Sylvie Monniaux
Depending of the geometry of the domain, one can define --at least-- three different Stokes operators with Dirichlet boundary conditions. We describe how the resolvents of these Stokes operators converge with respect to a converging sequence of domains.
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We construct pullback attractors to the weak solutions of the three-dimensional Dirichlet problem for the incompressible Navier-Stokes equations in the case when the external force may become unbounded as time goes to plus or minus infinity.
We consider a coupled two-phase Navier-Stokes/Mullins-Sekerka system describing the motion of two immiscible, incompressible fluids inside a bounded container. The moving interface separating the liquids meets the boundary of the container at a constant ninety degree angle. This common interface is unknown and has to be determined as a part of the problem. We show well-posedness and investigate the long-time behaviour of solutions starting close to certain equilibria. We prove that for equal densities these solutions exist globally in time, are stable, and converge to an equilibrium solution a…
We address the global-in-time existence, stability and long time behaviour of weak solutions of the three-dimensional compressible Navier-Stokes equations with potential force. We show the details of the $α$-dependence of different smoothing rates for weak solutions near $t=0$ under the assumption on the initial velocity $u_0$ that $u_0\in H^α$ for $α\in(\frac{1}{2},1]$ and obtain long time convergence of weak solutions in various norms. We then make use of the Lagrangean framework in comparing the instantaneous states of corresponding fluid particles in two different solutions. The present wo…
The paper is aimed at analysing a singular perturbation of the Navier-Stokes equations on a compact closed manifold. The case of compact smooth manifolds with boundary under the Dirichlet conditions is also included. Global existence and uniqueness is established for the weak solutions of the Cauchy problem. The solution of the regularised system is shown to converge to the solution of the conventional Navier-Stokes equations provided it is uniformly bounded in parameter.
We consider a two-phase problem for two incompressible, viscous and immiscible fluids which are separated by a sharp interface. The problem arises as a sharp interface limit of a diffuse interface model. We present results on local existence of strong solutions and on the long-time behavior of solutions which start close to an equilibrium. To be precise, we show that as time tends to infinity, the velocity field converges to zero and the interface converges to a sphere at an exponential rate.
This paper is concerned with the asymptotic stability of a composite wave of two viscous shocks under spatially periodic perturbations for the 1-D full compressible Navier-Stokes equations. It is proved that as time increases, the solution approaches the background composite wave with a shift for each shock, where the shifts can be uniquely determined if both the periodic perturbations and strengths of two shocks are small. The key of the proof is to construct a suitable ansatz such that the anti-derivative method works.
In this paper, we show that Landau solutions to the Navier-Stokes system are asymptotically stable under $L^3$-perturbations. We give the local well-posedness of solutions to the perturbed system with initial data in $L_σ^3$ space and the global well-posedness with small initial data in $L_σ^3$ space, together with a study of the $L^q$ decay for all $q>3.$ Moreover, we have also studied the local well-posedness, global well-posedness and stability in $L^p$ spaces for $3<p<\infty$.
This paper is dedicated to the construction of a pseudo-norm, for which small shock profiles of the barotropic Navier-Stokes equation have a contraction property. This contraction property holds in the class of any large 1D weak solutions to the barotropic Navier-Stokes equation. It implies a stability condition which is independent of the strength of the viscosity. The proof is based on the relative entropy method, and is reminiscent to the notion of a-contraction first introduced by the authors in the hyperbolic case.
On the long time behaviour of solutions to the Navier-Stokes-Fourier system on unbounded domains
本文研究 $R^3$ 无界区域上满足远场条件的 NSF 系统，证明其任意全局弱解按遍历平均收敛至零密度平衡态，并据此推出总动量不可能全局守恒。
We investigate the large-time behavior of solutions to an outflow problem of the full compressible Navier-Stokes equations in the half line. The non-degenerate stationary solution is shown to be asymptotically stable under large initial perturbation with no restriction on the adiabatic exponent $γ$, provided that the boundary strength is sufficiently small. The proofs are based on the standard energy method and the crucial step is to obtain positive lower and upper bounds of the density and the temperature uniformly in time and space.
