We study the three-dimensional compressible Navier-Stokes equations coupled with the $Q$-tensor equation perturbed by a multiplicative stochastic force, which describes the motion of nematic liquid crystal flows. The local existence and uniqueness of strong pathwise solution up to a positive stopping time is established where ``strong" is in both PDE and probability sense. The proof relies on the Galerkin approximation scheme, stochastic compactness, identification of the limit, uniqueness and a cutting-off argument. In the stochastic setting, we develop an extra layer approximation to overcome the difficulty arising from the stochastic integral while constructing the approximate solution. Due to the complex structure of the coupled system, the estimates of the high-order items are also the challenging part in the article.
The connection between the compressible flow of liquid crystals with low Mach number and the incompressible flow of liquid crystals is studied in a bounded domain. In particular, the convergence of weak solutions of the compressible flow of liquid crystals to the weak solutions of the incompressible flow of liquid crystals is proved when the Mach number approaches zero; that is, the incompressible limit is justified for weak solutions in a bounded domain.
The present paper is dedicated to the global large solutions and incompressible limit for the compressible flow of liquid crystals under the assumption on almost constant density and large volume viscosity. The result is based on Fourier analysis and involved so-called critical Besov norm.
In this paper, we consider the initial and boundary value problem of a simplified compressible nematic liquid crystal flow in $Ω\subset\mathbb R^3$. We establish the existence of global weak solutions, provided the initial orientational director field $d_0$ lies in the hemisphere $\mathbb S^2_+$.
This paper is concerned with the incompressible limit of the compressible hydrodynamic flow of liquid crystals with periodic boundary conditions in R^N(N = 2, 3). It is rigorously shown that the local (and global) strong solution of the compressible system converges to the local (and global) strong solution of the incompressible system. Furthermore, the convergence rates are also obtained in some sense.
In this paper, we investigate the Cauchy problem for the compressible nematic liquid crystal flows in three-dimensional whole space. First of all, we establish the time decay rates for compressible nematic liquid crystal flows by the method of spectral analysis and energy estimates. Furthermore, we enhance the convergence rates for the higher-order spatial derivatives of density, velocity and director. Finally, the time decay rates of mixed space-time derivatives of solution are also established.
The three-dimensional equations for the compressible flow of liquid crystals are considered. An initial-boundary value problem is studied in a bounded domain with large data. The existence and large-time behavior of a global weak solution are established through a three-level approximation, energy estimates, and weak convergence for the adiabatic exponent $γ>\frac32$.
In this paper, we consider the short time strong solution to a simplified hydrodynamic flow modeling the compressible, nematic liquid crystal materials in dimension three. We establish a criterion for possible breakdown of such solutions at finite time in terms of the temporal integral of both the maximum norm of the deformation tensor of velocity gradient and the square of maximum norm of gradient of liquid crystal director field.
This article is devoted to the study of the so-called incompressible limit for solutions of the compressible liquid crystals system. We consider the problem in the whole space $\mathbb{R}^{\mathbb{N}}$ and a bounded domain of $\mathbb{R}^{\mathbb{N}}$ with Dirichlet boundary conditions. Here the number of dimension $\mathbb{N}=2$ or 3.
We study the hydrodynamics of compressible flows of active liquid crystals in the Beris-Edwards hydrodynamics framework, using the Landau-de Gennes $Q$-tensor order parameter to describe liquid crystalline ordering. We prove the existence of global weak solutions for this active system in three space dimensions by the three-level approximations and weak convergence argument. New techniques and estimates are developed to overcome the difficulties caused by the active terms.
The present paper considers a homogeneous bubble inside an unbounded polytropic compressible liquid with viscosity. The system is governed by the Navier-Stokes equation with free boundary which is determined by the kinematic and dynamic boundary conditions on the bubble-liquid interface. The global existence of solution is proved, and the $\dot{H}^1$ asymptotic stability of the spherical equilibrium in terms of viscous damping together with a explicit decay rate is given in bare energy methods.
In this paper, we prove a Beale--Kato--Majda blow-up criterion in terms of the gradient of the velocity only for the strong solution to the 3-D compressible nematic liquid crystal flows with nonnegative initial densities. More precisely, the strong solution exists globally if the $L^{1}(0,T;L^{\infty})$-norm of the gradient of the velocity $u$ is bounded. Our criterion improves the recent result of X. Liu and L. Liu (\cite{LL}, A blow-up criterion for the compressible liquid crystals system, arXiv:1011.4399v2 [math-ph] 23 Nov. 2010).
