A new blowup criterion of strong solutions to the two-dimensional equations of compressible nematic liquid crystal flows

In this paper, we concern the Cauchy problem of two-dimensional (2D) compressible nematic liquid crystal flows with vacuum as far-field density. Under a geometric condition for the initial orientation field, we establish a blowup criterion in terms of the integrability of the density for strong solutions to the compressible nematic liquid crystal flows. This criterion generalizes previous results of compressible nematic liquid crystal flows with vacuum, which concludes the initial boundary problem and Cauchy problem.     

A blowup criterion for the compressible nematic liquid crystal flows in dimension two

In this paper, we establish a blowup criterion for the two-dimensional compressible nematic liquid crystal flows. The criterion is given in terms of the density and the gradient of direction field, where the later satisfies the Serrin-type blowup criterion. For this result, we do not need the initial density to be positive.     

Blow up criterion for three‐dimensional nematic liquid crystal flows with partial viscosity

In this paper, we investigate the Cauchy problem for the three‐dimensional nematic liquid crystal flows with partial viscosity, and a blow up criterion of smooth solutions is established. This result is analogous to the celebrated Beale‐Kato‐Majda breakdown criterion for the incompressible Euler equations. Copyright © 2012 John Wiley & Sons, Ltd.     

On the well‐posedness of the Euler equations in the Triebel‐Lizorkin spaces

We prove local‐in‐time unique existence and a blowup criterion for solutions in the Triebel‐Lizorkin space for the Euler equations of inviscid incompressible fluid flows in ?ⁿ, n ≥ 2. As a corollary we obtain global persistence of the initial regularity characterized by the Triebel‐Lizorkin spaces for the solutions of two‐dimensional Euler equations. To prove the results, we establish the logarithmic inequality of the Beale‐Kato‐Majda type, the Moser type of inequality, as well as the commutator estimate in the Triebel‐Lizorkin spaces. The key methods of proof used are the Littlewood‐Paley decomposition and the paradifferential calculus by J. M. Bony. © 2002 John Wiley & Sons, Inc.     

A Blow-up Criterion for 2D Compressible Nematic Liquid Crystal Flows in Terms of Density

This paper is concerned with the Cauchy problem for compressible nematic liquid crystal flows in the two-dimensional space (2D). We establish a blow-up criterion in terms of the density only, provided the macroscopic average of the nematic liquid crystal orientation field satisfies a geometric condition. In particular, the initial vacuum is allowed and the compatibility condition is removed.     

On weak solution to the steady compressible flow of nematic liquid crystals

In this paper, we study the three‐dimensional‐simplified Ericksen‐Leslie system for the steady compressible flow of nematic liquid crystals in a bounded domain. It is proved that the existence of a weak solution for the adiabatic exponent γ > 1 provided the initial direction field in the upper hemisphere.     

Blowup Solutions to the Semilinear Wave Equation with a Stylized Pyramid as a Blowup Surface

We consider the semilinear wave equation with subconformal power nonlinearity in two space dimensions. We construct a finite‐time blowup solution with an isolated characteristic blowup point at the origin and a blowup surface that is centered at the origin and has the shape of a stylized pyramid, whose edges follow the bisectrices of the axes in ℝ². The blowup surface is differentiable outside the bisectrices. As for the asymptotic behavior in similarity variables, the solution converges to the classical one‐dimensional soliton outside the bisectrices. On the bisectrices outside the origin, it converges (up to a subsequence) to a genuinely two‐dimensional stationary solution, whose existence is a by‐product of the proof. At the origin, it behaves like the sum of four solitons localized on the two axes, with opposite signs for neighbors. This is the first example of a blowup solution with a characteristic point in higher dimensions, showing a really two‐dimensional behavior. Moreover, the points of the bisectrices outside the origin give us the first example of noncharacteristic points where the blowup surface is nondifferentiable. © 2018 Wiley Periodicals, Inc.     

Blow-up of the Smooth Solution to the Compressible Nematic Liquid Crystal System

In this paper, we investigate the blow-up of the smooth solutions to a simplified Ericksen-Lesile system for compressible flows of nematic liquid crystals in different dimensional case. We prove that whether the smooth solution of the Cauchy problem or the initial-boundary problem to the nematic liquid crystal system will blow up in finite time. The main technique is the construction of the functional differential inequality.     

BKM's criterion for the 3D nematic liquid crystal flows via two velocity components and molecular orientations

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 _?<+∞ if and only if where u ^h =(u ¹,u ²), ?_h =(? ₁,? ₂). This result can be regarded as the generalization of the well‐known Beale‐Kato‐Majda (BKM) type criterion and is even new for the three‐dimensional incompressible Navier–Stokes equations. Copyright © 2016 John Wiley & Sons, Ltd.     

A new blowup criterion for strong solutions to the three‐dimensional compressible magnetohydrodynamic equations with vacuum in a bounded domain

In this paper, we establish a new blowup criterions for the strong solution to the Dirichlet problem of the three‐dimensional compressible MHD system with vacuum. Specifically, we obtain the blowup criterion in terms of the concentration of density in BMO norm or the concentration of the integrability of the magnetic field at the first singular time. The BMO‐type estimate for the Lam system 2.6 and a variant of the Brezis‐Waigner's inequality 2.3 play a critical role in the proof. Copyright © 2017 John Wiley & Sons, Ltd.     

