Hand Lindblad, Karl Hakan Nordgren
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.
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A free boundary problem for the incompressible neo-Hookean elastodynamics is studied in two and three spatial dimensions. The a priori estimates in Sobolev norms of solutions with the physical vacuum condition are established through a geometrical point of view. Some estimates on the second fundamental form and velocity of the free surface are also obtained.
In this paper, we prove the a priori estimates in Sobolev spaces for the free-boundary compressible inviscid magnetohydrodynamics equations with magnetic diffusion under the Rayleigh-Taylor physical sign condition. Our energy estimates are uniform in the sound speed. As a result, we can prove the convergence of solutions of the free-boundary compressible resistive MHD equations to the solution of the free-boundary incompressible resistive MHD equations, i.e., the incompressible limit. The key observation is that the magnetic diffusion together with elliptic estimates directly controls the Lore…
In the present paper, we prove the a priori estimates of Sobolev norms for a free boundary problem of the incompressible inviscid MHD equations in all physical spatial dimensions $n=2$ and 3 by adopting a geometrical point of view used in Christodoulou-Lindblad CPAM 2000, and estimating quantities such as the second fundamental form and the velocity of the free surface. We identify the well-posedness condition that the outer normal derivative of the total pressure including the fluid and magnetic pressures is negative on the free boundary, which is similar to the physical condition (Taylor sig…
We consider the three-dimensional incompressible free-boundary magnetohydrodynamics (MHD) equations in a bounded domain with surface tension on the boundary. We establish a priori estimate for solutions in the Lagrangian coordinates with $H^{3.5}$ regularity. To the best of our knowledge, this is the first result focusing on the incompressible ideal free-boundary MHD equations with surface tension. It is worth pointing out that the $1/2$-extra spatial regularity for the flow map $η$ is no longer required in this manuscript thanks to the presence of the surface tension on the boundary.
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 consider the motion of the interface separating a vacuum from an inviscid, incompressible, and irrotational fluid, subject to the self-gravitational force and neglecting surface tension, in two space dimensions. The fluid motion is described by the Euler-Poission system in moving bounded simply connected domains. A family of equilibrium solutions of the system are the perfect balls moving at constant velocity. We show that for smooth data which are small perturbations of size $ε$ of these static states, measured in appropriate Sobolev spaces, the solution exists and remains of size $ε$ on a…
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…
Incompressible limit for the 3D compressible FENE dumbbell model
本文针对三维环面上可压缩FENE哑铃模型，通过引入基于动量的新型估计克服大体积黏性抑制高频衰减的困难，建立了全局时间的不可压缩极限及时间递减收敛速率。
We consider the three-dimensional incompressible free-boundary Euler equations in a bounded domain and with surface tension. Using Lagrangian coordinates, we establish a priori estimates for solutions with minimal regularity assumptions on the initial data.
We consider two hydrodynamic model problems (one incompressible and one compressible) with three dimensional fluid flow on the torus and temperature-dependent viscosity and conductivity. The ambient heat for the fluid is transported by the flow and fed by the local energy dissipation, modeling the transfer of kinetic energy into thermal energy through fluid friction. Both the viscosity and conductivity grow with the local temperature. We prove a strong a priori bound on the enstrophy of the velocity weighed against the temperature for initial data of arbitrary size, requiring only that the con…
In a recent work [1] the authors studied the dynamics of the interface separating a vacuum from an inviscid incompressible fluid, subject to the self-gravitational force and neglecting surface tension, in two space dimensions. The fluid is additionally assumed to be irrotational, and we proved that for data which are size $ε$ perturbations of an equilibrium state, the lifespan $T$ of solutions satisfies $T \gtrsim ε^{-2}$. The key to the proof is to find a nonlinear transformation of the unknown function and a coordinate change, such that the equation for the new unknown in the new coordinate …
A priori estimates and a blow-up criterion for the incompressible ideal MHD equations with surface tension and a closed free surface
本文针对具有表面张力和闭合自由曲面的三维不可压缩理想MHD方程，利用参考曲面和新能量泛函建立了不损失正则性的先验估计，并证明了包含自由边界自相交、平均曲率破坏和法向速度爆破在内的完整爆破判据。
Under the action of a time-periodic external forces we prove the existence of at least one time-periodic weak solution for the interaction between a three-dimensional incompressible fluid, governed by the Navier- Stokes equation and a two dimensional elastic plate. The challenge is that the Eulerian domain for the fluid changes in time and is a part of the solution. We introduce a two fixed-point methodology: First we construct a time-periodic solutions for a given variable time-periodic geometry. Then in a second step a (set-valued) fixed point is performed w.r.t.\ the geometry of the domain.…
In this paper we study the evolution of a small rigid body in a viscous incompressible fluid, in particular we show that a small particle is not accelerated by the fluid in the limit when its size converges to zero under a lower bound on its mass. This result is based on a new a priori estimate on the velocities of the centers of mass of rigid bodies that holds in the case when their masses are also allowed to decrease to zero.
