We study the long-time behavior of global strong solutions to a hydrodynamic system for nonhomogeneous incompressible nematic liquid crystal flows driven by two types of external forces in a smooth bounded domain in $\mathbb{R}^2$. For arbitrary large regular initial data with the initial density being away from vacuum, we prove the decay of the velocity field for both cases. Furthermore, for the case with asymptotically autonomous external force, we can prove the convergence of the density function and the director vector as time goes to infinity. Estimates on convergence rate are also provided.
This paper concerns the Cauchy problem of the two-dimensional (2D) nonhomogeneous incompressible nematic liquid crystal flows on the whole space $\mathbb{R}^{2}$ with vacuum as far field density. It is proved that the 2D nonhomogeneous incompressible nematic liquid crystal flows admits a unique global strong solution provided the initial data density and the gradient of orientation decay not too slow at infinity, and the initial orientation satisfies a geometric condition (see \eqref{eq1.3}). In particular, the initial data can be arbitrarily large and the initial density may contain vacuum st…
In this paper, we establish the mathematical validity of the Prandtl boundary layer theory for a class of nonlinear plane parallel flow of nonhomogeneous incompressible Navier-Stokes equations. The convergence for the density and velocity are shown under various Sobolev norms, including the physically important space-time uniform norm, as well as the $L^\infty(H^1)$ norm. It is mentioned that the mathematical validity of the Prandtl boundary layer theory for nonlinear plane parallel flow is generalized to the nonhomogeneous case.
This paper is concerned with a Cauchy problem for the three-dimensional (3D) nonhomogeneous incompressible heat conducting magnetohydrodynamic (MHD) equations in the whole space. First of all, we establish a weak Serrin-type blowup criterion for the strong solutions. It is shown that for the Cauchy problem of the 3D nonhomogeneous heat conducting MHD equations, the strong solution exists globally if the velocity satisfies the weak Serrin's condition. In particular, this criterion is independent of the absolute temperature and magnetic field. Then as an immediate application, we prove the globa…
This paper is concerned with the Rayleigh-Taylor instability for the nonhomogeneous incompressible Navier-Stokes equations with Navier-slip boundary conditions around a steady-state in an infinite slab, where the Navier-slip coefficients do not have defined sign and the slab is horizontally periodic. Motivated by [18], we extend the result from Dirichlet boundary condition to Navier-slip boundary conditions. Our results indicate the factor that "heavier density with increasing height" still plays a key role in the instability under Navier-slip boundary conditions.
In this paper, we investigate the vanishing viscosity limit for the 3D nonhomogeneous incompressible Navier-Stokes equations with a slip boundary condition. We establish the local well-posedness of the strong solutions for initial boundary value problems for such systems. Furthermore, the vanishing viscosity limit process is established and a strong rate of convergence is obtained as the boundary of the domain is flat. In addition, it is needed to add some additional condition for density to match well the boundary condition.
We consider the spectral semi-Galerkin method applied to the nonhomogeneous Navier-Stokes equations. Under certain conditions it is known that the approximate solutions constructed through this method converge to a global strong solution of these equations. Here, we derive an optimal uniform in time error estimate in the $H^1$ norm for the velocity. We also derive an error estimate for the density in some spaces $L^r$.
In this paper, we investigate the Cauchy problem of the nonhomogeneous incompressible non-resistive MHD on $\mathbb{R}^2$ with vacuum as far field density and prove that the 2D Cauchy problem has a unique local strong solution provided that the initial density and magnetic field decay not too slow at infinity. Furthermore, if the initial data satisfy some additional regularity and compatibility conditions, the strong solution becomes a classical one.