We consider a system of equations governing the motion of a viscous, compressible, and heat conducting liquid-like fluid, with a general EOS of Mie-Grueneisen type. In addition, we suppose that the viscosity coefficients may decay to zero for large values of the temperature. We show the existence of global-in-time weak solution, derive a relative energy inequality, and compare the weak solutions with strong one emanating from the same initial data - the weak strong uniqueness property.
We prove the existence of a weak solution to a non-isothermal compressible model for nematic liquid crystals. An initial-boundary value problem is studied in a bounded domain with large data. The existence of a global weak solution is established through a three-level approximation, energy estimates, and weak convergence for the adiabatic exponent γ>3/2.
We consider the weak solution of the simplified Ericksen-Leslie system modeling compressible nematic liquid crystal flows in $\mathbb R^3$. When the initial data is small in $L^2$ and initial density is positive and essentially bounded, we first prove the existence of a global weak solution in $\mathbb R^3$. The large-time behavior of a global weak solution is also established.
We establish the local well-posedness for the free boundary problem for the compressible Euler equations describing the motion of liquid under the influence of Newtonian self-gravity. We do this by solving a tangentially-smoothed version of Euler's equations in Lagrangian coordinates which satisfies uniform energy estimates as the smoothing parameter goes to zero. The main technical tools are delicate energy estimates and optimal elliptic estimates in terms of boundary regularity, for the Dirichlet problem and Green's function.
We study the Ericksen-Leslie's parabolic-hyperbolic liquid crystal model in compressible flow. Inspired by our study for incompressible case \cite{Jiang-Luo-arXiv-2017} and some techniques from compressible Navier-Stokes equations, we prove the local-in-time existence of the classical solution to the system with finite initial energy, under some constraints on the Leslie coefficients which ensure the basic energy law is dissipative. Furthermore, with an additional assumption on the coefficients which provides a damping effect, and the smallness of the initial energy, the global classical solut…
We derive a priori estimates for the compressible free-boundary Euler equations with surface tension in three spatial dimensions in the case of a liquid. These are estimates for local existence in Lagrangian coordinates when the initial velocity and initial density belong to $H^3$, with an extra regularity condition on the moving boundary, thus lowering the regularity of the initial data. Our methods are direct and involve two key elements: the boundary regularity provided by the mean curvature, and a new compressible Cauchy invariance.
The Cauchy problem for the compressible flow of nematic liquid crystals in the framework of critical spaces is considered. We first establish the existence and uniqueness of global solutions provided that the initial data are close to some equilibrium states. This result improves the work by Hu and Wu [SIAM J. Math. Anal., 45 (2013), pp. 2678-2699] through relaxing the regularity requirement of the initial data in terms of the director field. We then consider the incompressible limit problem for ill prepared initial data. We prove that as the Mach number tends to zero, the global solution to t…
In this paper, we prove a new type of energy estimates for the compressible Euler's equation with free boundary, with a boundary part and an interior part. These can be thought of as a generalization of the energies in Christodoulou and Lindblad [CL] to the compressible case and do not require the fluid to be irrotational. In addition, we show that our estimates are in fact uniform in the sound speed. As a consequence, we obtain convergence of solutions of compressible Euler equations with a free boundary to solutions of the incompressible equations, generalizing the result of Ebin [Eb] to whe…
We study strong solutions of the simplified Ericksen-Leslie system modeling compressible nematic liquid crystal flows in a domain $Ω\subset\mathbb R^3$. We first prove the local existence of unique strong solutions provided that the initial data $ρ_0, u_0, d_0$are sufficiently regular and satisfy a natural compatibility condition. The initial density function $ρ_0$ may vanish on an open subset (i.e., an initial vacuum may exist). We then prove a criterion for possible breakdown of such a local strong solution at finite time in terms of blow up of the quantities $\|ρ\|_{L^\infty_tL^\infty_x}$ a…
In this paper, we study the uniform regularity and vanishing viscosity limit for the compressible nematic liquid crystal flows in three dimensional bounded domain. It is shown that there exists a unique strong solution for the compressible nematic liquid crystal flows with boundary condition in a finite time interval which is independent of the viscosity coefficient. The solutions are uniform bounded in a conormal Sobolev space. Furthermore, we prove that the density and velocity are uniform bounded in $W^{1, \infty}$, and the director field is uniform bounded in $W^{3,\infty}$ respectively. B…
In this paper, we consider the Cauchy problem to the three dimensional heat conducting compressible nematic liquid crystal system in the presence of vacuum and with vacuum far fields. Global well-posedness of strong solutions is established under the condition that the scaling invariant quantity $ (\|ρ_0\|_\infty+1)\big[\|ρ_0\|_3+(\|ρ_0\|_\infty+1)^2(\|\sqrt{ρ_0}u_0\|_2^2+ \|\nabla d_0\|_2^2)\big] \big[\|\nabla u_0\|_2^2+(\|ρ_0\|_\infty+1)(\|\sqrt{ρ_0}E_0\|_2^2 + \|\nabla^2 d_0\|_2^2)\big]$ is sufficiently small with the smallness depending only on the parameters appeared in the system.
