We introduce a new concept of statistical solution in the framework of weak solutions to the barotropic Navier--Stokes system with inhomogeneous boundary conditions. Statistical solution is a family $\{ M_t \}_{t \geq 0}$ of Markov operators on the set of probability measures $\mathfrak{P}[\mathcal{D}]$ on the data space $\mathcal{D}$ containing the initial data $[\varrho_0, \mathbf{m}_0]$ and the boundary data $\mathbf{d}_B$.
(1) $\{ M_t \}_{t \geq 0}$ possesses a.a. semigroup property, $ M_{t + s}(ν) = M_t \circ M_s(ν)$ for any $t \geq 0$, a.a. $s \geq 0$, and any $ν\in \mathfrak{P}[\mathcal{D}]$.
(2) $\{ M_t \}_{t \geq 0}$ is deterministic when restricted to deterministic data, specifically $ M_t(δ_{[\varrho_0, \mathbf{m}_0, \mathbf{d}_B]}) = δ_{[\varrho(t, \cdot), \mathbf{m}(t, \cdot), \mathbf{d}_B]}, $ where $[\varrho, \mathbf{m}]$ is a finite energy weak solution of the Navier--Stokes system corresponding to the data $[\varrho_0, \mathbf{m}_0, \mathbf{d}_B] \in \mathcal{D}$.
(3) $M_t: \mathfrak{P}[\mathcal{D}] \to \mathfrak{P}[\mathcal{D}]$ is continuous in a suitable Bregman--Wasserstein metric at measures supported by the data giving rise to regular solutions.
We are concerned about the barotropic compressible Navier-Stokes equations with density-dependent viscosities which may degenerate in vacuum. We show that any classical solution to barotropic compressible Navier-Stokes equations in bounded domains will blow up, when the initial density admits an isolated mass group and the viscousity coefficients satisfy some conditions. A new condition on viscosities is first put forward in this paper.
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.
We consider the barotropic Navier--Stokes system describing the motion of a viscous compressible fluid interacting with the outer world through general in/out flux boundary conditions. We consider a hard--sphere type pressure EOS and show that all trajectories eventually enter a bounded absorbing set. In particular, the associated omega-limit sets are compact and support a stationary statistical solution.
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.
In this paper we consider the barotropic compressible quantum Navier-Stokes equations with a linear density dependent viscosity and its limit when the scaled Planck constant vanish. Following recent works on degenerate compressible Navier-Stokes equations, we prove the global existence of weak solutions by the use of a singular pressure close to vacuum. With such singular pressure, we can use the standard definition of global weak solutions which also allows to justify the limit when the scaled Planck constant denoted by $ε$ tends to 0.
We study the large-time behavior of the 1D barotropic Navier-Stokes flow perturbed from Riemann data generating a composition of two shock waves with small amplitudes. We prove that the perturbed Navier-Stokes flow converges, uniformly in space, towards a composition of two viscous shock waves as time goes to infinity, up to dynamical shifts. Especially, the strengths of the two waves can be chosen independently. This is the first result for the convergence to a composite wave of two viscous shocks with independently small amplitudes.
This paper concerns the existence of global weak solutions to the barotropic compressible Navier-Stokes equations with degenerate viscosity coefficients. We construct suitable approximate system which has smooth solutions satisfying the energy inequality, the BD entropy one, and the Mellet-Vasseur type estimate. Then, after adapting the compactness results due to Mellet-Vasseur [Comm. Partial Differential Equations 32 (2007)], we obtain the global existence of weak solutions to the barotropic compressible Navier-Stokes equations with degenerate viscosity coefficients in two or three dimensiona…
For barotropic compressible Navier-Stokes equations in three-dimensional (3D) bounded domains, we prove that any finite energy weak solution obtained by Lions [Mathematical topics in fluid mechanics, Vol. 2. Compressible models(1998)] and Feireisl-Novotný-Petzeltová [J. Math. Fluid Mech. 3(2001), 358-392] decays exponentially to the equilibrium state. This result is established by both using the extra integrability of the density due to Lions and constructing a suitable Lyapunov functional just under the framework of Lions-Feireisl's weak solutions.