The present paper deals with the nonhomogeneous incompressible asymmetric fluids equations in dimension $d= 2,3$. The aim is to prove the unique global solvability of the system with only bounded nonnegative initial density and $H^{1}$ initial velocities. We first construct the global existence of the solution with large data in 2-D. Next, we establish the existence of local in time solution for arbitrary large data and global in time for some smallness conditions in 3-D. Finally, the uniqueness of the solution is proved under quite soft assumptions about its regularity through a Lagrangian ap…
In [F. Jiang, S. Jiang, On instability and stability of three-dimensional gravity driven viscous flows in a bounded domain, Adv. Math., 264 (2014) 831--863], Jiang et.al. investigated the instability of Rayleigh--Taylor steady-state of a three-dimensional nonhomogeneous incompressible viscous flow driven by gravity in a bounded domain $Ω$ of class $C^2$. In particular, they proved the steady-state is nonlinearly unstable under a restrictive condition of that the derivative function of steady density possesses a positive lower bound. In this article, by exploiting a standard energy functional a…
In this paper, the authors first establish the global well-posedness of strong solutions of the simplified Ericksen-Leslie model for nonhomogeneous incompressible nematic liquid crystal flows in two dimensions if the initial data satisfies some smallness condition. It is worth pointing out that the initial density is allowed to contain vacuum states and the initial velocity can be arbitrarily large. We also present a Serrin's type criterion, depending only on $\nabla d$, for the breakdown of local strong solutions. As a byproduct, the global strong solutions with large initial data are obtaine…
We propose a model of a density-dependent compressible-incompressible fluid, which is intended as a simplified version of models based on mixture theory as, for instance, those arising in the study of biofilms, tumor growth and vasculogenesis. Though our model is, in some sense, close to the density-dependent incompressible Euler equations, it presents some differences that require a different approach from an analytical point of view. In this paper, we establish a result of local existence and uniqueness of solutions in Sobolev spaces to our model, using paradifferential techniques. Besides, …
We investigate the nonlinear instability of a smooth Rayleigh-Taylor steady-state solution (including the case of heavier density with increasing height) to the three-dimensional incompressible nonhomogeneous magnetohydrodynamic (MHD) equations of zero resistivity in the presence of a uniform gravitational field. We first analyze the linearized equations around the steady-state solution. Then we construct solutions of the linearized problem that grow in time in the Sobolev space $H^k$, thus leading to the linear instability. With the help of the constructed unstable solutions of the linearized…
In this paper, the three-dimensional stochastic nonhomogeneous incompressible Navier-Stokes equations driven by Lévy process consisting of the Brownian motion, the compensated Poisson random measure and the Poisson random measure are considered in a bounded domain. We obtain the existence of martingale solutions. The construction of the solution is based on the classical Galerkin approximation method, stopping time, the compactness method and the Jakubowski-Skorokhod theorem.
In this paper, we first prove the unique global strong solution with vacuum to the two dimensional nonhomogeneous incompressible MHD system, as long as the initial data satisfies some compatibility condition. As a corollary, the global existence of strong solution with vacuum to the 2D nonhomogeneous incompressible Navier-Stokes equations is also established. Our main result improves all the previous results where the initial density need to be strictly positive. The key idea is to use some critical Sobolev inequality of logarithmic type, which is originally due to Brezis-Wainger.
In this paper, we consider the non-isothermal model for incompressible flow of nematic liquid crystals in three dimensions and prove the local existence and uniqueness of the strong solution with periodic initial conditions on $ \mathbb{T}^3$ .
We are concerned with the Cauchy problem of the two-dimensional (2D) nonhomogeneous incompressible nematic liquid crystal flows on the whole space $\mathbb{R}^{2}$ with vacuum as far field density. It is proved that the 2D nonhomogeneous incompressible nematic liquid crystal flows admits a unique global strong solution provided the initial data density and the gradient of orientation decay not too slow at infinity, and the basic energy $\|\sqrt{ρ_0}\mathbf{u}_0\|_{L^2}^2+\|\nabla\mathbf{d}_0\|_{L^2}^2$ is small. In particular, the initial density may contain vacuum states and even have compact…
Global Weak Solutions of a Thermodynamically Consistent Diffuse Interface Model for Nonhomogeneous Incompressible Two-phase Flows with a Soluble Surfactant
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 incompressible nematic liquid crystal flows in three-dimensional whole space. First of all, we establish the global existence of solution by energy method under assumption of small initial data. Furthermore, the time decay rates of velocity and director are built when the initial data belongs to $L^1(\mathbb{R}^3)$ additionally. Finally, one also constructs the time convergence rates for the mixed space-time derivatives of velocity and director.