A regularity condition of strong solutions to the two‐dimensional equations of compressible nematic liquid crystal flows

This paper is concerned with the short time strong solutions for Cauchy problem to a simplified Ericksen–Leslie system of compressible nematic liquid crystals in two dimensions with vacuum as far field density. We establish a blow‐up criterion for possible breakdown of such solutions at a finite time, which is analogous to the well‐known Serrin's blow‐up criterion for the incompressible Navier–Stokes equations. Copyright © 2016 John Wiley & Sons, Ltd.     

Blowup of solutions for the initial boundary value problem of the 3‐dimensional compressible damped Euler equations

The blowup phenomenon of solutions is investigated for the initial boundary value problem of the 3‐dimensional compressible damped Euler equations. It is shown that if the initial functional associated with a general test function is large enough, the solutions of the damped Euler equations will blow up on finite time. Hence, a class of blowup conditions is established. Moreover, the blowup time could be estimated.     

On the temporal decay of solutions to the two‐dimensional nematic liquid crystal flows

We consider the temporal decay estimates for weak solutions to the two‐dimensional nematic liquid crystal flows, and we show that the energy norm of a global weak solution has non‐uniform decay under suitable conditions on the initial data. We also show the exact rate of the decay (uniform decay) of the energy norm of the global weak solution.     

On the Finite‐Time Blowup of a One‐Dimensional Model for the Three‐Dimensional Axisymmetric Euler Equations

In connection with the recent proposal for possible singularity formation at the boundary for solutions of three‐dimensional axisymmetric incompressible Euler's equations (Luo and Hou, Proc. Natl. Acad. Sci. USA (2014)), we study models for the dynamics at the boundary and show that they exhibit a finite‐time blowup from smooth data. © 2017 Wiley Periodicals, Inc.     

Blowup mechanism for viscous compressible heat-conductive magnetohydrodynamic flows in three dimensions

We investigate initial-boundary-value problem for three-dimensional magnetohydrodynamic(MHD)system of compressible viscous heat-conductive flows and the three-dimensional full compressible Navier-Stokes equations. We establish a blowup criterion only in terms of the derivative of velocity field, similar to the Beale-Kato-Majda type criterion for compressible viscous barotropic flows by Huang et al.(2011). The results indicate that the nature of the blowup for compressible MHD models of viscous media is similar to the barotropic compressible Navier-Stokes equations and does not depend on further sophistication of the MHD model, in particular, it is independent of the temperature and magnetic field. It also reveals that the deformation tensor of the velocity field plays a more dominant role than the electromagnetic field and the temperature in regularity theory. Especially, the similar results also hold for compressible viscous heat-conductive Navier-Stokes flows,which extend the results established by Fan et al.(2010), and Huang and Li(2009). In addition, the viscous coefficients are only restricted by the physical conditions in this paper.     

The Cauchy problem of a focusing energy-critical nonlinear Schrödinger equation

In this paper, we combine variational methods and harmonic analysis to discuss the Cauchy problem of a focusing nonlinear Schrödinger equation. We study the global well-posedness, finite time blowup and asymptotic behavior of this problem. By Hamiltonian property, we establish two types of invariant evolution flows. Then from one flow and the stability of classical energy-critical nonlinear Schrödinger equation, we find that the solution exists globally and scattering occurs. Finally, we get a precise blowup criterion of this problem for positive energy initial data via the other flow.     

On center singularity for compressible spherically symmetric nematic liquid crystal flows

The paper is concerned with a simplified system, proposed by Ericksen [6] and Leslie [20], modeling the flow of nematic liquid crystals. In the first part, we give a new Serrin's continuation principle for strong solutions of general compressible liquid crystal flows. Based on new observations, we establish a localized Serrin's regularity criterion for the 3D compressible spherically symmetric flows. It is proved that the classical solution loses its regularity in finite time if and only if, either the concentration or vanishing of mass forms or the norm inflammation of gradient of orientation field occurs around the center.     

A blow‐up criterion for incompressible hydrodynamic flow of liquid crystals in dimension two

In the paper, we establish a Serrin type criterion for strong solutions to a simplified density‐dependent Ericksen‐ Leslie system modeling incompressible, nematic liquid crystal materials in dimension two. Copyright © 2013 John Wiley & Sons, Ltd.     

Blowup in a three‐dimensional vector model for the Euler equations

We present a three‐dimensional vector model given in terms of an infinite system of nonlinearly coupled ordinary differential equations. This model has structural similarities with the Euler equations for incompressible, inviscid fluid flows. It mimics certain important properties of the Euler equations, namely, conservation of energy and divergence‐free velocity. It is proven for certain families of initial data that the model system permits local existence in time for initial conditions in Sobolev spaces H^s, s > ; and blowup occurs in the sense that the H^{3/2 + ?} norm becomes unbounded in finite time. © 2004 Wiley Periodicals, Inc.     

Low Mach number limit of global solutions to 3‐D compressible nematic liquid crystal flows with Dirichlet boundary condition

In this paper, we consider the uniform estimates of strong solutions in the Mach number ? and t ∈ [0,∞) for the compressible nematic liquid crystal flows in a 3‐D bounded domain , provided the initial data are small enough and the density is close to the constant state. Here, we consider the case that the velocity field satisfies the Dirichlet boundary condition. Based on the uniform estimates, we obtain the global convergence of the compressible nematic liquid crystal system to the incompressible nematic liquid crystals system as the Mach number tends to zero.