This paper concerns the Cauchy problem of the barotropic compressible Navier-Stokes equations on the whole two-dimensional space with vacuum as far field density. In particular, the initial density can have compact support. When the shear and the bulk viscosities are a positive constant and a power function of the density respectively, it is proved that the two-dimensional Cauchy problem of the compressible Navier-Stokes equations admits a unique local strong solution provided the initial density decays not too slow at infinity. Moreover, if the initial data satisfy some additional regularity …
We consider the local well-posedness of strong and classical solutions to the three-dimensional barotropic compressible Navier-Stokes equations with density containing vacuum initially. We first prove the local existence and uniqueness of the strong solutions, where the initial compatibility condition proposed in \cite{cho1,K2,coi1} is removed under suitable sense. Then, the continuous of strong solutions on the initial data is derived under an additional compatibility condition. Moreover, for the initial data satisfying some additional regularity and compatibility condition, the strong soluti…
We prove the global existence and uniqueness of smooth solutions to the one-dimensional barotropic Navier-Stokes system with degenerate viscosity $μ(ρ)=ρ^α$. We establish that the smooth solutions have possibly two different far-fields, and the initial density remains positive globally in time, for the initial data satisfying the same conditions. In addition, our result works for any $α>0$, i.e., for a large class of degenerate viscosities. In particular, our models include the viscous shallow water equations. This extends the result of Constantin-Drivas-Nguyen-Pasqualotto \cite[Theorem 1.5]{C…
In the present paper we study the incompressible and fast rotation limit for the barotropic Navier-Stokes equations with Coriolis force, in the case when the Mach number $\rm Ma$ is large with respect to the Rossby number $\rm Ro$: namely, we focus on the regime ${\rm Ro}\ll{\rm Ma}$. For this, we follow a recent approach by Danchin and Mucha in \cite{D-M} and take also a large bulk viscosity coefficient. We prove that the limit dynamics is described by an incompressible Navier-Stokes type equation, recasted in the vorticy formulation, where however an additional unknown, linked to density osc…
Regularity and uniqueness of weak solutions of the compressible barotropic Navier-Stokes equations with constant viscosity coefficients is proven for small time in dimension $N=2,3$ under periodic boundary conditions. In this paper, the initial density is not required to have a positive lower bound and the pressure law is assumed to satisfy a condition that reduces to $P(ρ)=aρ^γ$ with $γ>1$ (in dimension three, additional conditions of size will be ask on $γ$). The second part of the paper is devoted to blow-up criteria for slightly subcritical initial data for the scaling of the equations whe…
Nonlinearly Exponential Stability for Lions-Feireisl's Weak Solutions to the Barotropic Compressible Navier-Stokes Equations with Large Potential External Forces
We are concerned with the inviscid limit of the Navier-Stokes equations to the Euler equations for barotropic compressible fluids in $\mathbb{R}^3$. When the viscosity coefficients obey a lower power-law of the density (i.e., $ρ^δ$ with $0<δ<1$), we identify a quasi-symmetric hyperbolic--singular elliptic coupled structure of the Navier-Stokes equations to control the behavior of the velocity of the fluids near the vacuum. Then this structure is employed to prove that there exists a unique regular solution to the corresponding Cauchy problem with arbitrarily large initial data and far-field va…
Asymptotic behavior toward viscous shock for the outflow problem of barotropic Navier-Stokes equations
Korea Advanced Institute of Mathematical Sciences（韩国高等数学研究院）；State Key Laboratory of Mathematical Sciences and Institute of Applied Mathematics, AMSS, CAS（中国科学院数学与系统科学研究院应用数学研究所）；University of Chinese Academy of Sciences（中国科学院大学）
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.
In this paper, we rigorously derive the compressible one-fluid Navier-Stokes equation from the scaled compressible two-fluid Navier-Stokes-Maxwell equations locally in time under the assumption that the initial data are well prepared. We justify the singular limit by proving the uniform decay of the error system, which is obtained by elaborate energy estimates.
The three-dimensional Navier-Stokes-$α$ model for fast rotating geophysical fluids is considered. The Navier-Stokes-$α$ model is a nonlinear dispersive regularization of the exact Navier-Stokes equations obtained by Lagrangian averaging and tend to the Navier-Stokes equations as $α\rightarrow 0^+$. We estimate upper bounds for the dimensions of global attractors and study the dependence of the dimensions on the parameter $α$. All the estimates are uniform in $α$, and our estimate of attractor dimensions remain finite when $α\rightarrow 0^+$.
In this paper, we study some model problem associated to the free boundary value problem of the barotropic compressible Navier-Stokes equations in general smooth domain with taking surface tension into account. To obtain the maximal $L_p-L_q$ regularity property of the model problem, we prove the existence of $\mathcal{R}-$bounded operator families of the resolvent problem via Weis' theory on operator valued Fourier multipliers.
We study Caccioppoli's inequalities of the non-stationary Stokes equations and Navier-Stokes equations. Our analysis is local near boundary and we prove that, in contrast to the interior case, the Caccioppoli's inequalities of the Stokes equations and the Navier-Stokes equations, in general, fail near boundary.
This paper studies two well-known models for two-phase fluid flow at constant temperature, the isothermal Navier-Stokes-Allen-Cahn and the isothermal Navier-Stokes-Cahn-Hilliard equations, both of which consist of equations for the (total) fluid density rho, the (mass-averaged)velocity u and the concentration (of one of the phases,) c. Assuming in either case that both phases are incompressible with different densities, each of the models is shown to reduce to a system of evolution equations in rho and u alone. In the case of the Navier-Stokes-Allen-Cahn model, this reduced system is the class…
An algebraic upper bound for the decay rate of solutions to the Navier-Stokes and Navier-Stokes-Coriolis equations in the critical space $\dot{H} ^{\frac{1}{2}} (\mathbb{R} ^3)$ is derived using the Fourier Splitting Method. Estimates are framed in terms of the decay character of initial data, leading to solutions with algebraic decay and showing in detail the roles played by the linear and nonlinear parts.
