Related papers: Blow up criterion for nematic liquid crystal flows
A blowup criteria along maximum point of the 3D-Navier-Stokes flow in terms of function spaces with variable growth condition is constructed. This criterion is different from the Beale-Kato-Majda type and Constantin-Fefferman type…
We establish a new BKM-type blow-up criterion for solutions of the incompressible Euler equations that belong to Sobolev or H\" older spaces. Our criterion involves the $L^2$ norm in time of the $L^\infty$ norm of the first order tangential…
In this paper, we obtain optimal time-decay rates in $L^r(\mathbb R^3_+)$ for $r\ge 1$ of global strong solutions to the nematic liquid crystal flows in $\mathbb R^3_+$, provided the initial data has small $L^3(\mathbb R^3_+)$-norm.
In this paper, we obtain a blow up criterion for classical solutions to the 3-D compressible Naiver-Stokes equations just in terms of the gradient of the velocity, similar to the Beal-Kato-Majda criterion for the ideal incompressible flow.…
In this paper we establish the uniformity property of a simplified Ericksen-Leslie system modelling the hydrodynamics of nematic liquid crystals on the two dimensional unit sphere $S^2$, namely the uniform convergence in $L^2$ to a steady…
In this paper we deal with the existence of local strong solution for a perfect compressible viscous fluid, heat conductive and self gravitating, coupled with a first order kinetics used in astrophysical hydrodynamical models. In our…
In this paper, we obtain upper bounds for the critical time $T^*$ of the blow-up for the parabolic-elliptic Patlak-Keller-Segel system on the 2D-Euclidean space. No moment condition or/and entropy condition are required on the initial data;…
We consider the compressible Navier-Stokes system in three dimensions with general inflow-outflow boundary conditions, meaning that we prescribe a boundary velocity which has non-zero normal component and accordingly the density is…
As is well known, for the harmonic heat flow or liquid crystal flow in two-dimension, the solution may blow up when the initial energy is greater than $8\pi$. Motivated by Lai--Lin--Wang--Wei--Zhou (CPAM, 2022), where singular solutions…
This paper is concerned with the blowup criterion for mild solution to the incompressible Navier-Stokes equation in higher spatial dimensions $d \geq 4$. By establishing an $\epsilon$ regularity criterion, we show that if the mild solution…
Consider the (simplified) Leslie-Erickson model for the flow of nematic liquid crystals in a bounded domain $\Omega \subset \mathbb{R}^n$ for n > 1$. This article develops a complete dynamic theory for these equations, analyzing the system…
We extend recent existence and uniqueness results for maximal solutions of SPDEs through an improved blow-up criterion. Whilst the maximal time of existence is typically characterised by blow-up in the energy norm of solutions, we show…
We establish a blow-up criterion in terms of the upper bound of the density and temperature for the strong solution to 2D compressible viscous heat-conductive flows. The initial vacuum is allowed.
In this paper, we obtain a blow up criterion for strong solutions to the 3-D compressible Naveri-Stokes equations just in terms of the gradient of the velocity, similar to the Beal-Kato-Majda criterion for the ideal incompressible flow. The…
In this paper, we consider a simplified Ericksen-Leslie model for the nematic liquid crystal flow. The evolution system consists of the Navier-Stokes equations coupled with a convective Ginzburg-Landau type equation for the averaged…
In this paper we develop two different types of criteria for the finite time blow-up solutions to the combined nonlinear Schr\"odinger equation in 1D. The first one is a negative energy criterion developed for triple combined nonlinearity…
This paper deals with unbounded solutions to the following zero--flux chemotaxis system \begin{equation}\label{ProblemAbstract} \tag{$\Diamond$} \begin{cases} % about u u_t=\nabla \cdot [(u+\alpha)^{m_1-1} \nabla u-\chi u(u+\alpha)^{m_2-2}…
We study the finite-time blow-up in two variants of the parabolic-elliptic Keller-Segel system with nonlinear diffusion and logistic source. In $n$-dimensional balls, we consider \begin{align*} \begin{cases} u_t = \nabla \cdot…
In this paper, we revisit the blow-up criteria for the simplest parabolic-elliptic (PKS) system in the 2D Euclidean space, including a consumption term. In the supercritical mass case M > 8pi, and under an additional global assumption on…
In this paper we consider nonnegative solutions of the following parabolic-elliptic cross-diffusion system \begin{equation*} \left\{ \begin{array}{l} \begin{aligned} &u_t = \Delta u - \nabla(u f(|\nabla v|^2 )\nabla v), \\[6pt] &0= \Delta v…