Hans Lindblad
We study the problem of the motion of the free surface of a liquid. We prove existence and stability for the linearized equations.
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We study the asymptotic behaviour, in the sense of $Γ$-convergence, of a thin incompressible magnetoelastic plate, as its thickness goes to zero. We focus on the linearized von Kármán regime. The model features a mixed Eulerian-Lagrangian formulation, as magnetizations are defined on the deformed confguration.
We consider the linearized and nonlinear stationary incompressible flow around rotating and translating body in the exterior domain B with Lipschitz boundary. We derive the pointwise estimates for the pressure in both cases. Moreover, we consider the linearized problem in a truncation domain B_R of the exterior domain B under certain artificial boundary conditions on the truncating boundary and then compare this solution with the solution in the exterior domain B to get the truncation error estimate.
We study an unbounded operator arising naturally after linearizing the system modelling the motion of a rigid body in a viscous incompressible fluid. We show that this operator is $\mathcal{R}$ sectorial in $L^q$ for every $q\in (1,\infty)$, thus it has the maximal $L^p$-$L^q$ regularity property. Moreover, we show that the generated semigroup is exponentially stable with respect to the $L^q$ norm. Finally, we use the results to prove the global existence for small initial data, in an $L^p$-$L^q$ setting, for the original nonlinear problem.
We study the well-posedness theory for the linearized free boundary problem of incompressible ideal magnetohydrodynamics equations in a bounded domain. We express the magnetic field in terms of the velocity field and the deformation tensors in the Lagrangian coordinates, and substitute the magnetic field into the momentum equation to get an equation of the velocity in which the initial magnetic field serves only as a parameter. Then, we linearize this equation with respect to the position vector field whose time derivative is the velocity, and obtain the local-in-time well-posedness of the sol…
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.
We study the problem of the motion of the free surface of a compressible fluid. We prove existence for the linearized equations.
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$ .
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.
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.
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…
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.
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.
The initial boundary value problem for the three-dimensional incompressible flow of liquid crystals is considered in a bounded smooth domain. The existence and uniqueness is established for both the local strong solution with large initial data and the global strong solution with small data. It is also proved that when the strong solution exists, a weak solution must be equal to the unique strong solution with the same data.
In this paper, we prove a priori estimates in Lagrangian coordinates for the equations of motion of an incompressible, inviscid, self-gravitating fluid with free boundary. The estimates show that on a finite time interval we control five derivatives of the fluid velocity and five and a half derivatives of the coordinates of the moving domain.
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.
The initial-boundary value problem for the density-dependent incompressible flow of liquid crystals is studied in a three-dimensional bounded smooth domain. For the initial density away from vacuum, the existence and uniqueness is established for both the local strong solution with large initial data and the global strong solution with small data. It is also proved that when the strong solution exists, a weak solution with the same data must be equal to the unique strong solution.
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…
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 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 $p\in(3,+\infty]$
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 provid…
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…
In this work, we analyze a Stokes problem arising in the study of the Navier-Stokes flow of a liquid jet. The analysis is accomplished by showing that the relevant Stokes operator accounting for a free surface gives rise to a sectorial operator which generates an analytic semigroup of contractions. Estimates on solutions are established using Fourier methods. The result presented is the key ingredient in a local existence and uniqueness proof for solutions of the full nonlinear problem.
We study the Ericksen-Leslie's hyperbolic incompressible liquid crystal model. Under some constraints on the Leslie coefficients which ensure the basic energy law is dissipative, we prove the local-in-time existence and uniqueness of the classical solution to the system with finite initial energy. Furthermore, with an additional assumption on the coefficients which provides a damping effect, and the smallness of the initial energy, the unique global classical solution can be established.
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 …
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…
Double Porosity Models for Liquid Filtration in Incompressible Poroelastic Media
本文基于均匀化理论，通过迭代均匀化方法推导了不可压缩多孔弹性介质中液体过滤的双孔隙度模型，得到了Biot-Terzaghi方程组及绝对刚性体的对应模型。
Nonuniqueness of Leray-Hopf solutions to the unforced incompressible 3D Navier-Stokes Equation
本文通过计算机辅助证明首次展示了拉耶-霍夫解的非唯一性，通过构造自相似解并分析线性化算子稳定性，证明了存在无限多个拉耶-霍夫解。
We prove that the incompressible, density dependent, Navier-Stokes equations are globally well posed in a low Froude number regime. The density is supposed to be increasing in depth and linearized around a stable state. Moreover if the Froude number tends to zero we prove that such system converges (strongly) to a two-dimensional, stratified Navier-Stokes system with full diffusivity. No smallness assumption is made on the initial data.
