Inhomogeneous Navier Stokes System Research Articles

Global well-posedness of 3D inhomogeneous Navier–Stokes system with small unidirectional derivative

In Liu and Zhang (2020 Arch. Ration. Mech. Anal. 235 1405–44); Liu et al (2020 Arch. Ration. Mech. Anal. 238 805–43), the authors proved that as long as the one-directional derivative of the initial velocity is sufficiently small in some scaling invariant spaces, then the (anisotropic) Navier–Stokes (NS) system has a unique global solution. The goal of this paper is to extend this type of result to the 3D inhomogeneous (density-dependent) NS system. More precisely, given initial density such that and the initial velocity with belonging to , then the inhomogeneous NS system has a unique global solution provided that being sufficiently small for some bounded function f depending on and . This provide a more general result that of Chemin et al (2014 Commun. Math. Phys. 272 529–66); Chemin and Zhang (2015 Commun. PDE 40 878–96).

Global Solutions of 3-D Inhomogeneous Navier-Stokes System with Large Viscosity in One Variable

Global Solutions of 3-D Inhomogeneous Navier-Stokes System with Large Viscosity in One Variable

On the Global Well-Posedness of 3-D Density-Dependent MHD System

In this paper we study existence and uniqueness of solutions for the magneto-hydrodynamic system with variable density, variable viscosity and variable conductivity, which describes the coupling between the inhomogeneous Navier-Stokes system and the Maxwell equation.

Wave-Packet Models for Jet Dynamics and Sound Radiation

Organized structures in turbulent jets can be modeled as wavepackets. These are characterized by spatial amplification and decay, both of which are related to stability mechanisms, and they are coherent over several jet diameters, thereby constituting a noncompact acoustic source that produces a distinctive directivity in the acoustic field. In this review, we use simplified model problems to discuss the salient features of turbulent-jet wavepackets and their modeling frameworks. Two classes of model are considered. The first, that we refer to as kinematic, is based on Lighthill's acoustic analogy, and allows an evaluation of the radiation properties of sound-source functions postulated following observation of jets. The second, referred to as dynamic, is based on the linearized, inhomogeneous Ginzburg–Landau equation, which we use as a surrogate for the linearized, inhomogeneous Navier–Stokes system. Both models are elaborated in the framework of resolvent analysis, which allows the dynamics to be viewed in terms of an input–ouput system, the input being either sound-source or nonlinear forcing term, and the output, correspondingly, either farfield acoustic pressure fluctuations or nearfield flow fluctuations. Emphasis is placed on the extension of resolvent analysis to stochastic systems, which allows for the treatment of wavepacket jitter, a feature known to be relevant for subsonic jet noise. Despite the simplicity of the models, they are found to qualitatively reproduce many of the features of turbulent jets observed in experiment and simulation. Sample scripts are provided and allow calculation of most of the presented results.

Between homogeneous and inhomogeneous Navier–Stokes systems: The issue of stability

We construct large velocity vector solutions to the three dimensional inhomogeneous Navier–Stokes system. The result is proved via the stability of two dimensional solutions with constant density, under the assumption that initial density is point-wisely close to a constant. Key elements of our approach are estimates in the maximal regularity regime and the Lagrangian coordinates. Considerations are done in the whole R3.

Global well-posedness of 3-D inhomogeneous Navier–Stokes system with initial velocity being a small perturbation of 2-D solenoidal vector field

Motivated by [21], we consider the global wellposedness to the 3-D incompressible inhomogeneous Navier–Stokes equations with large horizontal velocity. In particular, we proved that when the initial density is close enough to a positive constant, then given divergence free initial velocity field of the type (v0h,0)(xh)+(w0h,w03)(xh,x3), we shall prove the global wellposedness of (1.1). The main difficulty here lies in the fact that we will have to obtain the L1(R+;Lip(R3)) estimate for convection velocity in the transport equation of (1.1). Toward this and due to the strong anisotropic properties of the approximate solutions, we will have to work in the framework of anisotropic Littlewood–Paley theory here.

Global well-posedness of 3-D inhomogeneous Navier–Stokes system with ill-prepared initial data

In this paper, we investigate the global well-posedness of 3-D incompressible inhomogeneous Navier–Stokes system with ill-prepared initial data of the form \bigl(1+\epsilon^{\beta}a_0(x_{\rm h},\epsilon x_3),(\epsilon^{1-\alpha}\,v^{\rm h}_0, \epsilon^{-\alpha}v_0^3)(x_{\rm h},\epsilon x_3)\bigr) for any \alpha \in ]0,1/3[, \beta > 2\alpha , and \epsilon being sufficiently small. This result improves the global well-posedness result for so-called well-prepared initial data, which corresponds to the case of \alpha=0 .

