Vaclav Macha
We use Feireisl-Lions theory to deduce the existence of weak solutions to a system describing the dynamics of a linear oscillator containing a Newtonian compressible fluid. The appropriate Navier-Stokes equation is considered on a domain whose movement has one degree of freedom. The equation is paired with the Newton law and we assume a no-slip boundary condition.
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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.
Incompressible Limit of Compressible Ideal MHD Flows inside a Perfectly Conducting Wall
本文证明了磁场与边界相切时理想MHD流动的不可压缩极限，通过Lorentz力揭示的隐藏结构构造各向异性Sobolev能量泛函，并据此获得关于Mach数一致的能量估计。
Brinkman's law as $Γ$-limit of compressible low Mach Navier-Stokes equations and application to randomly perforated domains
该研究证明低马赫数下可压缩Navier-Stokes方程的极限为不可压缩Navier-Stokes-Brinkman方程，并补充了随机穿孔区域临界情形的随机均匀化结果。
Let $\mathcal{S} = \{ τ_n \}_{n=1}^\infty \subset (0,T)$ be an arbitrary countable (dense) set. We show that for any given initial density and momentum, the compressible Euler system admits (infinitely many) admissible weak solutions that are not strongly continuous at each $τ_n$, $n=1,2,\dots$. The proof is based on a refined version of the oscillatory lemma of De Lellis and Sz\' ekelyhidi with coefficients that may be discontinuous on a set of zero Lebesgue measure.
Multiple scales and singular limits for compressible rotating fluids with general initial data
本文研究了由尺度正压Navier-Stokes方程描述的旋转可压缩流体的奇异极限，其中Rossby数、马赫数和Froude数在某一特定的相互速率下趋于0，而雷诺数趋于无穷大。将无粘平面欧拉方程组归结为极限问题。证明是基于应用相对熵的方法和仔细分析的振荡积分描述的Rossby声波的传播。
Maximal dissipation and well-posedness for the compressible Euler system
在弱解的框架下讨论了可压缩（正压）Euler方程组的适定性问题。最大耗散原理是C.M.对Dafermos进行了改进，并结合了容许弱解的概念.我们使用的方法，凸积分的精神，最近的工作C.DeLellis和L.Szekelyhidi显示各种反例的适定性。另一方面，我们推测，最大耗散的原则应保留作为唯一性的可能的标准，因为它是违反了在凸积分过程中获得的振荡解。
We consider a sequence of approximate solutions to the compressible Euler system admitting uniform energy bounds and/or satisfying the relevant field equations modulo an error vanishing in the asymptotic limit. We show that such a sequence either (i) converges strongly in the energy norm, or (ii) the limit is not a weak solution of the associated Euler system. This is in sharp contrast to the incompressible case, where (oscillatory) approximate solutions may converge weakly to solutions of the Euler system. Our approach leans on identifying a system of differential equations satisfied by the a…
Hydrostatic bubbles of compressible fluid in an incompressible fluid
针对恒定重力场下不可压缩流体中可压缩正压流体的稳态构型问题，通过在勒让德算子定义的加权索伯列夫空间中运用分岔论证构建近似球形轴对称光滑解，并开发了具独立价值的泛函分析工具。
Collision/No-collision results of a solid body with its container in a 3D compressible viscous fluid
本文研究了三维可压缩粘性流体中刚体与容器的碰撞问题，利用物体表面粗糙度证明了有限时间内的碰撞，并分析了$C^{1,α}$-正则性边界下的碰撞条件及光滑物体的无碰撞结果。
We consider a system of multiple insulating rigid bodies moving inside of an electrically conducting compressible fluid. In this system we take into account the interaction of the fluid with the bodies as well as with the electromagnetic fields trespassing both the fluid and the solids. The main result of this article yields the existence of weak solutions to the system. While the mechanical part of the problem can be dealt with via a classical penalization method, the electromagnetic part requires an approximation by means of a hybrid discrete-continuous in time system: The discrete part of t…
In this work we study the coupled system of partial and ordinary differential equations describing the interaction between a compressible isentropic viscous fluid and a rigid body moving freely inside the fluid. In particular the position and velocity of the rigid body in the fluid are unknown and the motion of the rigid body is driven by the normal stress forces of the fluid acting on the boundary of the body. We prove that the strong solution, which is known to exist under certain smallness assumptions, is unique in the class of weak solutions to the problem. The proof relies on a correct de…
Singular limits of anisotropic weak solutions to compressible magnetohydrodynamics
本文严格证明了强各向异性可压缩磁流体动力学弱解收敛至简化磁流体动力学模型弱解的奇异极限，揭示了系统在垂直磁场方向不可压缩而在平行方向可压缩的特征。
In this paper, we consider the Cauchy problem for the one-dimensional compressible isentropic magnetohydrodynamic (MHD) equations with no vacuum at infinity, but the initial vacuum can be permitted inside the region. By deriving a priori $ν$ (resistivity coefficient)-independent estimates, we establish the non-resistive limit of the global strong solutions with large initial data. Moreover, as a by-product, the global well-posedness of strong solutions for both the compressible resistive MHD equations and non-resistive MHD equations are also established, respectively.