The stability of solutions under periodic perturbations for both inviscid and viscous conservation laws is an interesting and important problem. In this paper, a large-amplitude viscous shock under space-periodic perturbation for the isentropic Navier-Stokes equations is considered. It is shown that if the initial perturbation around the shock is suitably small and satisfies a zero-mass type condition (2.17), then the solution of the N-S equations tends to the viscous shock with a shift, which is partially determined by the periodic oscillations. In other words, the viscous shock is nonlinearl…
We consider the incompressible Navier-Stokes equations with the Dirichlet boundary condition in an exterior domain of $\mathbb{R}^n$ with $n\geq2$. We compare the long-time behaviour of solutions to this initial-boundary value problem with the long-time behaviour of solutions of the analogous Cauchy problem in the whole space $\mathbb{R}^n$. We find that the long-time asymptotics of solutions to both problems coincide either in the case of small initial data in the weak $L^{n}$-space or for a certain class of large initial data.
Asymptotic behaviour of solutions of linearized Navier Stokes equations in the long waves regime
本文研究粘度消失时线性化Navier-Stokes方程在长波区域的渐近行为，发现特定$α$范围内存在唯一不稳定模态，并给出解的完整描述，为非线性不稳定性研究奠定基础。
The two-phase free boundary value problem for the isothermal Navier-Stokes system is studied for general bounded geometries in absence of phase transitions, external forces and boundary contacts. It is shown that the problem is well-posed in an Lp-setting, and that it generates a local semiflow on the induced phase manifold. If the phases are connected, the set of equilibria of the system forms a (n+1)-dimensional manifold, each equilibrium is stable, and it is shown that global solutions which do not develop singularities converge to an equilibrium as time goes to infinity. The latter is prov…
Considering the space-periodic perturbations, we prove the time-asymptotic stability of the composite wave of a viscous contact wave and two rarefaction waves for the Cauchy problem of 1-D compressible Navier-Stokes equations in this paper. This kind of perturbations keep oscillating at the far field and are not integrable. The key is to construct a suitable ansatz carrying the same oscillation %eliminating the oscillation of the solution as in \cite{HuangXuYuan2020,HuangYuan2021}, but due to the degeneration of contact discontinuity, the construction is more subtle. We find a way to use the s…
Perturbation theory of the compressible Navier-Stokes equations and its application
本文提出了一种可压缩纳维-斯托克斯方程的扰动理论，用于研究解的衰减估计，并应用于阻尼波动方程的衰减率分析。
Nonlinear Stability of Linearly Expanding Goldreich-Weber Solutions for the Navier-Stokes-Poisson System with Degenerate Viscosity Under Radial Perturbations
该研究针对带退化黏性的引力纳维-斯托克斯-泊松系统，在特定剪切黏性条件下，证明了线性膨胀Goldreich-Weber解在径向扰动下的非线性稳定性。
We are concerned with the barotropic compressible Navier-Stokes system in a bounded domain of $\mathbb{R}^d$ (with $d\geq2$). In a critical regularity setting, we establish local well-posedness for large data with no vacuum and global well-posedness for small perturbations of a stable constant equilibrium state.Our results rely on new maximal regularity estimates - of independent interest - for the semigroup of the Lam\{é} operator, and of the linearized compressible Navier-Stokes equations.
We investigate the size of the regular set for small perturbations of some classes of strong large solutions to the Navier--Stokes equation. We consider perturbations of the data which are small in suitable weighted $L^{2}$ spaces but can be arbitrarily large in any translation invariant critical Banach space. We give similar results in the small data setting.
In this paper, we study a hyperbolic version of the Navier-Stokes equations, obtained by using the approximation by relaxation of the Euler system, evolving in a thin strip domain. The formal limit of these equations is a hyperbolic Prandtl type equation, our goal is to prove the existence and uniqueness of a global solution to these equations for analytic initial data in the tangential variable, under a uniform smallness assumption. Then we justify the limit from the anisotropic hyperbolic Navier-Stokes system to the hydrostatic hyperbolic Navier-Stokes system with small analytic data.