INHOMOGENEOUS INCOMPRESSIBLE VISCOUS FLOWS WITH SLOWLY VARYING INITIAL DATA

The purpose of this paper is to provide a large class of initial data which generates global smooth solution of the 3D inhomogeneous incompressible Navier–Stokes system in the whole space $\mathbb{R}^{3}$. This class of data is based on functions which vary slowly in one direction. The idea is that 2D inhomogeneous Navier–Stokes system with large data is globally well-posed and we construct the 3D approximate solutions by the 2D solutions with a parameter. One of the key point of this study is the investigation of the time decay properties of the solutions to the 2D inhomogeneous Navier–Stokes system. We obtained the same optimal decay estimates as the solutions of 2D homogeneous Navier–Stokes system.

On the global well-posedness of 2-D inhomogeneous incompressible Navier–Stokes system with variable viscous coefficient

Given solenoidal vector u0∈H˙−2δ∩H1(R2), ρ0−1∈L2(R2), and ρ0∈L∞∩W˙1,r(R2) with a positive lower bound for δ∈(0,12) and 2<r<21−2δ, we prove that 2-D incompressible inhomogeneous Navier–Stokes system (1.1) has a unique global solution provided that the viscous coefficient μ(ρ0) is close enough to 1 in the L∞ norm compared to the size of δ and the norms of the initial data. With smoother initial data, we can prove the propagation of regularities for such solutions. Furthermore, for 1<p<4, if (ρ0−1,u0) belongs to the critical Besov spaces B˙p,12p(R2)×(B˙p,1−1+2p∩L2(R2)) and the B˙p,12p(R2) norm of ρ0−1 is sufficiently small compared to the exponential of ‖u0‖L22+‖u0‖B˙p,1−1+2p, we prove the global well-posedness of (1.1) in the scaling invariant spaces. Finally for initial data in the almost critical Besov spaces, we prove the global well-posedness of (1.1) under the assumption that the L∞ norm of ρ0−1 is sufficiently small.

Inhomogeneous Navier–Stokes equations in the half-space, with only bounded density

In this paper, we establish the global existence and uniqueness of solutions to the inhomogeneous Navier–Stokes system in the half-space. The initial density only has to be bounded and close enough to a positive constant, the initial velocity belongs to some critical Besov space, and the L∞ norm of the inhomogeneity plus the critical norm to the horizontal components of the initial velocity has to be very small compared to the exponential of the norm to the vertical component of the initial velocity. With a little bit more regularity for the initial velocity, those solutions are proved to be unique. In the last section of the paper, our results are partially extended to the bounded domain case.

Global large solutions to 3-D inhomogeneous Navier–Stokes system with one slow variable

Under a nonlinear smallness condition on the isotropic critical Besov norm to the fluctuation of the initial density and the critical anisotropic Besov norm of the horizontal components of the initial velocity, which have to be exponentially small compared with the critical anisotropic Besov norm to the third component of the initial velocity, we investigate the global wellposedness of 3-D inhomogeneous incompressible Navier–Stokes equations (1.2) in the critical Besov spaces. The novelty of this results is that the isotropic space structure to the inhomogeneity of the initial density function is consistent with the propagation of anisotropic regularity for the velocity field. In the second part, we apply the same idea to prove the global wellposedness of (1.2) with some large data which are slowly varying in one direction.

Global solutions to 2-D inhomogeneous Navier–Stokes system with general velocity

In this paper, we are concerned with the global wellposedness of 2-D density-dependent incompressible Navier–Stokes equations (1.1) with variable viscosity, in a critical functional framework which is invariant by the scaling of the equations and under a nonlinear smallness condition on fluctuation of the initial density which has to be doubly exponential small compared with the size of the initial velocity. In the second part of the paper, we apply our methods combined with the techniques in Danchin and Mucha (2012) [10] to prove the global existence of solutions to (1.1) with constant viscosity and with piecewise constant initial density which has small jump at the interface and is away from vacuum. In particular, this latter result removes the smallness condition for the initial velocity in a corresponding theorem of Danchin and Mucha (2012) [10].

Global solutions to the 3-D incompressible inhomogeneous Navier–Stokes system

In this paper, we consider the global well-posedness of the 3-D incompressible inhomogeneous Navier–Stokes equations with initial data in the critical Besov spaces a0∈Bq,13q(R3), u0=(u0h,u03)∈Bp,1−1+3p(R3) for p, q satisfying 1<q⩽p<6 and 1q−1p⩽13. More precisely, we prove that there exist two positive constants c0, C0 such that if (μ‖a0‖Bq,13q+‖u0h‖Bp,1−1+3p)exp(C0‖u03‖Bp,1−1+3p2/μ2)⩽c0μ, then (1.3) has a unique global solution a∈L˜∞(R+;Bq,13q(R3)), u∈L˜∞(R+;Bp,1−1+3p(R3))∩L1(R+;Bp,11+3p(R3)). In particular, this result implies the global well-posedness result in Abidi and Paicu (2007) [2] for the inhomogeneous Navier–Stokes system with small initial data.