We are concerned with an initial boundary value problem for the nonhomogeneous heat conducting Navier-Stokes flows with non-negative density. First of all, we show that for the initial density allowing vacuum, the strong solution exists globally if the velocity satisfies the Serrin's condition. Then, under some smallness condition, we prove that there is a unique global strong solution to the 3D viscous nonhomogeneous heat conducting Navier-Stokes flows. Our method relies upon the delicate energy estimates and regularity properties of Stokes system and elliptic equation.
In this paper we derive LPS's criterion for the breakdown of classical solutions to the incompressible nematic liquid crystal flow, a simplified version of Ericksen-Leslie system modeling the hydrodynamic evolution of nematic liquid crystals in $\mathbb R^3$. We show that if $0<T<+\infty$} is the maximal time interval for the unique smooth solution $u\in C^\infty([0,T),\mathbb R^3)$, then $|u|+|\nabla d|\notin L^q([0,T],L^p(\mathbb R^3))$, where $p$ and $q$ safisfy the Ladyzhenskaya-Prodi-Serrin's condition: $\frac{3}{p}+\frac{2}{q}=1$ and $p\in(3,+\infty]$
In this paper, we investigate the density-dependent incompressible nematic liquid crystal flows in $n(n=2$ or $3)$ dimensional bounded domain. More precisely, we obtain the local existence and uniqueness of the solutions when the viscosity coefficient of fluid depends on density. Moreover, we establish blowup criterions for the regularity of the strong solutions in dimension two and three respectively. In particular, we build a blowup criterion just in terms of the gradient of density if the initial direction field satisfies some geometric configuration. For these results, the initial density …
We investigate the nonlinear instability of a smooth steady density profile solution of the threedimensional nonhomogeneous incompressible Navier-Stokes equations in the presence of a uniform gravitational field, including a Rayleigh-Taylor steady-state solution with heavier density with increasing height (referred to the Rayleigh-Taylor instability). We first analyze the equations obtained from linearization around the steady density profile solution. Then we construct solutions of the linearized problem that grow in time in the Sobolev space Hk, thus leading to a global instability result fo…
In this paper, we prove some a priori estimates for a system of partial differential equations arising in the nonstationary flow of a nonhomogeneous incompressible asymmetric fluid in a bounded domain with smooth boundary. The unknowns of the system are the velocity field of the fluid particles, the angular velocity of rotation of the fluid particles, the mass density of the fluid and the pressure distribution. For the density functions we consider the application of the Helmholtz decomposition.
This paper concerns the Cauchy problem of the nonhomogeneous incompressible magnetohydrodynamic (MHD) equations on the whole two-dimensional (2D) space with vacuum as far field density. In particular, the initial density can have compact support. We prove that the 2D Cauchy problem of the nonhomogeneous incompressible MHD equations admits a unique local strong solution provided the initial density and the initial magnetic decay not too slow at infinity.