Incompressible limit of the Ericksen-Leslie hyperbolic liquid crystal model in compressible flow
本文在经典解框架下研究可压缩Ericksen-Leslie抛物-双曲液晶模型的不可压缩极限，通过引入惯性常数并建立关于Mach数一致的能量估计，证明了不可压缩系统全局经典解的存在性并给出收敛速率。
Koch-Tataru theorem for 3D incompressible active nematic liquid crystals
本文基于Beris-Edwards框架研究了三维不可压缩活性向列液态晶体的流体动力学系统，通过Kato策略和Banach压缩原理，证明了在小初始数据下解的存在性和唯一性，首次在临界空间中获得该系统的适定性结果。
We consider a hydrodynamic system that models the Smectic-A liquid crystal flow. The model consists of the Navier-Stokes equation for the fluid velocity coupled with a fourth-order equation for the layer variable $\vp$, endowed with periodic boundary conditions. We analyze the long-time behavior of the solutions within the theory of infinite-dimensional dissipative dynamical systems. We first prove that in 2D, the problem possesses a global attractor $\mathcal{A}$ in certain phase space. Then we establish the existence of an exponential attractor $\mathcal{M}$ which entails that the global att…
Exponential stability of a free boundary problem with spherical symmetry for a gas bubble immersed in a bounded incompressible liquid
本文研究了有界不可压缩粘性液体中气泡的球对称自由边界问题，证明了平衡态在小扰动下的指数稳定性，并推导了小液体体积下的最优指数衰减率。
On a dryout point for a stationary incompressible thermal fluid with phase transition in a pipe
本文研究管道中定常不可压缩热流体的干涸点问题，通过求解一维 Navier-Stokes-Fourier 方程导出了干涸点存在的充要条件，并基于热力学确定了干涸点温度与气相密度。
We consider pure traction problems and we show that incompressible linearized elasticity can be obtained as variational limit of incompressible finite elasticity under suitable conditions on external loads.
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…
We derive geometrically linearized theories for incompressible materials from nonlinear elasticity theory in the small displacement regime. Our nonlinear stored energy densities may vary on the same (small) length scale as the typical displacements. This allows for applications to multiwell energies as, e.g., encountered in martensitic phases of shape memory alloys and models for nematic elastomers. Under natural assumptions on the asymptotic behavior of such densities we prove Gamma-convergence of the properly rescaled nonlinear energy functionals to the relaxation of an effective model. The …
The aim of this paper is to show time-decay estimates of solutions to linearized two-phase Navier-Stokes equations with surface tension and gravity. The original two-phase Navier-Stokes equations describe the two-phase incompressible viscous flow with a sharp interface that is close to the hyperplane $x_N=0$ in the $N$-dimensional Euclidean space, $N \geq 2$. It is well-known that the Rayleigh-Taylor instability occurs when the upper fluid is heavier than the lower one, while this paper assumes that the lower fluid is heavier than the upper one and proves time-decay estimates of $L_p-L_q$ type…
We are interested in studying an unsteady fluid-structure interaction problem in a three-dimensional space. We consider a homogeneous Newtonian fluid which is modeled by the Navier-Stokes equations. Whereas the motion of the structure is described by the quasi-incompressible non-linear Saint Venant-Kirchhoff model. We establish the local in time existence and uniqueness of solution for this model. For this sake, first we rewrite the non-linearity of the elastodynamic equation in an explicit way. Then, a linearized problem is introduced in the Lagrangian reference configuration and we prove tha…
In this paper, we derive sharp bounds on the semigroup of the linearized incompressible Navier-Stokes equations near a stationary shear layer in the half plane and in the half space ($\mathbb{R}_+^2$ or $\mathbb{R}_+^3$), with Dirichlet boundary conditions, assuming that this shear layer in spectrally unstable for Euler equations. In the inviscid limit, due to the prescribed no-slip boundary conditions, vorticity becomes unbounded near the boundary. The novelty of this paper is to introduce boundary layer norms that capture the unbounded vorticity and to derive sharp estimates on this vorticit…
The equations for the three-dimensional incompressible flow of liquid crystals are considered in a smooth bounded domain. The existence and uniqueness of the global strong solution with small initial data are established. It is also proved that when the strong solution exists, all the global weak solutions constructed in [16] must be equal to the unique strong solution.
This paper is the first part of a work which consists in proving the stabilization to zero of a fluid-solid system, in dimension 2 and 3. The considered system couples a deformable solid and a viscous incompressible fluid which satisfies the incompressible Navier-Stokes equations. By deforming itself, the solid can interact with the environing fluid and then move itself. The control function represents nothing else than the deformation of the solid in its own frame of reference. We there prove that the velocities of the linearized system are stabilizable to zero with an arbitrary exponential d…
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 $
We study the Rayleigh-Taylor problem for two incompressible, immiscible, viscous magnetohydrodynamic (MHD) flows, with zero resistivity, surface tension (or without surface tenstion) and special initial magnetic field, evolving with a free interface in the presence of a uniform gravitational field. First, we reformulate in Lagrangian coordinates MHD equations in a infinite slab as one for the Navier-Stokes equations with a force term induced by the fluid flow map. Then we analyze the linearized problem around the steady state which describes a denser immiscible fluid lying above a light one wi…
Initial-boundary value problem for linearized equations of motion of viscous barotropic fluid in a bounded domain is considered. Existence, uniqueness and estimates of weak solutions to this problem are derived. Convergence of the solutions towards the incompressible limit when compressibility tends to zero is studied.
