We study the two-phase Stokes flow driven by surface tension with two fluids of equal viscosity, separated by an asymptotically flat interface with graph geometry. The flow is assumed to be two-dimensional with the fluids filling the entire space. We prove well-posedness and parabolic smoothing in Sobolev spaces up to critical regularity. The main technical tools are an analysis of nonlinear singular integral operators arising from the hydrodynamic single-layer potential and abstract results on nonlinear parabolic evolution equations.
We study the inhomogeneous incompressible Navier-Stokes system endowed with a general capillary term. Thanks to recent methods based on Lagrangian change of variables, we obtain local well-posedness in critical Besov spaces (even if the integration index p is different from 2) and for variable viscosity and capillary terms. In the case of constant coefficients and for initial data that are perturbations of a constant state, we are able to prove that the lifespan goes to infinity as the capillary coefficient goes to zero, connecting our result to the global existence result obtained by Danchin …
In the first main result of this paper we prove that one can approximate discontinious solutions of the 1d Navier Stokes system with solutions of the 1d Navier-Stokes-Korteweg system as the capilarity parameter tends to 0. Moreover, we allow the viscosity coefficients $μ$ = $μ$ ($ρ$) to degenerate near vaccum. In order to obtain this result, we propose two main technical novelties. First of all, we provide an upper bound for the density verifing NSK that does not degenerate when the capillarity coefficient tends to 0. Second of all, we are able to show that the positive part of the effective v…
Global existence and uniqueness of the density-dependent incompressible Navier-Stokes-Korteweg system with variable capillarity and viscosity coefficients
In this article, we consider the compressible Navier-Stokes equation with density dependent viscosity coefficients and a term of capillarity introduced by Coquel et al in \cite{5CR}. This model includes at the same time the barotropic Navier-Stokes equations with variable viscosity coefficients, shallow-water system and the model of Rohde. We first study the well-posedness of the model in critical regularity spaces with respect to the scaling of the associated equations. In a functional setting as close as possibleto the physical energy spaces, we prove global existence of solutions close to a…
We are concerned with an isothermal model of viscous and capillary compressible fluids derived by J. E. Dunn and J. Serrin (1985), which can be used as a phase transition model. Compared with the classical compressible Navier-Stokes equations, there is a smoothing effect on the density that comes from the capillary terms. First, we prove that the global solutions with critical regularity that have been constructed in [11] by the second author and B. Desjardins (2001), are Gevrey analytic. Second, we extend that result to a more general critical L p framework. As a consequence, we obtain algebr…
In this paper we study a singular limit problem for a Navier-Stokes-Korteweg system with Coriolis force, in the domain $\R^2\times\,]0,1[\,$ and for general ill-prepared initial data. Taking the Mach and the Rossby numbers to be proportional to a small parameter $\veps$ going to $0$, we perform the incompressible and high rotation limits simultaneously. Moreover, we consider both the constant capillarity and vanishing capillarity regimes. In this last case, the limit problem is identified as a $2$-D incompressible Navier-Stokes equation in the variables orthogonal to the rotation axis. If the …
Bresch-Desjardins-Gisclon-Sart have derived that the capillarity can slow {down} the growth rate of Rayleigh-Taylor (RT) instability in the capillary fluids based on the linearized two-dimensional (2D) Navier-Stokes-Korteweg equations in 2008. Motivated by their linear theory, we further investigate the nonlinear RT problem for the 2D incompressible case in a horizontally periodic slab domain with Navier boundary condition, and rigorously verify that the RT instability can be inhibited by capillarity under our 2D setting. More precisely, if the RT density profile $\barρ$ satisfies an additiona…
This is the first of a series of papers devoted to the initial value problem for the Euler system of compressible fluids and augmented versions containing higher-order terms. We encompass solutions that have finite total energy and enjoy a certain symmetry (for instance, plane symmetry); these solutions may have unbounded amplitude and contain cavitation regions in which the mass density vanishes. In the present paper, we are interested in dispersive shock waves and analyze the zero viscosity-capillarity limit associated with the Navier-Stokes-Korteweg system. Specifically, we establish the ex…
We consider a nonlocal version of the quasi-static Navier-Stokes-Korteweg equations with a non-monotone pressure law. This system governs the low-Reynolds number dynamics of a compressible viscous fluid that may take either a liquid or a vapour state. For a porous domain that is perforated by cavities with diameter proportional to their mutual distance the homogenization limit is analyzed. We extend the results for compressible one-phase flow with polytropic pressure laws and prove that the effective motion is governed by a nonlocal version of the Cahn-Hilliard equation. Crucial for the analys…
A combined incompressible and vanishing capillarity limit in the barotropic compressible Navier-Stokes equations for smooth solutions is proved. The equations are considered on the two-dimensional torus with well prepared initial data. The momentum equation contains a rotational term originating from a Coriolis force, a general Korteweg-type tensor modeling capillary effects, and a density-dependent viscosity. The limiting model is the viscous quasi-geostrophic equation for the "rotated" velocity potential. The proof of the singular limit is based on the modulated energy method with a careful …
In this paper we consider the Navier-Stokes-Korteweg equations for a viscous compressible fluid with capillarity effects in three space dimensions. We prove compactness of finite energy weak solutions for large initial data. In contrast with previous results regarding this system, vacuum regions are allowed in the definition of weak solutions and no additional damping terms are considered. The compactness is obtained by introducing suitable truncations of the velocity field and the mass density at different scales and use only the a priori bounds obtained by the energy and the BD entropy.