Stability analysis of time-periodic solutions to the Navier-Stokes-Fourier system in 3D whole space
本文研究了Navier-Stokes-Fourier系统在三维全空间中时间周期解的稳定性，通过推导扰动的时间衰减估计，分析了外部力对解的影响。
We study the asymptotic stability of a planar rarefaction wave (in the $ x_1 $- direction) for the 3-d isentropic Navier-Stokes equations, where the initial perturbation is periodic on the torus $ \mathbb{T}^3 $ with zero average. To solve this Cauchy problem in which the initial data is periodic with respect to only $ x_2 $ and $ x_3 $ but not to $ x_1, $ we construct a suitable ansatz carrying the oscillations of the solution in the $ x_1 $- direction, but remaining to be periodic in the transverse $ x_2 $- and $ x_3 $- directions. In such a way, the difference between the ansatz and the sol…
Asymptotic stability of homogeneous solutions to Navier-Stokes equations under $L^{p}$-perturbations
本文证明了三维 Navier-Stokes 方程中除 Landau 解外的最小奇异解在 $L^{3}$-扰动下的渐近稳定性，并建立了任意 $q>3$ 时的 $L^{q}$ 衰减估计，同时研究了扰动方程的局部与整体适定性。
In this note, we study the behaviour of Lebesgue norms $\|v(\cdot,t)\|_p$ of solutions $v$ to the Cauchy problem for the Stokes system with drift $u$, which is supposed to be a divergence free smooth vector valued function satisfying a scale invariant condition.
This paper is concerned with the inflow problem for the one-dimensional compressible Navier-Stokes equations. For such a problem, Matsumura and Nishihara showed in [A. Matsumura and K. Nishihara, Large-time behaviors of solutions to an inflow problem in the half space for a one-dimensional system of compressible viscous gas. Comm. Math. Phys. 222 (2001), 449-474] that there exists boundary layer solution to the inflow problem and both the boundary layer solution, the rarefaction wave, and the superposition of boundary layer solution and rarefaction wave are nonlinear stable under small initial…
We show that the Navier-Stokes as well as a random perturbation of this equation can be derived from a stochastic variational principle where the pressure is introduced as a Lagrange multiplier. Moreover we describe how to obtain corresponding constants of the motion.
Long-time behaviour of two-dimensional Navier-Stokes equations in the presence of Couette flow on the half plane
本文研究了在半平面存在科氏流条件下二维纳维-斯托克斯方程解的长期行为，通过证明总涡量趋近于一个包含科氏流涡量和福克-计划类型算子核的表达式，揭示了涡量的渐进行为。
Stability of strong viscous shock wave under periodic perturbation for 1-D isentropic Navier-Stokes system in the half space
本文证明半空间一维等熵Navier-Stokes系统在周期扰动下，小扰动解渐近收敛到带位移的粘性激波，且激波强度可任意大，改进了先前仅适用于弱激波且无周期振荡的结果。
The goal of the present paper is to study the weak--strong uniqueness problem for the compressible Navier--Stokes system with a general barotropic pressure law. Our results include the case of a hard sphere pressure of Van der Waals type with a non--monotone perturbation and a Lipschitz perturbation of a monotone pressure. Although the main tool is the relative energy inequality, the results are conditioned by the presence of viscosity and do not seem extendable to the Euler system.
We introduce a rough perturbation of the Navier-Stokes system and justify its physical relevance from balance of momentum and conservation of circulation in the inviscid limit. We present a framework for a well-posedness analysis of the system. In particular, we define an intrinsic notion of solution based on ideas from the rough path theory and study the system in an equivalent vorticity formulation. In two space dimensions, we prove that well-posedness and enstrophy balance holds. Moreover, we derive rough path continuity of the equation, which yields a Wong-Zakai result for Brownian driving…
Asymptotic stability of shock profiles and rarefaction waves to the Navier-Stokes-Poisson system under space-periodic perturbations
本文证明了一维可压缩 Navier-Stokes-Poisson 系统在空间周期扰动下，粘性激波剖面和稀疏波分别具有渐近稳定性，其中激波情形需满足零质量条件，证明依赖二次拟设和能量方法。
It was proved by Karch and Pilarzyc that Landau solutions are asymptotically stable under any $L^2$-perturbation. In our earlier work with L. Li, we have classified all $(-1)$-homogeneous axisymmetric no-swirl solutions of incompressible stationary Navier-Stokes equations in three dimension which are smooth on the unit sphere minus the south and north poles. In this paper, we study the asymptotic stability of the least singular solutions among these solutions other than Landau solutions, and prove that such solutions are asymptotically stable under any $L^2$-perturbation.
