Related papers: Blow up criterion for nematic liquid crystal flows

In this paper we consider the nonlinear dispersive wave equation on the real line, $u_t-u_{txx}+[f(u)]_x-[f(u)]_{xxx}+\bigl[g(u)+\frac{f''(u)}{2}u_x^2\bigr]_x=0$, that for appropriate choices of the functions $f$ and $g$ includes well known…

We investigate the blowup criterion of the barotropic compressible viscous fluids for the Cauchy problem, Dirichlet problem and Navier-slip boundary condition. The main novelty of this paper is two-fold: First, for the Cauchy problem and…

A sufficient integral criterion for a blow-up solution of the Hopf equations (the Euler equations with zero pressure) is found. This criterion shows that a certain positive integral quantity blows up in a finite time under specific initial…

We consider the scaling critical Lebesgue norm of blow-up solutions to the semilinear heat equation $u_t=\Delta u+|u|^{p-1}u$ in an arbitrary smooth domain of $\mathbf{R}^n$. In the range $p>p_S:=(n+2)/(n-2)$, we show that the critical norm…

We study an initial boundary value problem for the Navier-Stokes equations of compressible viscous heat-conductive fluids in a 2-D periodic domain or the unit square domain. We establish a blow-up criterion for the local strong solutions in…

This paper is devoted to the study of blow-up phenomenon for a fouth-order nonlocal parabolic equation with Neumann boundary condition, \begin{equation*} \left\{\begin{array}{ll}\ds u_{t}+u_{xxxx}=|u|^{p-1}u-\frac{1}{a}\int_{0}^a|u|^{p-1}u\…

We study the blowup criterion of smooth solution to the Oldroyd models. Let $(u(t,x), F(t,x)$ be a smooth solution in $[0,T)$, it is shown that the solution $(u(t,x), F(t,x)$ does not appear breakdown until $t=T$ provided $\nabla u(t,x)\in…

We consider the $L^2$ critical inhomogeneous nonlinear Schr\"odinger (INLS) equation in $\mathbb{R}^N$ $$ i \partial_t u +\Delta u +|x|^{-b} |u|^{\frac{4-2b}{N}}u = 0, $$ where $N\geq 1$ and $0<b<2$. We prove that if $u_0\in…

We show that the solution to the Cauchy problem of the 3D nematic liquid crystal flows, with initial data belongs to a critical Besov space, belongs to a Gevrey class. More precisely, it is proved that for any $(u_{0},d_{0} -…

We study the strong solution to the 3-D compressible Navier--Stokes equations. We propose a new blow up criterion for barotropic gases in terms of the integral norm of density $\rho$ and the divergence of the velocity $\bu$ without any…

In this paper we study the zero inertia limit that is from the hyperbolic to parabolic Ericksen-Leslie's liquid crystal flow. By introducing an initial layer and constructing an energy norm and energy dissipation functional depending on the…

We consider the Poiseuille flow of nematic liquid crystals via the full Ericksen-Leslie model. The model is described by a coupled system consisting of a heat equation and a quasilinear wave equation. In this paper, we will construct an…

In this paper we study the magneto-micropolar fluid equations in $\R^3$, prove the existence of the strong solution with initial data in $H^s(\R^3)$ for $s> {3/2}$, and set up its blow-up criterion. The tool we mainly use is…

Perhaps the most classical diffusion model for chemotaxis is the Keller-Segel system $\begin{equation} \begin{cases} u_{t} =\Delta u - \nabla \cdot(u \nabla v) \ \ \ \text{in } \mathbb{R}^2\times(0,T),\\[5pt] v =…

We construct a two-parameter continuum of type II blow up solutions for the energy-critical focusing NLS in dimension $ d = 3$. The solutions collapse to a single energy bubble in finite time, precisely they have the form $ u(t,x) = e^{i…

The temperature-dependent incompressible nematic liquid crystal flows in a bounded domain $\Omega\subset\mathbb{R}^N$ ($N=2, 3$) are studied in this paper. Following Danchin's method in [J. Math. Fluid Mech., 2006], we use a localization…

We establish quantitative blow-up criteria below the scaling threshold for radially symmetric solutions to the defocusing nonlinear Schr\"odinger equation with nonlinearity $|u|^6u$. This provides to our knowledge the first generic results…

For the Keller-Segel system \[ \left\{\, \begin{aligned} u_t &= \Delta u - \nabla \cdot ( u \nabla v ), \\ v_t &= \Delta v - v + u \end{aligned} \right. \tag{$\star$} \] posed in a planar domain $\Omega$ with Neumann boundary conditions,…

The Ericksen-Leslie model for nematic liquid crystal flows in case of an isothermal and incompressible fluid with general Leslie stress and anisotropic elasticity, i.e. with general Ericksen stress tensor, is shown for the first time to be…

The aim of this paper is to apply the modified potential well method and some new differential inequalities to study the asymptotic behavior of solutions to the initial homogeneous $\hbox{Neumann}$ problem of a nonlinear diffusion equation…