In this paper, we first prove the global well-posedness of a scaled anisotropic Navier-Stokes system and the hydrostatic Navier-Stokes system in a 2-D striped domain with small analytic data in the tangential variable. Then we justify the limit from the anisotropic Navier-Stokes system to the hydrostatic Navier-Stokes system with analytic data.
In this paper, we shall establish the global well-posedness, the space-time analyticity of the Navier-Stokes equations for a class of large periodic data $u_0 \in BMO^{-1}(\mathbb{R}^3)$. This improves the classical result of Koch \& Tataru \cite{koch-tataru}, for the global well-posedness with small initial data $u_0 \in BMO^{-1}(\mathbb{R}^n)$.
We study the stationary Stokes and Navier-Stokes equations with non-homogeneous Navier boundary condition in a bounded domain $Ω\subset\mathbb{R}^{3}$ of class $\mathcal{C}^{1,1}$. We prove existence, uniqueness of weak and strong solutions in $\mathbf{W}^{1,p}(Ω)$ and $\mathbf{W}^{2,p}(Ω)$ for all $1<p<\infty$ considering minimal regularity on the friction coefficient $α$. Moreover, we deduce uniform estimates on the solution with respect to $α$ which enables us to analyze the behavior of the solution when $α\rightarrow \infty$.
In this paper we prove a global well-posedness result for tridimensional Navier-Stokes-Boussinesq system with axisymmetric initial data. This system couples Navier-Stokes equations with a transport equation governing the density.
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.
In the note, a local regularity condition for axisymmetric solutions to the non-stationary 3D Navier-Stokes equations is proven. It reads that axially symmetric energy solutions to the Navier-Stokes equations have no Type I blowups.
We introduce a concept of space-time holomorphic solutions of partial differential equations and construct a meromorphic solution of Navier-Stokes equations.
Fundamental solutions to the time-periodic Stokes and Oseen linearizations of the Navier-Stokes equations in dimension $n\geq 2$ are investigated. Integrability properties and pointwise estimates are established.
Multiple spatial zero formations for Stokes and Navier-Stokes equations in three dimensions are shown to occur according to nodal sets of solenoidal Hermite polynomials. Extensions to well-posed Burnett equations with the bi-harmonic viscosity operator are also discussed.
In this paper, we justify the limit from the Navier-Stokes system in a thin domain to the hydrostatic Navier-Stokes/Prandtl system for the convex initial data with Gevrey 9/8 regualrity in $x$.
These notes are dedicated to the analysis of the one-dimensional free-congested Navier-Stokes equations. After a brief synthesis of the results obtained in [4] related to the existence and the asymptotic stability of partially congested profiles associated to the soft congestion Navier-Stokes system, we present a first local well-posedness result for the one-dimensional free-congested Navier-Stokes equations.
An enstrophy cascade is exhibited for the Navier-Stokes equations in physical scales independently of boundary conditions under physically reasonable assumptions on the flow.
In this paper, we prove existence of smooth solutions of the Navier-Stokes equations that gives a positive answer to the problem proposed by Fefferman [3].
In this paper, we study the upper bound of the time decay rate of solutions to the Navier-Stokes equations and generalized Navier-Stokes equations with damping term $|u|^{β-1}u$ ($β>1$) in $\mathbb{R}^3$.
We study 2D Navier-Stokes equations with a constraint on $L^2$ energy of the solution. We prove the existence and uniqueness of a global solution for the constrained Navier-Stokes equation on $\R^2$ and $\T$, by a fixed point argument. We also show that the solution of constrained Navier-Stokes converges to the solution of Euler equation as viscosity $ν$ vanishes.
We discuss the regularity of solutions of 2D incompressible Navier-Stokes equations forced by singular forces. The problem is motivated by the study of complex fluids modeled by the Navier-Stokes equations coupled to a nonlinear Fokker-Planck equation describing microscopic corpora embedded in the fluid. This leads naturally to bounded added stress and hence to $W^{-1,\infty}$ forcing of the Navier-Stokes equations.
We address the issue of existence of weak solutions for the non-homogeneous Navier-Stokes system with Navier friction boundary conditions allowing the presence of vacuum zones and assuming rough conditions on the data. We also study the convergence, as the viscosity goes to zero, of weak solutions for the non-homogeneous Navier-Stokes system with Navier friction boundary conditions to the strong solution of the Euler equations with variable density, provided that the initial data converge in $L^{2}$ to a smooth enough limit.
We prove in this paper a decay estimate for scaling invariant local energy solutions for some toy-models related to the incompressible Navier-Stokes system.