We consider the compressible (barotropic) Navier-Stokes system on time-dependent domains, supplemented with slip boundary conditions. Our approach is based on penalization of the boundary behaviour, viscosity, and the pressure in the weak formulation. Global-in-time weak solutions are obtained.
Proof of the transverse instability of Stokes waves
该研究针对Stokes波的横向不稳定性问题，通过严格数学证明，首次证实小振幅Stokes波线性化后横向扰动会指数增长，填补了此前仅有二维相关严格结果的空白。
Stability of Anomalous Dissipation for the Forced 3D Navier--Stokes Equations under Geometric Perturbations
本文研究了强制三维纳维-斯托克斯方程在几何扰动下的异常耗散稳定性问题，通过准自相似混合方法证明了中心曲线在纯法向扰动下的结构稳定性，建立了映射的C²稳定性及局部场的C¹稳定性，并获得了Hölder估计和高频能量集中，从而得到与扰动无关的正耗散下界，进一步在(2+1/2)维框架下证明了C⁶结构稳定性，为K41理论提供了严谨的基础。
In this paper, we introduce a finite propagation speed perturbation of the incompressible Navier-Stokes equations (NS). The model we consider is inspired by a hyperbolic perturbation of the heat equation due to Cattaneo (Sulla conduzione del calore. Atti Semin. Mat. Fis. Univ., Modena, 1949, 3:83-101) and by an equation that Vishik and Fursikov (Solutions statistiques homogènes des systèmes différentiels paraboliques et du système de Navier-Stokes. Ann. Scuola Norm. Sup. Pisa Cl. Sci., 1977, 4:531-576) investigated in order to find statistical solutions to (NS). We prove that the solutions to …
Low Mach number limit for the compressible Navier-Stokes equation with a stationary force
研究三维全空间中带定常力的可压缩Navier-Stokes方程在低马赫数极限下的行为，证明了定常解收敛到不可压缩流，且小扰动全局收敛。
We prove global in time well-posedness for perturbations of the 2D stochastic Navier-Stokes equations \begin{equation*} \partial_t u + u \cdot \nabla u = Δu - \nabla p + ζ+ ξ \;, \quad u (0, \cdot) = u_{0}(\cdot) \;, \quad \mathrm{div} (u) = 0 \;, \end{equation*} driven by additive space-time white noise $ ξ$, with perturbation $ ζ$ in the Hölder-Besov space $\mathcal{C}^{-2 + 3κ} $, periodic boundary conditions and initial condition $ u_{0} \in \mathcal{C}^{-1 + κ} $ for any $ κ>0 $. The proof relies on an energy estimate which in turn builds on a dynamic high-low frequency decomposition and …
Asymptotic stability of planar entropy wave for 3-d Navier-Stokes equations in Eulerian coordinates
针对欧拉坐标系下三维Navier-Stokes方程平面熵波的大时间渐近行为问题，引入新变换满足结构条件，结合加权能量估计控制慢衰减项，解决了两类初始扰动下的渐近稳定性并推导最优衰减率。
Nonlinear Stability for the Superposition of Viscous Contact Wave and Rarefaction Waves to Non-isentropic Compressible Navier-Stokes System with General Initial Perturbations
本文研究一维可压缩 Navier-Stokes 系统的 Cauchy 问题，证明了在一般初始扰动下粘性接触波与稀疏波叠加的非线性渐近稳定性。
Linear stability analysis of the Couette flow for 2D compressible Navier-Stokes-Poisson system
针对高 Reynolds 数下二维可压缩 Navier-Stokes-Poisson 系统的 Couette 流，本文在环面—实轴区域内建立线性化解上界，揭示无旋扰动先瞬态增长后指数衰减的稳定机制。
Convergence of a two-parameter hyperbolic relaxation system toward the incompressible Navier-Stokes equations