In the setting of general initial data and whole space we perform a rigorous analysis of the quasineutral limit for a hydrodynamical model of a viscous plasma with capillarity tensor represented by the Navier Stokes Korteweg Poisson system. We shall provide a detailed mathematical description of the convergence process by analyzing the dispersion of the fast oscillating acoustic waves. However the standard acoustic wave analysis is not sufficient to control the high frequency oscillations in the electric field but it is necessary to estimates the dispersive properties induced by the capillarit…
We introduce a framework to justify hydrodynamic limits of the Vlasov-Navier-Stokes system. We specifically study high friction regimes, which take into account the fact that particles of the dispersed phase are light (resp. small) compared to the fluid part, and lead to the derivation of Transport-Navier-Stokes (resp. Inhomogeneous Navier-Stokes) systems.
As is well-known that the general radiation hydrodynamics models include two mainly coupled parts: one is macroscopic fluid part, which is governed by the compressible Navier-Stokes-Fourier equations, another is radiation field part, which is described by the transport equation of photons. Under the two physical approximations: "gray" approximation and P1 approximation, one can derive the so-called Navier-Stokes-Fourier-P1 approximation radiation hydrodynamics model from the general one. In this paper we study the non-relativistic limit problem for the Navier-Stokes-Fourier-P1 approximation mo…
A coupled system of self-organized hydrodynamics and Navier-Stokes equations (SOH-NS), which models self-propelled particles in a viscous fluid, was recently derived by Degond et al. \cite{DMVY-2017-arXiv}, starting from a micro-macro particle system of Vicsek-Navier-Stokes model, through an intermediate step of a self-organized kinetic-kinetic model by multiple coarse-graining processes. We first transfer SOH-NS into a non-singular system by stereographic projection, then prove the local in time well-posedness of classical solutions by energy method. Furthermore, employing the Hilbert expansi…
We consider the isentropic Navier-Stokes-Korteweg equations with a non-decreasing pressure on the whole space $\mathbb{R}^n$ $(n \ge 2)$, where the system describes the motion of compressible fluids such as liquid-vapor mixtures with phase transitions including a variable internal capillarity effect. We prove the existence of a unique global strong solution to the system in the $\mathrm{L}^p$-in-time and $\mathrm{L}^q$-in-space framework, especially in the maximal regularity class, by assuming $(p, q) \in (1, 2) \times (1, \infty)$ or $(p, q) \in \{2\} \times (1, 2]$. We show that the system i…
A Quantitative Hydrodynamic Limit for the Vlasov-Fokker-Planck/Navier-Stokes System in the Fine Particle Regime
School of Mathematics and Information Science, Guangzhou University（广州大学数学与信息科学学院）；School of Mathematics, South China University of Technology（华南理工大学数学学院）；Department of Computational Mathematics, Science and Engineering and Department of Mathematics, Michigan State University（密歇根州立大学计算数学、科学与工程系及数学系）
We prove a global-in-time limit from the two-species Vlasov-Maxwell-Boltzmann system to the two-fluid incompressible Navier-Stokes-Fourier-Maxwell system with Ohm's law. Besides the techniques developed for the classical solutions to the Vlasov-Maxwell-Boltzmann equations in the past years, such as the nonlinear energy method and micro-macro decomposition are employed, key roles are played by the decay properties of both the electric field and the wave equation with linear damping of the divergence free magnetic field. This is a companion paper of [N. Jiang and Y.-L. Luo, \emph{Ann. PDE} 8 (20…
We consider the zero-electron-mass limit for the Navier-Stokes-Poisson system in unbounded spatial domains. Assuming smallness of the viscosity coefficient and ill-prepared initial data, we show that the asymptotic limit is represented by the incompressible Navier-Stokes system, with a Brinkman damping, in the case when viscosity is proportional to the electron-mass, and by the incompressible Euler system provided the viscosity is dominated by the electron mass. The proof is based on the RAGE theorem and dispersive estimates for acoustic waves, and on the concept of suitable weak solutions for…
This paper addresses the issue of global existence of so-called $κ$-entropy solutions to the Navier-Stokes equations for viscous compressible and barotropic fluids with degenerate viscosities. We consider the three dimensional space domain with periodic boundary conditions. Our solutions satisfy the weak formulation of the mass and momentum conservation equations and also a generalization of the BD-entropy identity called: $κ$-entropy. This new entropy involves a mixture parameter $κ\in (0,1)$ between the two velocities $\bf{u}$ and $\bf{u} + 2\nablaφ(\varrho)$ (the latter was introduced by th…