In this paper we prove that the Euler equation describing the motion of an ideal fluid in $\R^d$ is well-posed in a class of functions allowing spatial asymptotic expansions as $|x|\to\infty$ of any a priori given order. These asymptotic expansions can involve log terms and lead to a family of conservation laws. Typically, the solutions of the Euler equation with initial data in the Schwartz class develop non-trivial spatial asymptotic expansions of the type considered here.
Local Well-posedness of the Free-Boundary Incompressible Magnetohydrodynamics with Surface Tension
本文研究带表面张力的三维自由边界不可压缩理想 MHD 方程组，通过人工粘性近似和极限论证证明其局部适定性，并克服速度—磁场强耦合及对称结构丧失带来的困难。
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…
Global weak solutions and incompressible limit to the isentropic compressible magnetohydrodynamic equations in 2D bounded domains with ripped density and large initial data
本文将等熵可压缩磁流体动力学方程的全局弱解存在性与不可压缩极限推广至带Navier-slip边界条件的二维有界凸区域，并获得了与体粘性系数无关的一致先验估计。
For the free boundary problem of the plasma-vacuum interface to ideal incompressible magnetohydrodynamics (MHD) in two-dimensional space, the a priori estimates of solutions are proved in Sobolev norms by adopting a geometrical point of view. In the vacuum region, the magnetic field is described by the div-curl system of pre-Maxwell dynamics, while at the interface the total pressure is continuous and the magnetic field is tangent to the boundary. We prove that the $L^2$ norms of any order covariant derivatives of the magnetic field in vacuum and on the boundaries are bounded in terms of initi…
Stabilization of an incompressible fluid-elastic structure system using a vacuum bubble
本研究通过引入真空气泡，稳定了不可压缩流体-弹性结构系统的动态行为，实现了压力平均值的控制，保证了小数据下的全局解存在与指数衰减。
We consider the three-dimensional incompressible magnetohydrodynamics (MHD) equations in a bounded domain with small volume and free moving surface boundary. We establish a priori estimate for solutions with minimal regularity assumptions on the initial data in Lagrangian coordinates. In particular, due to the lack of the Cauchy invariance for MHD equations, the smallness assumption on the fluid domain is required to compensate a loss of control of the flow map. Moreover, we show that the magnetic field has certain regularizing effect which allows us to control the vorticity of the fluid and t…
In this paper, we consider a free boundary problem of the incompressible elatodynamics, a coupling system of the Euler equations for the fluid motion with a transport equation for the deformation tensor. Under a natural force balance law on the free boundary with the surface tension, we establish its well-posedness theory on a short time interval. Our method is the vanishing viscosity limit by establishing a uniform a priori estimates with respect to the viscosity. As a by-product, the inviscid limit of the incompressible viscoelasticity (the system coupling with the Navier-Stokes equations) i…
This paper is dedicated to the study of the initial value problem for density dependent incompressible viscous fluids in $\R^{N}$ with $N\geq2$. We address the question of well-posedness for {\it large} data having critical Besov regularity and we aim at stating well-posedness in functional spaces as close as possible to the ones imposed in the incompressible Navier Stokes system by Cannone, Meyer and Planchon where $u_{0}\in B^{\NN-1}_{p,\infty}$ with $1\leq p<+\infty$. This improves the analysis of H. Abidi, R. Danchin and M. Paicu where $u_{0}$ is considered belonging to $B^{\NN-1}_{p,1}$ w…
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.
