Incompressible Inhomogeneous Navier Stokes Equations Research Articles

Mathematical analysis of a finite difference method for inhomogeneous incompressible Navier–Stokes equations

Mathematical analysis of a finite difference method for inhomogeneous incompressible Navier–Stokes equations

Solvability of the two-dimensional stationary incompressible inhomogeneous Navier–Stokes equations with variable viscosity coefficient

We show the existence and the regularity properties of (a class of) weak solutions to the two-dimensional stationary incompressible inhomogeneous Navier–Stokes equations with density-dependent viscosity coefficients, by analyzing a fourth-order nonlinear elliptic equation for the stream function. For some stationary symmetric flows, we reformulate the Navier–Stokes equations as ordinary differential equations and give explicit examples of weak solutions. We present some further (ir-)regularity results in the case of piecewise-constant viscosity coefficients.

Energy equality for the 3D inhomogeneous Navier–Stokes equations in Lorentz–Besov spaces

We establish the energy equality for Leray–Hopf weak solutions of the 3D incompressible inhomogeneous Navier–Stokes equations in Lorentz–Besov spaces. This result may be regarded as an extension of that by Cheskidov and Luo (2020).

Global well-posedness for 2D fractional inhomogeneous Navier–Stokes equations with rough density

This paper is concerned with the global well-posedness issue of the two-dimensional (2D) incompressible inhomogeneous Navier–Stokes equations with fractional dissipation and rough density. By establishing the -maximal regularity estimate for the generalized Stokes system and using the Lagrangian approach, we prove the global existence and uniqueness of regular solutions for the 2D fractional inhomogeneous Navier–Stokes equations with large velocity field, provided that the initial density is sufficiently close to the constant 1 in and in the norm of some multiplier spaces. Moreover, we also consider the associated density patch problem, and show the global persistence of -regularity of the density patch boundary when the piecewise jump of density is small enough.

Global strong solution of 3D inhomogeneous incompressible Navier-Stokes equations with degenerate viscosity

In this paper, we investigate the Cauchy problem of the three-dimensional inhomogeneous incompressible Navier-Stokes equations with degenerate viscosity. For the case μ(ρ)=ρ, we prove the global existence of strong solution for small initial data satisfying the compatibility condition. Furthermore, we also obtain the large time algebraic decay rates of the velocity. The innovation of this article is that it does not require a positive lower bound for viscosity.

The hydrodynamic limit for the inhomogeneous Vlasov–Navier–Stokes system

In this paper, we study the hydrodynamic limit for the fluid–particle model, which consists of the inhomogeneous incompressible Navier–Stokes equations coupled with the Vlasov equation through a drag force in a bounded domain of [Formula: see text] with a homogeneous Dirichlet boundary condition on the fluid velocity field and Maxwell boundary condition on the kinetic distribution function. The proof relies on the relative entropy argument, which extends the work of El Ghani [Asymptotic analysis for a Vlasov–Navier–Stokes system in a bounded domain, J. Hyperbolic Differ. Equ. 7 (2010) 191–210] to inhomogeneous incompressible Navier–Stokes–Vlasov equations.

Fujita-Kato Theorem for the Inhomogeneous Incompressible Navier-Stokes Equations with Nonnegative Density

Fujita-Kato Theorem for the Inhomogeneous Incompressible Navier-Stokes Equations with Nonnegative Density

Liouville-type theorems for the stationary compressible barotropic and incompressible inhomogeneous Navier–Stokes equations

The present paper is dedicated to the Liouville-type problem for the three-dimensional stationary compressible barotropic Navier–Stokes equations and incompressible inhomogeneous Navier–Stokes equations. We show that velocity u is trivial under some suitable assumptions stated in terms of BMO−1 spaces.