In this paper, we prove the local uniqueness of an inverse problem arising in the nonstationary flow of a nonhomogeneous incompressible asymmetric fluid in a bounded domain with smooth boundary. The direct problem is an initial-boundary value problem for a system for the velocity field, the angular velocity of rotation of the fluid particles, the mass density and the pressure distribution. The inverse problem consists in the external force recover assuming an integral measurements on the boundary. We characterize the inverse problem solutions using an operator equation of second kind, which is…
Motivated by the works of Cheskidov, Lopes Filho, Nussenzveig Lopes and Shvydkoy in [8, Commun. Math. Phys. 348: 129-143, 2016] and Chen and Yu in [5, J. Math. Pures Appl. 131: 1-16, 2019], we address how the $L^p$ control of vorticity could influence the energy conservation for the incompressible homogeneous and nonhomogeneous Euler equations in this paper. For the homogeneous flow in the periodic domain or whole space, we provide a self-contained proof for the criterion $ω=\text{curl}u\in L^{3}(0,T;L^{\frac{3n}{n+2}}(Ω))\,(n=2,3)$, which generalizes the corresponding result in [8] and can be…
We consider the global existence and large-time asymptotic behavior of strong solutions to the Cauchy problem of the three-dimensional nonhomogeneous incompressible Navier-Stokes equations with density-dependent viscosity and vacuum. We establish some key a priori exponential decay-in-time rates of the strong solutions. Then after using these estimates, we also obtain the global existence of strong solutions in the whole three-dimensional space, provided that the initial velocity is suitably small in the $\dot H^β$-norm for some $β\in(1/2,1].$ Note that this result is proved without any smalln…
We prove the existence of large-data global-in-time weak solutions to a general class of coupled bead-spring chain models with finitely extensible nonlinear elastic (FENE) type spring potentials for nonhomogeneous incompressible dilute polymeric fluids in a bounded domain in $\mathbb{R}^d$, $d=2$ or $3$. The class of models under consideration involves the Navier--Stokes system with variable density, where the viscosity coefficient depends on both the density and the polymer number density, coupled to a Fokker--Planck equation with a density-dependent drag coefficient. The proof is based on co…
In this paper, we investigate the uniform regularity and vanishing limit for the incompressible nematic liquid crystal flows in three dimensional bounded domain. It is shown that there exists a unique strong solution for the incompressible nematic liquid crystal flows with boundary condition in a finite time interval which is independent of the viscosity. The solution is uniformly bounded in a conormal Sobolev space. Finally, we also study the convergence rate of the viscous solutions to the inviscid ones.
In this paper, we consider the short time classical solution to a simplified hydrodynamic flow modeling incompressible, nematic liquid crystal materials in dimension three. We establish a criterion for possible breakdown of such solutions at a finite time. More precisely, if $(u,d)$ is smooth up to time $T$ provided that $\int_0^T|\nabla\times u(t,\cdot)|_{BMO(\mathbb R^3)}+|\nabla d(t,\cdot)|^8_{L^4(\mathbb R^3)}dt<\infty $
Global Well-posedness of Strong Solutions to the Cauchy Problem of 2D Nonhomogeneous Navier-Stokes Equations with Density-Dependent Viscosity and Vacuum
This paper is devoted to the existence of a weak solution to a system describing a self-propelled motion of a rigid body in a viscous fluid in the whole $\mathbb{R}^3$. The fluid is modelled by the incompressible nonhomogeneous Navier-Stokes system with a nonnegative density. The motion of the rigid body is described by the balance of linear and angular momentum. We consider the case where slip is allowed at the fluid-solid interface through Navier condition and prove the global existence of a weak solution.