This paper is concerned with the rigorous analysis of the zero electron mass limit of the full Navier-Stokes-Poisson. This system has been introduced in the literature by Anile and Pennisi (see [5]) in order to describe a hydrodynamic model for charge-carrier transport in semiconductor devices. The purpose of this paper is to prove rigorously zero electron mass limit in the framework of general ill prepared initial data. In this situation the velocity field and the electronic fields develop fast oscillations in time. The main idea we will use in this paper is a combination of formal asymptotic…
We derive two systems of boundary-domain integral equations (BDIEs) equivalent to the Dirichlet problem for the compressible Stokes system using the potential method with an explicit parametrix (Levi function). The BDIEs are given in terms of surface and volume hydrodynamic potentials. We study the mapping properties of these integral potential operators and used them to prove existence and uniqueness of solution of the two systems of BDIEs obtained taking into account the non-trivial kernels of the single layer and hypersingular hydrodynamic surface potentials.
This paper is devoted to the global well-posedness of two Diffuse Interface systems modeling the motion of an incompressible two-phase fluid mixture in presence of capillarity effects in a bounded smooth domain $Ω\subset \mathbb{R}^d$, $d=2,3$. We focus on dissipative mixing effects originating from the mass-conserving Allen-Cahn dynamics with the physically relevant Flory-Huggins potential. More precisely, we study the mass-conserving Navier-Stokes-Allen-Cahn system for nonhomogeneous fluids and the mass-conserving Euler-Allen-Cahn system for homogeneous fluids. We prove existence and uniquen…
We derive macroscopic dynamics for self-propelled particles in a fluid. The starting point is a coupled Vicsek-Stokes system. The Vicsek model describes self-propelled agents interacting through alignment. It provides a phenomenological description of hydrodynamic interactions between agents at high density. Stokes equations describe a low Reynolds number fluid. These two dynamics are coupled by the interaction between the agents and the fluid. The fluid contributes to rotating the particles through Jeffery's equation. Particle self-propulsion induces a force dipole on the fluid. After coarse-…
In this paper, we give an overview of the results established in [3] which provides the first rigorous derivation of hydrodynamic equations from the Boltzmann equation for inelastic hard spheres in 3D. In particular, we obtain a new system of hydrodynamic equations describing granular flows and prove existence of classical solutions to the aforementioned system. One of the main issue is to identify the correct relation between the restitution coefficient (which quantifies the rate of energy loss at the microscopic level) and the Knudsen number which allows us to obtain non trivial hydrodynamic…
We present a new hydrodynamic model consisting of the pressureless Euler equations and the isentropic compressible Navier-Stokes equations where the coupling of two systems is through the drag force. This coupled system can be derived, in the hydrodynamic limit, from the particle-fluid equations that are frequently used to study the medical sprays, aerosols and sedimentation problems. For the proposed system, we first construct the local-in-time classical solutions in an appropriate $L^2$ Sobolev space. We also establish the \emph{a priori} large-time behavior estimate by constructing a Lyapun…
Global well-posedness and optimal decay rates of classical solutions to the compressible Navier-Stokes-Fourier-P$_1$ approximation model in radiation hydrodynamics
We prove that any weak space-time $L^2$ vanishing viscosity limit of a sequence of strong solutions of Navier-Stokes equations in a bounded domain of ${\mathbb{R}}^2$ satisfies the Euler equation if the solutions' local enstrophies are uniformly bounded. We also prove that $t-a.e.$ weak $L^2$ inviscid limits of solutions of 3D Navier-Stokes equations in bounded domains are weak solutions of the Euler equation if they locally satisfy a scaling property of their second order structure function. The conditions imposed are far away from boundaries, and wild solutions of Euler equations are not a p…
We demonstrate that the three dimensional incompressible magneto-hydrodynamics (MHD) system is ill-posed due to the discontinuity of weak solutions in a wide range of spaces. Specifically, we construct initial data which has finite energy and is small in certain spaces, such that any Leray-Hopf type of weak solution to the MHD system starting from this initial data is discontinuous at time $t=0$ in such spaces. An analogous result is also obtained for the Navier-Stokes equation which extends the previous result of ill-posedness in $\dot B^{-1}_{\infty,\infty}$ by Cheskidov and Shvydkoy to spac…