In this paper, we consider the Cauchy problem for the incompressible Navier-Stokes equations in $\mathbb{R}^n$ for $n\geq 3 $ with smooth periodic initial data and derive a priori estimtes of the maximum norm of all derivatives of the solution in terms of the maximum norm of the initial data. This paper is a special case of a paper by H-O Kreiss and J. Lorenz which also generalizes the main result of their paper to higher dimension.
We derive a priori estimates for the incompressible free-boundary Euler equations with surface tension in three spatial dimensions. Working in Lagrangian coordinates, we provide a priori estimates for the local existence when the initial velocity, which is rotational, belongs to $H^3$ and the trace of initial velocity on the free boundary to $H^{3.5}$, thus lowering the requirement on the regularity of initial data in the Lagrangian setting. Our methods are direct and involve three key elements: estimates for the pressure, the boundary regularity provided by the mean curvature, and the Cauchy …
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 …
In this work we study the asymptotic behavior of viscous incompressible 2D flow in the exterior of a small material obstacle. We fix the initial vorticity $ω_0$ and the circulation $γ$ of the initial flow around the obstacle. We prove that, if $γ$ is sufficiently small, the limit flow satisfies the full-plane Navier-Stokes system, with initial vorticity $ω_0 + γδ$, where $δ$ is the standard Dirac measure. The result should be contrasted with the corresponding inviscid result obtained by the authors in [Comm P.D.E. 28 (2003) 349-379], where the effect of the small obstacle appears in the coeffi…
On a model of an elastic body fully immersed in a viscous incompressible fluid with small data
针对浸没在两层不可压缩粘性流体间的弹性体模型，由阻尼波动方程与Navier-Stokes方程耦合，建立了小初始数据下的全局时间适定性与指数衰减，并证明最终状态除水平小位移外必为平凡。
Several fluid systems are characterised by time reversal and parity breaking. Examples of such phenomena arise both in quantum and classical hydrodynamics. In these situations, the viscosity tensor, often dubbed ``odd viscosity'', becomes non-dissipative. At the mathematical level, this fact translates into a loss of derivatives at the level of \textsl{a priori} estimates: while the odd viscosity term depends on derivatives of the velocity field, no parabolic smoothing effect can be expected. In the present paper, we establish a well-posedness theory in Sobolev spaces for a system of incompres…
We consider the regularity of an interface between two incompressible and inviscid fluids flows in the presence of surface tension. We obtain local in time estimates on the interface in $H^{\frac32k +1}$ and the velocity fields in $H^{\frac32k}$. These estimates are obtained using geometric considerations which show that the Kelvin-Helmholtz instabilities are a consequence of a curvature calculation.
Yudovich theory under geometric regularity for density-dependent incompressible fluids
本文研究d=2下密度依赖不可压缩Euler方程在非Lipschitz初始数据下的Yudovich型解，通过对流体速度沿密度梯度向量场的方向导数进行先验几何控制，证明了低正则解的稳定性和唯一性，并改进了先前唯一性结果。
In this paper, we consider the global wellposedness of 2-D incompressible magneto-hydrodynamical system with small and smooth initial data. It is a coupled system between the Navier-Stokes equations and a free transport equation with an universal nonlinear coupling structure. The main difficulty of the proof lies in exploring the dissipative mechanism of the system due to the fact that there is a free transport equation in the system. To achieve this and to avoid the difficulty of propagating anisotropic regularity for the free transport equation, we first reformulate our system \eqref{1.1} in…
The motion of an elastic solid inside of an incompressible viscous fluid is ubiquitous in nature. Mathematically, such motion is described by a PDE system that couples the parabolic and hyperbolic phases, the latter inducing a loss of regularity which has left the basic question of existence open until now. In this paper, we prove the existence and uniqueness of such motions (locally in time), when the elastic solid is the linear Kirchhoff elastic material. The solution is found using a topological fixed-point theorem that requires the analysis of a linear problem consisting of the coupling be…