We consider a nonlocal version of the quasi-static Navier-Stokes-Korteweg equations with a non-monotone pressure law. This system governs the low-Reynolds number dynamics of a compressible viscous fluid that may take either a liquid or a vapour state. For a porous domain that is perforated by cavities with diameter proportional to their mutual distance the homogenization limit is analyzed. We extend the results for compressible one-phase flow with polytropic pressure laws and prove that the effective motion is governed by a nonlocal version of the Cahn-Hilliard equation. Crucial for the analys…
The equations of stationary compressible flows of active liquid crystals are considered in a bounded three-dimensional domain. The system consists of the stationary Navier-Stokes equations coupled with the equation of Q-tensors and the equation of the active particles. The existence of weak solutions to the stationary problem is established through a two-level approximation scheme, compactness estimates and weak convergence arguments. Novel techniques are developed to overcome the difficulties due to the lower regularity of stationary solutions, a Moser-type iteration is used to deal with the …
We consider weak solutions to a two-dimensional simplified Ericksen-Leslie system of compressible flow of nematic liquid crystals. An initial-boundary value problem is first studied in a bounded domain. By developing new techniques and estimates to overcome the difficulties induced by the supercritical nonlinearity in the equations of angular momentum on the direction field, and adapting the standard three-level approximation scheme and the weak convergence arguments for the compressible Navier-Stokes equations, we establish the global existence of weak solutions under a restriction imposed on…
We study the global existence of weak solutions to a multi-dimensional simplified Ericksen-Leslie system for compressible flows of nematic liquid crystals with large initial energy in a bounded domain $Ω\subset \mathbb{R}^N$, where N=2 or 3. By exploiting a maximum principle, Nirenberg's interpolation inequality and a smallness condition imposed on the $N$-th component of initial direction field $\mf{d}_0$ to overcome the difficulties induced by the supercritical nonlinearity $|\nabla{\mathbf d}|^2{\mathbf d}$ in the equations of angular momentum, and then adapting a modified three-dimensional…
This paper is concerned with the three-dimensional equations of a simplified hydrodynamic flow modeling the motion of compressible, nematic liquid crystal materials. The authors establish the global existence of classical solution to the Cauchy problem with smooth initial data which are of small energy but possibly large oscillations with constant state as far-field condition which could be either vacuum or non-vacuum. The initial density is allowed to vanish and the spatial measure of the set of vacuum can be arbitrarily large, in particular, the initial density can even have compact support.…
This paper is devoted to establishing the optimal decay rate of the global large solution to compressible nematic liquid crystal equations when the initial perturbation is large and belongs to $L^1(\mathbb R^3)\cap H^2(\mathbb R^3)$. More precisely, we show that the first and second order spatial derivatives of large solution $(ρ-1, u, \nabla d)(t)$ converges to zero at the $L^2-$rate $(1+t)^{-\frac54}$ and $L^2-$rate $(1+t)^{-\frac74}$ respectively, which are optimal in the sense that they coincide with the decay rates of solution to the heat equation. Thus, we establish optimal decay rate fo…
Global Existence and Large-Time Behavior of Strong Solutions to the Two-Dimensional Compressible Nematic Liquid Crystal Flows with Large Initial Data and Vacuum
University of Chinese Academy of Sciences（中国科学院大学）；Institute of Applied Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院应用数学研究所）
In this paper we establish the incompressible limit for the compressible free-boundary Euler equations with surface tension in the case of a liquid. Compared to the case without surface tension treated recently, the presence of surface tension introduces severe new technical challenges, in that several boundary terms that automatically vanish when surface tension is absent now contribute at top order. Combined with the necessity of producing estimates uniform in the sound speed in order to pass to the limit, such difficulties imply that neither the techniques employed for the case without surf…
In this paper we study existence and regularity of solutions to the capillarity problem for compressible liquids in a tube. We introduce an appropriate space of functions of bounded variation, in which the energy functional recently introduced by Robert Finn can be defined. We prove existence of a locally Lipschitz minimiser in this class.
