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.     

A free boundary problem for compressible hydrodynamic flow of liquid crystals in one dimension

In this paper, we consider the free boundary problem for a simplified version of Ericksen–Leslie equations modeling the compressible hydrodynamic flow of nematic liquid crystals in dimension one. We obtain both existence and uniqueness of global classical solutions provided that the initial density is away from vacuum.     

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.     

Global well-posedness for the dynamical Q-tensor model of liquid crystals

We consider a complex fluid modeling nematic liquid crystal flows, which is described by a system coupling Navier-Stokes equations with a parabolic Q-tensor system. We first prove the global existence of weak solutions in dimension three. Furthermore, the global well-posedness of strong solutions is studied with sufficiently large viscosity of fluid. Finally, we show a continuous dependence result on the initial data which directly yields the weak-strong uniqueness of solutions.     

A time dependent coupled system related to the tridimensional equations of a nematic liquid crystal

In this paper we prove some results about the existence, uniqueness and regularity of the solutions of a tridimensional variational approximated model for the Leslie's equations of an incompressible nematic liquid crystal (cf. [7]). This model was introduced in [1] where the bidimensional case is studied. Here we use the methods developed in [6], [4] and [8].     

Global well‐posedness of the nonhomogeneous incompressible liquid crystals systems

This paper examines the initial‐value problem for the nonhomogeneous incompressible nematic liquid crystals system with vacuum. This paper establishes two main results. The first result is involved with the global strong solutions to the 2D liquid crystals system in a bounded smooth domain. Our second result is concerned with the small data global existence result about the 3D system in the whole space. In addition, the local existence and a blow‐up criterion of strong solutions are also mentioned. Copyright © 2016 John Wiley & Sons, Ltd.     

Global well-posedness for the dynamical <Emphasis Type="Italic">Q</Emphasis>-tensor model of liquid crystals

We consider a complex fluid modeling nematic liquid crystal flows, which is described by a system coupling Navier-Stokes equations with a parabolic Q-tensor system. We first prove the global existence of weak solutions in dimension three. Furthermore, the global well-posedness of strong solutions is studied with sufficiently large viscosity of fluid. Finally, we show a continuous dependence result on the initial data which directly yields the weak-strong uniqueness of solutions.     

Global Existence and the Optimal Decay Rates for the Three Dimensional Compressible Nematic Liquid Crystal Flow

The present paper is dedicated to the study of the Cauchy problems for the three-dimensional compressible nematic liquid crystal flow. We obtain the global existence and the optimal decay rates of smooth solutions to the system under the condition that the initial data in lower regular spaces are close to the constant equilibrium state. Our main method is based on the spectral analysis and the smooth effect of dissipative operator.     

On temporal decay estimates for the compressible nematic liquid crystal flow in

We use a general energy method recently developed by [Guo Y, Wang Y. Decay of dissipative equations and negative sobolev spaces. Commun. Partial Differ. Equ. 2012;37:2165–2208.] to prove the global existence and temporal decay rates of solutions to the three-dimensional compressible nematic liquid crystal flow in the whole space. In particular, the negative Sobolev norms of solutions are shown to be preserved along time evolution, and then the optimal decay rates of the higher order spatial derivatives of solutions are obtained by energy estimates and the interpolation inequalities.     

(Co)bordism groups in quantum super PDE's. III: Quantum super Yang–Mills equations

In this third part of a series of three papers devoted to the study of geometry of quantum super PDE's [A. Prástaro, (Co)bordism groups in quantum super PDE's. I: quantum supermanifolds, Nonlinear Anal. Real World Appl., in press, doi:10.1016/j.nonrwa.2005.12.007; (Co)bordism groups in quantum super PDE's. II: quantum super PDE's, Nonlinear Anal. Real World Appl., in press, doi:10.1016/j.nonrwa.2005.12.008], we apply our theory, developed in the first two parts, to quantum super Yang–Mills equations and quantum supergravity equations. For such equations we determine their integral bordism groups, and by using some surgery techniques, we obtain theorems of existence of global solutions, also with nontrivial topology, for Cauchy problems and boundary value problems. Quantum tunnelling effects are described in this context. Furthermore, for quantum supergravity equations we prove existence of solutions of the type quantum black holes evaporation processes just by using an extension to quantum super PDEs of our theory of integral (co)bordism groups. Our proof is constructive, i.e., we give geometric methods to build such solutions. In particular a criterion to recognize quantum global (smooth) solutions with mass-gap, for the quantum super Yang–Mills equation, is given. Finally it is proved that quantum super PDE's contain also solutions that come from Dirac quantization of their superclassical counterparts. This proves that quantum super PDE's are (nonlinear) generalizations of Dirac quantized superclassical PDE's. Applications of this result to free quantum super Yang–Mills equations are given.     