本文研究了双参数双曲松弛系统向不可压缩纳维-斯托克斯方程的收敛性，通过构造中间仿射系统和调制能量结构，证明了速度和压力在初始扰动趋于零时的收敛性，并扩展了速度收敛分析到常数级初始扰动情况。
In this paper, we analyze the behavior of viscous shock profiles of one-dimensional compressible Navier-Stokes equations with a singular pressure law which encodes the effects of congestion. As the intensity of the singular pressure tends to 0, we show the convergence of these profiles towards free-congested traveling front solutions of a two-phase compressible-incompressible Navier-Stokes system and we provide a refined description of the profiles in the vicinity of the transition between the free domain and the congested domain. In the second part of the paper, we prove that the profiles are…
Stability of Poiseuille Flow of Navier-Stokes Equations on $\mathbb{R}^2$
该研究分析二维全空间上带粘度的Navier-Stokes方程Poiseuille流的稳定性，证明线性化问题高频扰动的增强耗散效应，并给出非线性情形下扰动保持有界的初始幅值条件。
We consider the coupled chemotaxis Navier-Stokes model with logistic source terms \[ n_t + u\cdot \nabla n = Δn - χ\nabla \cdot (n \nabla c) + κn - μn^2\] \[ c_t + u\cdot \nabla c = Δc - nc\] \[ u_t + (u\cdot \nabla)u = Δu +\nabla P + n\nabla Φ+ f, \quad\qquad \nabla \cdot u=0 \] in a bounded, smooth domain $Ω\subset \mathbb{R}^3$ under homogeneous Neumann boundary conditions for $n$ and $c$ and homogeneous Dirichlet boundary conditions for $u$ and with given functions $f\in L^\infty(Ω\times(0,\infty))$ satisfying certain decay conditions and $Φ\in C^{1+β}(\barΩ)$ for some $β\in(0,1)$. We cons…
Stability of planar shock wave for the 3-dimensional compressible Navier-Stokes-Poisson equations
本文针对三维可压缩 NSP 系统，通过零模/非零模分解、反导数技术及电子势非线性项估计，证明平面粘性激波在小三维扰动下稳定，且非零模指数衰减。
Time-asymptotic stability of planar Navier-Stokes shocks with spatial oscillations
该研究针对三维可压缩等熵Navier-Stokes方程，证明带空间振荡的平面粘性激波对零质量小扰动的时间渐近稳定性，通过反导数技术与不同方向Poincaré不等式结合消除激波压缩的不利影响。
We consider the compressible Navier - Stokes - Fourier - Poisson system describing the motion of a viscous heat conducting rotating fluid confined to a straight layer $ Ω_ε = ω\times (0,ε) $, where $ω$ is a 2-D domain. The aim of this paper is to show that the weak solutions in the 3D domain converge to the strong solution of the 2-D Navier - Stokes - Fourier - Poisson system $ω$ as $ε\to 0$ on the time interval, where the strong solution exists. We consider two different regimes in dependence on the asymptotic behaviour of the Froude number.
The well-known Stokes waves refer to periodic traveling waves under the gravity at the free surface of a two dimensional full water wave system. In this paper, we prove that small-amplitude Stokes waves with infinite depth are nonlinearly unstable under long-wave perturbations. Our approach is based on the modulational approximation of the water wave system and the instability mechanism of the focusing cubic nonlinear Schrödinger equation.