In this article, we consider the Ericksen-Leslie's hyperbolic system for incompressible liquid crystal model without kinematic transport in three spatial dimensions, which is a nonlinear coupling of incompressible Navier-Stokes equations with wave map to $\mathbb{S}^2$. Global regularity for small and smooth initial data near the equilibrium is proved. The proof relies on the idea of space-time resonance.
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.
We study the motion of an incompressible 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.
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…
Global well-posedness and decay rates for the three dimensional incompressible active liquid crystals
本文研究了在恒定活动性下三维不可压缩活性液晶的全局适定性及长时间行为，通过改进的对流算子估计，证明了小初始数据下的全局强解存在唯一性，并获得了混合衰减估计，揭示了高活动性下活性向列相趋于各向同性，且衰减率与活动性相关。
In this paper, we establish some local and global solutions for the two phase incompressible inhomogeneous flows with moving interfaces in $L_p-L_q$ maximal regularity class. Compared with previous results obtained by V.A.Solonnikov and by Y.Shibata \& S.Shimizu, we find the local solutions in $L_p-L_q$ class in some general uniform $W^{2-1\slash r}_r$ domain in $\BBR^N$ by assuming $(p, q)\in ]2,\infty[\times ]N,\infty[$ or $(p, q)\in ]1,2[ \times ]N,\infty[$ satisfying $1 \slash p + N \slash q>3\slash 2.$ In particular, less regular initial data are allowed by assuming $p<2.$ In addition, if…
Stability analysis of the incompressible porous media equation and the Stokes transport system via energy structure
本文利用能量结构直接分析二维周期通道中多孔介质方程与Stokes输运系统的非线性渐近稳定性，通过势能二阶导数的退化强制性结构改进扰动正则性假设与收敛速率。
We consider the free boundary problem for current-vortex sheets in ideal incompressible magneto-hydrodynamics. It is known that current-vortex sheets may be at most weakly (neutrally) stable due to the existence of surface waves solutions to the linearized equations. The existence of such waves may yield a loss of derivatives in the energy estimate of the solution with respect to the source terms. However, under a suitable stability condition satisfied at each point of the initial discontinuity and a flatness condition on the initial front, we prove an a priori estimate in Sobolev spaces for s…
We study the spectral properties of the linearized Euler operator obtained by linearizing the equations of incompressible two dimensional fluid at a steady state with the vorticity that contains only two nonzero complex conjugate Fourier modes. We prove that the essential spectrum coincides with the imaginary axis, and give an estimate from above for the number of isolated nonimaginary eigenvalues. In addition, we prove that the spectral mapping theorem holds for the group generated by the linearized 2D Euler operator.
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…
On the Inhibition of Rayleigh Taylor Instability by Capillarity in the Navier Stokes Korteweg Model
本文严格证明了在非线性不可压缩Navier-Stokes-Korteweg方程中，当毛细系数超过阈值时，毛细作用可抑制Rayleigh-Taylor不稳定性，通过捕捉密度切向导数的耗散估计实现速度与密度扰动的衰减。
In this paper we study a coupled system modeling the movement of a deformable solid immersed in a fluid. For the solid we consider a given deformation that has to obey several physical constraints. The motion of the fluid is modeled by the incompressible Navier-Stokes equations in a time-dependent bounded domain of $\R^3$, and the solid satisfies the Newton's laws. Our contribution consists in adapting and completing some results of ARMA 2008 in dimension 3, in a framework where the regularity of the deformation of the solid is limited. We rewrite the main system in domains which do not depend…
In this paper we consider a slab of viscous incompressible fluid bounded above by a free boundary, bounded below by a flat rigid interface, and acted on by gravity. The unique equilibrium is a flat slab of quiescent fluid. It is well-known that equilibria are asymptotically stable but that the rate of decay to equilibrium depends heavily on whether or not surface tension forces are accounted for at the free interface. The aim of the paper is to better understand the decay rate by studying a generalization of the linearized dynamics in which the surface tension operator is replaced by a more ge…
The local existence and uniqueness of strong solutions for Cauchy problem of three-dimensional inhomogeneous incompressible Navier-Stokes-Vlasov equations
本文研究了三维不均匀不可压缩纳维-斯托克斯-维拉索方程柯西问题的强解的局部存在性和唯一性，通过线性化和近似解构造方法，丰富了强解的存在性结果。
We prove a continuation critereon for incompressible liquids with free surface boundary. We combine the energy estimates of Christodoulou and Lindblad with an analog of the estimate due to Beale, Kato, and Majda for the gradient of the velocity in terms of the vorticity, and use this to show solution can be continued so long as the second fundamental form and injectivity radius of the free boundary, the vorticity, and one derivative of the velocity on the free boundary remain bounded, assuming that the Taylor sign condition holds.
In this paper, we consider the Cauchy problem to the Ericksen-Leslie system of liquid crystals in $\mathbb R^3$. Global well-posedness of strong solutions are obtained under the condition that the product of $\|u_0\|_2+\|\nabla d_0\|_2$ and $\|\nabla u_0\|_2+\|\nabla^2d_0\|_2$ is suitably small. This result can be viewed as a supplement to the local existence and blow up criteria discussed in \cite{Hong1}.
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