For the free boundary problem of the plasma-vacuum interface to three-dimensional ideal incompressible magnetohydrodynamics (MHD), the a priori estimates of smooth solutions are proved in Sobolev norms by adopting a geometrical point of view and some quantities such as the second fundamental form and the velocity of the free interface are estimated. In the vacuum region, the magnetic fields are described by the div-curl system of pre-Maxwell dynamics, while at the interface the total pressure is continuous and the magnetic fields are tangent to the interface, but we do not need any restriction…
We introduce a diffuse interface model describing the evolution of a mixture of two different viscous incompressible fluids of equal density. The main novelty of the present contribution consists in the fact that the effects of temperature on the flow are taken into account. In the mathematical model, the evolution of the macroscopic velocity is ruled by the Navier-Stokes system with temperature-dependent viscosity, while the order parameter representing the concentration of one of the components of the fluid is assumed to satisfy a convective Cahn-Hilliard equation. The effects of the tempera…
Global weak solutions to a diffuse-interface model for quasi-incompressible two-phase flows with unmatched densities and singular potential
本文研究了描述两种宏观不混溶、不可压缩且粘性牛顿流体的热力学一致扩散界面模型，证明了在特定奇异自由能密度下，全局时间弱解的存在性。
Low-regularity a priori estimates, blow-up criterion, and self-intersection singularities for free-boundary ideal magnetohydrodynamics with surface tension
针对带表面张力的三维不可压缩自由边界理想MHD方程，在一般有界区域中建立了H^3先验估计，将门槛从H^6降低；提出了区分拓扑自交、边界正则性丧失等的有限时间爆破准则，揭示了磁场不对称性；构造了正则初始数据，证明表面张力和理想磁耦合不能阻止自由边界拓扑自交。
We consider the interface problem between two incompressible and inviscid fluids in the presence of surface tension. Following the geometric approach of [Shatah,J.;Zeng,C. A priori estimates for Fluid Interface Problems. CPAM, vol.16, no.6, June 2008] we show that solutions to this problem converge to solutions of the free-boundary Euler equations in vacuum as one of the densities goes to zero.
Existence of weak solutions to the two-dimensional incompressible Euler equations in the presence of sources and sinks
本文研究含源汇边界条件的二维不可压缩 Euler 系统，利用涡量先验界与粘性近似的紧性方法，证明了 $L^p$ 涡量在不同 p 范围下弱解的存在性。
Here we consider the 2D free boundary incompressible Euler equation with surface tension. We prove that the surface tension does not prevent a finite time splash or splat singularity, i.e. that the curve touches itself either in a point or along an arc. To do so, the main ingredients of the proof are a transformation to desingularize the curve and a priori energy estimates.
Vorticity interior trace estimates and higher derivative estimates via blow-up method
本研究针对三维不可压缩Navier-Stokes方程，运用爆破方法与新型平均算子推导了若干非线性先验迹估计，拓展了混合范数高阶导数估计的研究，相关方法可推广至具标度不变性、ε-正则性的偏微分方程。
An arbitrary order and pointwise divergence-free finite element scheme for the incompressible 3D Navier-Stokes equations
本文提出一种基于有限元外微积分和Lamb恒等式的任意阶有限元离散方案，用于不可压缩三维Navier-Stokes方程，实现离散速度逐点无散，保证压力鲁棒性，并通过理论分析与数值模拟验证。
Robust globally divergence-free Weak Galerkin finite element method for incompressible Magnetohydrodynamics flow
针对定常不可压缩磁流体方程，提出任意阶弱Galerkin有限元方法，可得到全局无散度的速度与磁场近似，证明了离散格式的适定性与最优误差估计，给出收敛线性化迭代算法并通过数值实验验证理论结果。
Time-periodic weak solutions for the interaction of an incompressible fluid with a linear Koiter type shell under dynamic pressure boundary conditions
该研究针对流固耦合中的时间周期运动问题，考虑动压边界条件下三维Navier-Stokes流体与边界二维线性Koiter壳的相互作用，通过新逼近技术和先验估计，证明了小周期外力下至少存在一个时间周期弱解。
We consider the two-dimensional water wave problem in the case where the free interface of the fluid meets a vertical wall at a possibly non-right angle; and where the free interface can be non-$C^1$ with angled crests. We assume that the air has density zero, the fluid is inviscid, incompressible, irrotational, and subject to the gravitational force, and the surface tension is zero. In this regime, only a degenerate Taylor stability criterion $-\frac{\partial P}{\partial \bf {n}} \ge 0$ holds, with degeneracies at the singularities on the interface and at the point where it meets the wall if …
We regard the Cauchy problem for a particular Whitham-Boussinesq system modelling surface waves of an inviscid incompressible fluid layer. The system can be seen as a weak nonlocal dispersive perturbation of the shallow water system. The proof of well-posedness relies on energy estimates. However, due to the symmetry lack of the nonlinear part, in order to close the a priori estimates one has to modify the traditional energy norm in use. Hamiltonian conservation provides with global well-posedness at least for small initial data in the one dimensional settings.