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 $T_{*}<+\infty$ if and only if \begin{align*} \int_{0}^{T_*} \big( \|\nabla_h u^h\|_{\dot{B}^0_{p,\frac{2p}{3}}}^q + \|\nabla d\|_{\dot{B}^0_{\infty,\infty}}^2 \big)dt = \infty\ \ \text{wit…
In this paper, we consider the Dirichlet problem of inhomogeneous incompressible nematic liquid crystal equations in bounded smooth domains of two or three dimensions. We prove the global existence and uniqueness of strong solutions with initial data being of small norm but allowed to have vacuum. More precisely, for two dimensional case, we only require that the basic energy $|\sqrt{ρ_0}u_0|_{L^2}^2+|\nabla d_0|_{L^2}^2$ is small, while for three dimensional case, we ask for the smallness of the production of the basic energy and the quantity $|\nabla u_0|_{L^2}^2+|\nabla^2d_0|_{L^2}^2$. Our …
Global Existence and Large-Time Asymptotic Behavior of Strong Solutions to the Two-Dimensional Nematic Liquid Crystal Flows with Large Initial Data and Vacuum
University of Chinese Academy of Sciences（中国科学院大学）
The temperature-dependent incompressible nematic liquid crystal flows in a bounded domain $Ω\subset\mathbb{R}^N$ ($N=2, 3$) are studied in this paper. Following Danchin's method in [J. Math. Fluid Mech., 2006], we use a localization argument to recover the maximal regularity of Stokes equation with variable viscosity, by which we first prove the local existence of strong solution, then extend it to a global one provided that the initial data is a sufficiently small perturbation around the trivial equilibrium state. This paper also generalizes Hu-Wang's result in [Commun. Math. Phys., 2010] to …
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 the existence of a trajectory attractor (in the sense of V.V. Chepyzhov and M.I. Vishik) for a nonlinear PDE system coming from a 3D liquid crystal model accounting for stretching effects. The system couples a nonlinear evolution equation for the director d (introduced in order to describe the preferred orientation of the molecules) with an incompressible Navier-Stokes equation for the evolution of the velocity field u. The technique is based on the introduction of a suitable trajectory space and of a metric accounting for the double-well type nonlinearity contained in t…
The Ericksen-Leslie model for nematic liquid crystals in a bounded domain with general Leslie and isotropic Ericksen stress is studied in the case of a non-isothermal and incompressible fluid. This system is shown to be locally well-posed in the $L_p$-setting, and a dynamic theory is developed. The equilibria are identified and shown to be normally stable. In particular, a local solution extends to a unique, global strong solution provided the initial data are close to an equilibrium or the solution is eventually bounded in the topology of the natural state manifold. In this case, the solution…
The present contribution investigates the well-posedness of a PDE system describing the evolution of a nematic liquid crystal flow under kinematic transports for molecules of different shapes. More in particular, the evolution of the {\em velocity field} $\ub$ is ruled by the Navier-Stokes incompressible system with a stress tensor exhibiting a special coupling between the transport and the induced terms. The dynamic of the {\em director field} $\bd$ is described by a variation of a parabolic Ginzburg-Landau equation with a suitable penalization of the physical constraint $|\bd|=1$. Such equat…
In this paper we prove the existence of global in time weak solutions for an evolutionary PDE system modelling nonisothermal Landau-de Gennes nematic liquid crystal (LC) flows in three dimensions of space. In our model, the incompressible Navier-Stokes system for the macroscopic velocity $\vu$ is coupled to a nonlinear convective parabolic equation describing the evolution of the Q-tensor $\QQ$, namely a tensor-valued variable representing the normalized second order moments of the probability distribution function of the LC molecules. The effects of the (absolute) temperature $\vt$ are prescr…
A model describing the evolution of a liquid crystal substance in the nematic phase is investigated in terms of three basic state variables: the {\it absolute temperature} $\teta$, the {\it velocity field} $\ub$, and the {\it director field} $\bd$, representing preferred orientation of molecules in a neighborhood of any point of a reference domain. The time evolution of the velocity field is governed by the incompressible Navier-Stokes system, with a non-isotropic stress tensor depending on the gradients of the velocity and of the director field $\bd$, where the transport (viscosity) coefficie…
In recent paper \cite{DW1} (Y. Du and K. Wang, Space-time regularity of the Kock $\&$ Tataru solutions to the liquid crystal equations, SIAM J. Math. Anal., \textbf{45}(6), 3838--3853.), the authors proved that the global-in-time Koch-Tataru type solution $(u,d)$ to the $n$-dimensional incompressible nematic liquid crystal flow with small initial data $(u_{0},d_{0})$ in $BMO^{-1}\times BMO$ has arbitrary space-time derivative estimates in the so called Koch-Tataru space norms. The purpose of this paper is to show that the Koch-Tataru type solution satisfies the decay estimates for any space-ti…
