ. Equ. 37 (2012) 2165--2208), Y. Guo and Y. Wang developed a general new energy method for proving the optimal time decay rates of the solutions to dissipative equations. In this paper, we generalize this method in the framework of homogeneous Besov spaces. Moreover, we apply this method to a model arising from electro-hydrodynamics, which is a coupled system of the Navier-Stokes equations and the Poisson-Nernst-Planck equations through charge transport and external forcing terms. We show that the negative Besov norms are preserved along time evolution, and obtain the optimal temporal decay rates of the higher-order spatial derivatives of solutions by the Fourier splitting approach and the interpolation techniques., Comment: 26 pages

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

In this paper we provide a sufficient condition, in terms of the horizontal gradient of two horizontal velocity components and the gradient of liquid crystal molecular orientation field, for the breakdown of local in time strong solutions to the three-dimensional incompressible nematic liquid crystal flows. More precisely, let $T_{*}$ be the maximal existence time of the local strong solution $(u, d)$, then $T_{*}<+\infty$ if and only if \begin{align*} \int_{0}^{T_*} \big( \|\nabla_h u^h\|_{\dot{B}^0_{p,\frac{2p}{3}}}^q + \|\nabla d\|_{\dot{B}^0_{\infty,\infty}}^2 \big)dt = \infty\ \ \text{with}\ \ \ \frac{3}{p} +\frac{2}{q} = 2,\ \ \frac{3}{2} < p\leq\infty, \end{align*} where $u^{h}=(u^{1},u^{2})$, $\nabla_{h}=(\partial_{1}, \partial_{2})$. This result can be regarded as the generalization of the BKM's criterion in \cite{HW12}, and is even new for the three-dimensional incompressible Navier-Stokes equations., Comment: 16pages

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 pages

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:1209.5623

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

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 pages

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

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 pages