Weak-strong uniqueness property in the class of finite energy weak solutions is established for two different compressible liquid crystal systems by the method of relative entropy. To overcome the difficulties caused by the molecular direction with inhomogeneous Dirichlet boundary condition, new techniques are introduced to build up the relative entropy inequalities.
We consider the compressible Ericksen-Leslie system of liquid crystal flows in one dimension. A global weak solution is constructed with initial density $ρ_0\geq 0$ and $ρ_0\in L^γ$ for $γ>1$.
We study the motion of a compressible perfect liquid body in vacuum. This can be thought of as a model for the motion of the ocean or a star. The free surface moves with the velocity of the liquid and the pressure vanishes on the free surface. This leads to a free boundary problem for Euler's equations, where the regularity of the boundary enters to highest order. We prove local existence in Sobolev spaces assuming a "physical condition", related to the fact that the pressure of a fluid has to be positive.
A Regularized Framework and Admissible Solutions for Liquid-Vapor Phase Transitions in Steady Compressible Flows
College of Mathematics and Physics, Beijing University of Chemical Technology（北京化工大学理学院）；School of Mathematical Science, Sichuan University School（四川大学数学学院）；School of Mathematical Science, Capital Normal University School（首都师范大学数学学院）
In this paper, we study a coupled compressible Navier-Stokes/Q-tensor system modeling the nematic liquid crystal flow in a three-dimensional bounded spatial domain. The existence and long time dynamics of globally defined weak solutions for the coupled system are established, using weak convergence methods, compactness and interpolation arguments. The symmetry and traceless properties of the Q-tensor play key roles in this process.
In this paper we prove the local well-posedness (LWP) for the 3D compressible Euler equations describing the motion of a liquid in an unbounded initial domain with moving boundary. The liquid is under the influence of gravity but without surface tension, and it is not assumed to be irrotational. We apply the tangential smoothing method introduced in [10,11] to construct the approximation system with energy estimates uniform in the smooth parameter. It should be emphasized that, when doing the nonlinear a priori estimates, we need neither the higher order wave equation of the pressure and delic…
We prove a priori estimates for the compressible Euler equations modeling the motion of a liquid with moving physical vacuum boundary in an unbounded initial domain. The liquid is under influence of gravity but without surface tension. Our fluid is not assumed to be irrotational. But the physical sign condition needs to be assumed on the free boundary. We generalize the method used in \cite{LL} to prove the energy estimates in an unbounded domain up to arbitrary order. To our knowledge, this result appears to be the first that concerns a priori energy bounds for the compressible water wave. In…
We prove a local existence theorem for the free boundary problem for a relativistic fluid in a fixed spacetime. Our proof involves an a priori estimate which only requires control of derivatives tangential to the boundary, which holds also in the Newtonian compressible case.
The global weak martingale solution is built through a four-level approximation scheme to stochastic compressible active liquid crystal system driven by multiplicative noise in a smooth bounded domain in $\mathbb{R}^{3}$ with large initial data. The coupled structure makes the analysis challenging, and more delicate arguments are required in stochastic case compared to the deterministic one \cite{11}.
Global existence for weak solutions to systems of nematic liquid crystals, with non-constant fluid density has been established by several authors. In this paper, we establish the regularity and uniqueness results for solutions to the density dependent nematic liquid crystals system.
In this paper, we consider the global existence and uniqueness of the classical solutions for the 3D viscous liquid-gas two-phase flow model. Initial data is only small in the energy-norm. Our main ideas come from [15] where the existence of global classical solutions to the compressible Navier-Stokes equations was obtained by using the continuity methods under the assumption that the initial energy is sufficiently small.
In this paper, we prove the stability of half-degree point defect profiles in $\mathbb{R}^2$ for the nematic liquid crystal within Landau-de Gennes model.
In this paper we construct a family of exact strong solutions to the two-dimensional incompressible liquid crystal equations with finite energy. The initial velocity is chosen to be rotationally symmetric and the image of the initial orientation of the liquid crystal is a non-trivial curve on the unit sphere. It turns out that this family of initial data evolves globally in time by liquid crystal flow and may shrink to a single point as time goes to infinity.