We obtain the local existence and uniqueness for a system describing interaction of an incompressible inviscid fluid, modeled by the Euler equations, and an elastic plate, represented by the fourth-order hyperbolic PDE. We provide a~priori estimates for the existence with the optimal regularity $H^{r}$, for $r>2.5$, on the fluid initial data and construct a unique solution of the system for initial data $u_0\in H^{r}$ for $r\geq3$. An important feature of the existence theorem is that the Taylor-Rayleigh instability does not occur.
In this paper we study the dynamics of eigenvalues of the deformation tensor for solutions of the 3D incompressible Euler equations. Using the evolution equation of the $L^2$ norm of spectra, we deduce new a priori estimates of the $L^2$ norm of vorticity. As an immediate corollary of the estimate we obtain a new sufficient condition of $L^2$ norm control of vorticity. We also obtain decay in time estimates of the ratios of the eigenvalues. In the remarks we discuss what these estimates suggest in the study of searching initial data leading to a possible finite time singularities. We find that…
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 consider the motion of a two-dimensional interface between air (above) and an irrotational, incompressible, inviscid, infinitely deep water (below), with surface tension present. We propose a new way to reduce the original problem into an equivalent quasilinear system which are related to the interface's tangent angle and a quantity related to the difference of tangential velocities of the interface in the Lagrangian and the arc-length coordinates. The new way is relatively simple because it involves only taking differentiation and the real and the imaginary parts. Then if assuming that wav…
The combined limits of large bulk viscosity and small angular viscosity for the 2D compressible micropolar fluid equations
本文研究二维可压缩微极流体方程大体黏度与小角黏度组合极限，证明解收敛到不可压缩系统并给出收敛速率，且无需额外技术假设。
A priori error estimates for the $θ$-method for the flow of nonsmooth velocity fields
本文研究非光滑速度场流动的θ-方法，建立先验误差估计，证明数值解以对数速率收敛于精确解，并通过数值实验验证结果。
Transport equation theory in the Triebel-Lizorkin spaces and its applications to the ideal fluid flows
本文在Triebel-Lizorkin空间中发展了输运方程理论，通过交换子估计和特征方法证明了局部适定性，并应用于理想流体流动模型。
We prove the existence and uniqueness of solutions to the time-dependent incompressible Navier-Stokes equations with a free-boundary governed by surface tension. The solution is found using a topological fixed-point theorem for a nonlinear iteration scheme, requiring at each step, the solution of a model linear problem consisting of the time-dependent Stokes equation with linearized mean-curvature forcing on the boundary. We use energy methods to establish new types of spacetime inequalities that allow us to find a unique weak solution to this problem. We then prove regularity of the weak solu…
We study the second spatial derivatives of suitable weak solutions to the incompressible Navier-Stokes equations in dimension three. We show that it is locally $L ^{\frac43, q}$ for any $q > \frac43$, which improves from the current result $L ^{\frac43, \infty}$. Similar improvements in Lorentz space are also obtained for higher derivatives of the vorticity for smooth solutions. We use a blow-up technique to obtain nonlinear bounds compatible with the scaling. The local study works on the vorticity equation and uses De Giorgi iteration. In this local study, we can obtain any regularity of the …
The Nordström-Vlasov system is a Lorentz invariant model for a self-gravitating collisionless gas. We establish suitable a-priori-bounds on the solutions of this system, which together with energy estimates and the smoothing effect of ``momentum averaging'' yield the existence of global weak solutions to the corresponding initial value problem. In the process we improve the continuation criterion for classical solutions which was derived recently. The weak solutions are shown to preserve mass.
New a priori estimate for stochastic 2D Navier-Stokes equation with applications to invariant measure
本文改造 Sobolevskiĭ-Kato-Fujita 方法研究加性噪声驱动的随机二维 Navier-Stokes 方程，借助新先验估计在 $\gamma\in(1/4,3/8]$ 范围内证明不变测度唯一、强混合及半群强 Feller 与不可约。
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