The incompressible Navier-Stokes-Fourier system with viscous heating was first derived from the Boltzmann equation in the form of the diffusive scaling by Bardos-Levermore-Ukai-Yang (2008). The purpose of this paper is to justify such an incompressible hydrodynamic approximation to the Boltzmann equation in $L^2\cap L^\infty$ setting in a periodic box. Based on an odd-even expansion of the solution with respect to the microscopic velocity, the diffusive coefficients are determined by the incompressible Navier-Stokes-Fourier system with viscous heating and the super Burnett functions. More impo…
Despite its conceptual and practical importance, the rigorous derivation of the steady incompressible Navier-Stokes-Fourier system from the Boltzmann theory has been {an} outstanding {open problem} for general domains in 3D. We settle this open question in {the} affirmative, in the presence of a small external field and a small boundary temperature variation for the diffuse boundary condition. We employ a recent quantitative $L^{2}-L^{\infty }$ approach with new $L^{6}$ estimates for the hydrodynamic part $\mathbf{P}f$ of the distribution function. Our results also imply the validity of Fourie…
We study a hydrodynamic limit of a system of coupled kinetic and fluid equations under a strong local alignment force and a strong Brownian motion. More precisely, we consider the Vlasov-Fokker-Planck type equation and compressible Navier-Stokes equations with a density-dependent viscosity. Based on a relative entropy argument, by assuming the existence of weak solutions to that kinetic-fluid system, we rigorously derive a two-phase fluid model consisting of isothermal Euler equations and compressible Navier-Stokes equations with a density-dependent viscosity.
In this paper, we study a hydrodynamical system modeling the deformation of vesicle membrane under external incompressible viscous flow fields. The system is in the Eulerian formulation and is governed by the coupling of the incompressible Navier-Stokes equations with a phase field equation. In the three dimensional case, we establish two logarithmically improved blow-up criteria for local smooth solutions of this system in terms of the vorticity field only in the homogeneous Besov spaces.
We provide local formulas for the pressure of incompressible fluids. The pressure can be expressed in terms of its average and averages of squares of velocity increments in arbitrary small neighborhoods. As application, we give a brief proof of the fact that $C^α$ velocities have $C^{2α}$ (or Lipschitz) pressures. We also give some regularity criteria for 3D incompressible Navier-Stokes equations.
In this paper, we extend some results proved in previous references for three-dimensional Navier-Stokes equations. We show that when the norm of the velocity field is small enough in $L^3({I\!\!R}^3)$, then a global smooth solution of the Navier-Stokes equations is ensured. We show that a similar result holds when the norm of the velocity field is small enough in $H^{\frac{1}{2}}({I\!\!R}^3)$. The scale invariance of these two norms is discussed.
We study the Boltzmann equation with external forces, not necessarily deriving from a potential, in the incompressible Navier-Stokes perturbative regime. On the torus, we establish local-in-time, for any time, Cauchy theories that are independent of the Knudsen number in Sobolev spaces. The existence is proved around a time-dependent Maxwellian that behaves like the global equilibrium both as time grows and as the Knudsen number decreases. We combine hypocoercive properties of linearized Boltzmann operators with linearization around a time-dependent Maxwellian that catches the fluctuations of …
Hydrodynamic systems for quantum fluids are systems for compressible fluid flows for which quantum effects are macroscopically relevant. We discuss how the presence of the dispersive tensor describing the quantum effects alters the acoustic dispersion at the example of the Quantum Hydrodynamic system (QHD). For the QHD system the dispersion relation is given by the Bogoliubov dispersion relation for weakly interacting Bose gases. We provide refined Strichartz estimates allowing for an accurate control of acoustic oscillations. Applications to the low Mach number limit for the quantum Navier-St…
We consider a modification of the three-dimensional Navier--Stokes equations and other hydrodynamical evolution equations with space-periodic initial conditions in which the usual Laplacian of the dissipation operator is replaced by an operator whose Fourier symbol grows exponentially as $\ue ^{|k|/\kd}$ at high wavenumbers $|k|$. Using estimates in suitable classes of analytic functions, we show that the solutions with initially finite energy become immediately entire in the space variables and that the Fourier coefficients decay faster than $\ue ^{-C(k/\kd) \ln (|k|/\kd)}$ for any $C<1/(2\ln…