Remarks on classical solutions to steady quantum Navier-Stokes equations

The paper of Dong [Dong, J. Classical solutions to one-dimensional stationary quantum Navier–Stokes equations, J. Math Pure Appl. 2011] which proved the existence of classical solutions to one-dimensional steady quantum Navier–Stokes equations, when the nonzero boundary value u ₀ satisfies some conditions. In this paper, we obtain a different version of existence theorem without restriction to u ₀. As a byproduct, we get the existence result of classical solutions to the stationary quantum Navier–Stokes equations.     

A family of global classical solutions to 3D incompressible nematic liquid crystal flows

In this paper we construct a family of finite energy smooth solutions to the three-dimensional incompressible nematic liquid crystal flows. We achieve this by choosing the steady state Beltrami flows which have infinite energies as the initial data and using a special cut-off technique.     

Global existence for Maxwell-Bloch systems

Maxwell-Bloch equations describe the propagation of an electromagnetic wave through a quantum medium. For any number of quantum levels, in space dimension 3, we show the global existence of weak (L²) solutions to the initial-value problem. In the case of smoother electromagnetic fields (with curl in L²), the solution is unique. For smooth data (H^s, s?2), the solutions remain smooth for all times.     

Smooth solutions of non-linear stochastic partial differential equations driven by multiplicative noises

In this paper, we study the regularity of solutions of nonlinear stochastic partial differential equations (SPDEs) with multiplicative noises in the framework of Hilbert scales. Then we apply our abstract result to several typical nonlinear SPDEs such as stochastic Burgers and Ginzburg-Landau equations on the real line, stochastic 2D Navier-Stokes equations (SNSEs) in the whole space and a stochastic tamed 3D Navier-Stokes equation in the whole space, and obtain the existence of their smooth solutions respectively. In particular, we also get the existence of local smooth solutions for 3D SNSEs.     

Vector solitary waves for a class of nonlocal nonlinear Schrödinger system

In this paper, we study a class of one-dimensional nonlocal nonlinear Schrödinger system, which describes two-color optical beams propagating through a cell with nematic liquid crystals. The existence of local and global solutions is derived first upon applying the Strichartz's estimates, conservation laws, and fixed points theorem. Then, we prove the existence of positive normalized vector solitary wave solutions by using variational approach and the concentration-compactness technique.     

Global existence of the finite energy weak solutions to a nematic liquid crystals model

In this paper, we are concerned with a simplified hydrodynamic equation, proposed by Ericksen and Leslie, modeling the flow of nematic liquid crystals. For a bounded domain in Ropf³, under the assumption that initial density belongs to , we show the global existence of weak solutions to the nematic liquid crystals model with a penalized system. Furthermore, we also obtain the energy inequality for weak solutions. Copyright © 2011 John Wiley & Sons, Ltd.     

微分代数方程去奇异化分析

利用去奇异化方法讨论了拟线性微分代数方程在奇点邻域内光滑解的性质.通过尺度参数的微分同胚变换,将拟线性微分代数方程转化为相应的常微分方程,从而构造出在孤立奇点邻域内的初始微分代数方程的光滑解,给出解存在的充分条件,并进一步讨论了解的性质.     

On the Cauchy problem of degenerate hyperbolic equations

In this paper, we study a class of degenerate hyperbolic equations and prove the existence of smooth solutions for Cauchy problems. The existence result is based on a priori estimates of Sobolev norms of solutions. Such estimates illustrate a loss of derivatives because of the degeneracy.

     

A simplified variational model for the bidimensional coupled evolution equations of a nematic liquid crystal

In Dias (C. R. Acad. Sci. Paris Sér. A. 285 (1977)) we have deduced, from Leslie's model (Arch. Rat. Mech. Anal. 28 (1968)), a weak formulation for the bidimensional coupled evolution equations of an incompressible nematic liquid crystal submitted to an homogeneous magnetic field. In this paper we prove some results about the existence, regularity and uniqueness of their solutions. This study extends the special case developed in Dias (J. Mécanique15 (1976)), where we assumed that the director field depends on time only.     

Global smooth solutions to relativistic Euler-Poisson equations with repulsive force

In this paper, we are concerned with the global existence of smooth solutions for the one dimen- sional relativistic Euler-Poisson equations： Combining certain physical background, the relativistic Euler-Poisson model is derived mathematically. By using an invariant of Lax＇s method, we will give a sufficient condition for the existence of a global smooth solution to the one-dimensional Euler-Poisson equations with repulsive force.