Nonlocal Hyperdissipative Perturbations of the Three Dimensional Navier-Stokes System
研究非局部超耗散项对三维不可压缩纳维-斯托克斯系统的扰动，证明了能量恒等式、全局弱解性和局部强适定性，并确定了临界能量增长阈值。
Existence and uniqueness of viscous-dispersive shock profiles for the Navier-Stokes-Fourier-Korteweg system
该研究针对满足Rankine-Hugoniot与Lax熵条件的小强度激波，证明了一维NSFK系统粘性-弥散激波剖面的存在唯一性，通过几何奇异摄动理论构造剖面并给出逐点估计，为后续黎曼解稳定性分析提供基础。
Improved stability threshold for 2D Navier-Stokes Couette flow in an infinite channel
该研究针对无限通道中带Navier滑移边界的二维Navier-Stokes Couette流的非线性稳定性，改进了初始涡量扰动的稳定性阈值，消除了已有结果的对数损失，确定$\nu^{1/2}$为临界扰动尺度。
In this paper we treat three problems on a two-dimensional `punctured periodic domain': we take $Ω_r=(-L,L)^2\setminus D_r$, where $D_r=B(0,r)$ is the disc of radius $r$ centred at the origin. We impose periodic boundary conditions on the boundary of the box $Ω=(-L,L)^2$, and Dirichlet boundary conditions on the circumference of the disc. In this setting we consider the Poisson equation, the Stokes equations, and the time-dependent Navier-Stokes equations, all with a fixed forcing function $f$ (which must satisfy $\int_Ωf=0$ for the stationary problems), and examine the behaviour of solutions …
This paper is concerned with the large time behavior of the Cauchy problem for Navier-Stokes/Allen-Cahn system describing the interface motion of immiscible two-phase flow in 3-D. The existence and uniqueness of global solutions and the stability of the phase separation state is proved under the small initial perturbations. Moreover, the optimal time decay rates are obtained for higher-order spatial derivatives of density, velocity and phase. Our results implies that if the immiscible two-phase flow is initially located near the phase separation state, then under small perturbation conditions,…
This paper is concerned with the large-time behavior of solutions for the one-dimensional compressible Navier-Stokes system. We show that the combination of viscous contact wave with rarefaction waves for the non-isentropic polytropic gas is stable under \emph{large} initial perturbation without the condition that the adiabatic exponent $γ$ is close to 1, provided the strength of the combination waves is suitably small.
Motivated by \cite{CG10,CZ6}, we prove the global existence of solutions to the two-dimensional isentropic compressible Navier-Stokes equations with smooth initial data which are slowly varying in one direction and with initial density being away from vacuum. In particular, we present examples of initial data which generate unique global smooth solutions to 2D compressible Navier-Stokes equations with constant viscosity and with initial data which are neither small perturbation of some constant equilibrium state nor of small energy.
Global well-posedness and large-time behavior of classical solutions to the Euler-Navier-Stokes system in R^3
本文研究可压缩等温欧拉方程与不可压缩纳维-斯托克斯方程耦合的两相流系统柯西问题，证明平衡态小扰动下经典解的全局适定性与收敛性，借助霍奇分解等得到最优L^2时间衰减率。
The two dimensional Navier-Stokes equation in a perforated domain with a dynamical slip boundary condition is considered. We assume that the dynamic is driven by a stochastic perturbation on the interior of the domain and another stochastic perturbation on the boundaries of the holes. We consider a scaling ($\e^2$ for the viscosity and 1 for the density) that will lead to a time dependent limit problem. However, the noncritical scaling ($\e^β,\quad β>1$) is considered in front of the nonlinear term. The homogenized system in the limit is obtained as a Darcy's law with memory with two permeabil…
Existence and non-uniqueness of classical solutions to the axially symmetric stationary Navier-Stokes equations in an exterior cylinder
研究外周期圆柱内轴对称稳态 Navier-Stokes 方程，构造满足边界与无穷远条件、具临界及超临界衰减的经典解，并证明径向汇足够强时解非唯一。
We study here a singular perturbation problem of biLaplacian type, which can be seen as the biharmonic counterpart of classical combustion models. We provide different results, that include the convergence to a free boundary problem driven by a biharmonic operator, as introduced in a previous paper of ours, and a monotonicity formula in the plane. For the latter result, an important tool is provided by an integral identity that is satisfied by solutions of the singular perturbation problem. We also investigate the quadratic behaviour of solutions near the zero level set, at least for small val…
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