Time-periodic solutions of the compressible Euler equations and the Nonlinear Theory of Sound
该研究证明了一维3×3可压缩欧拉方程存在所有频率的纯音非线性声波周期解，解决了声学领域百年问题，表明一般熵剖面下任意波数都存在无激波周期解。
Almost global existence and nonlinear asymptotic stability for bubble dynamics in inviscid compressible liquid
本文研究无界等熵可压缩无粘液体中气泡的非线性动力学，通过耦合广义加权时空估计与特征线分析克服拟线性导致的导数损失，证明了几乎全局存在性和非线性渐近稳定性。
Electrostatic effects on critical regularity and long-time behavior of viscous compressible fluids
本文研究了静电效应在粘性可压缩流体临界正则性和长时间行为中的影响，证明了在无额外假设下，初始数据接近平衡态的柯西问题具有全局解，并建立了更优的收敛率。
This paper studies the global feedback stabilization problem of a system with two pistons and the area between them containing a viscous compressible fluid (gas) modeled by the Navier-Stokes equations. The control input is the force applied on the left piston (boundary input) and the overall system consists of two nonlinear Partial Differential Equations and four nonlinear Ordinary Differential Equations. Global feedback stabilizers are designed for the overall system by means of the Control Lyapunov Functional methodology. The closed-loop system exhibits global asymptotic stability with an ex…
Existence of stationary weak solutions for the heat conducting flows
本文考虑定常可压缩Navier-Stokes-Fourier方程组，边界上的速度和热流与内外温差成正比，边界条件为Dirichlet或Navier边界条件。本文根据压力定律$p(\vr,\vt) \sim \vr^γ+ \vr \vt$中的绝热常数$γ$和导热系数中的增长指数$κ(\vt) \sim (1+ \vt^m)$等参数，在不限制数据大小的情况下，阐明了构造粘性系数随温度变化的三维流动的弱熵解和变熵解的主要思想。此外，当它是可能的情况下，证明有界密度的解决方案的存在性进行了审查。的二维流动的解决方案的建设中提到的主要变化，最后，更复杂的系统的结果进行审查，稳定的可压缩Navier-Stokes-Fourier方程起着重要的作用。
Existence and Stability of Compressible Current-Vortex Sheets in Three-Dimensional Magnetohydrodynamics
可压缩涡面是多维双曲守恒律方程组熵解的基本波。对于二维气体动力学中的欧拉方程，对可压缩涡面的经典线性化稳定性分析预测了当马赫数为$M>\sqrt{2}$时的稳定性和当$M<\sqrt{2}$时的不稳定性; Artola-Majda的分析表明，如果平面涡面受到高度振荡波的扰动，即使当$M>\sqrt{2}$时，也可能发生非线性不稳定性。对于三维Euler方程，每个可压缩涡面都是剧烈不稳定的，这种剧烈不稳定性类似于不可压缩流体的Kelvin-Helmholtz不稳定性。本文的目的是了解在纯气体动力学区域不稳定的三维可压缩涡面在三维磁流体动力学（MHD）中的磁场作用下是否变得稳定。其主要特点之一是将稳定性问题等价为自由边界为特征曲面的自由边界问题。另一个特点是，MHD中的电流涡面的线性化问题不满足一致的Kreiss-Lopatinskii条件。在本文中，我们开发了一个非线性的方法来处理这些困难的3-D磁流体。我们首先仔细制定的线性化问题的电流涡面严格表明，磁效应使问题弱稳定，并建立能量估计，特别是高阶能量估计，在不损失的顺序的非齐次项和可变系数。然后我们利用这些结果发展了一个合适的迭代格式的Nash-Moser-Hörmander型，并建立其收敛性，这导致可压缩的电流涡面的存在性和稳定性，局部的时间，在3-D MHD。
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