Global well‐posedness of 3D incompressible inhomogeneous magnetohydrodynamic equations

In this paper, we investigate the 3D inhomogeneous incompressible magnetohydrodynamic (MHD) system. By the assumption of the smallness of initial velocity and magnetic fluids in the critical Besov space, the local and global well‐posedness of 3D inhomogeneous incompressible equations is obtained. It improves some previous results of MHD equations by generalizing the range of exponent in Besov spaces with . Besides, the initial density belongs to the critical Besov space with , and it is removed the additional restriction of , which is an important condition in some previous results for both 3D inhomogeneous incompressible Navier–Stokes equations and MHD system.

Global unique solvability of inhomogeneous incompressible Navier–Stokes equations with nonnegative density

In this paper, we consider the initial-boundary value problem to the inhomogeneous incompressible Navier–Stokes equations in . The initial density is allowed to be nonnegative, and in particular, the initial vacuum is allowed. The global existence and uniqueness of solutions are proved, for any initial data with s > 0, which constitutes a positive answer to the question raised by Danchin and Mucha (2019 Commun. Pure Appl. Math. 72 1351–85), in which the initial velocity (see also Li (2017 J. Differ. Equ. 263 6512–36).

Maximal L1 regularity for solutions to inhomogeneous incompressible Navier-Stokes equations

This paper is devoted to the maximal L1 regularity and asymptotic behavior for solutions to the inhomogeneous incompressible Navier-Stokes equations under a scaling-invariant smallness assumption on the initial velocity. We obtain a new global L1-in-time estimate for the Lipschitz seminorm of the velocity field without any smallness assumption on the initial density fluctuation. In the derivation of this estimate, we study the maximal L1 regularity for a linear Stokes system with variable coefficients. The analysis tools are a use of the semigroup generated by a generalized Stokes operator to characterize some Besov norms and a new gradient estimate for a class of second-order elliptic equations of divergence form. Our method can be used to study some other issues arising from incompressible or compressible viscous fluids.

An inverse boundary value problem for the incompressible inhomogeneous Navier–Stokes equations

An inverse boundary value problem for the incompressible inhomogeneous Navier–Stokes equations

Regularity for 3D Inhomogeneous Naiver–Stokes Equations in Vishik Spaces

In this short paper, we consider the conditional regularity for the 3D inhomogeneous incompressible Navier–Stokes equations in Vishik spaces and give regularity criterion of strong solutions.

Global existence and exponential decay of strong solutions for the inhomogeneous incompressible Navier–Stokes equations with vacuum

The inhomogeneous incompressible Navier-Stokes equations with fractional Laplacian dissipations in the multi-dimensional whole space are considered. The existence and uniqueness of global strong solution with vacuum are established for large initial data. The exponential decay-in-time of the strong solution is also obtained, which is different from the homogeneous case. The initial density may have vacuum and even compact support.

Inviscid limit of the inhomogeneous incompressible Navier–Stokes equations under the weak Kolmogorov hypothesis in $\mathbb{R}^3$

In this paper, we consider the inviscid limit of inhomogeneous incompressible Navier-Stokes equations under the weak Kolmogorov hypothesis in $\mathbb{R}^3$. In particular, we first deduce the Kolmogorov-type hypothesis in $\mathbb{R}^3$, which yields the uniform bounds of $\alpha^{th}$-order fractional derivatives of $\sqrt{\rho^\mu}{\bf u}^\mu $ in $L^2_x$ for some $\alpha>0$, independent of the viscosity. The uniform bounds can provide strong convergence of $\sqrt{\rho^{\mu}}\bf u^{\mu}$ in $L^2$ space. This shows that the inviscid limit is a weak solution to the corresponding Euler equations.

Global well-posedness for 3D incompressible inhomogeneous asymmetric fluids with density-dependent viscosity

In this paper, we investigate the 3D inhomogeneous incompressible asymmetric fluids system with density-dependent viscosity. By the assumption of the smallness of initial velocity in the critical Besov space, B˙p,13/p−1 for 1<p<6 and the initial density in the critical Besov space and bounded away from vacuum, the local and global well-posedness of 3D inhomogeneous incompressible asymmetric fluids is obtained. By giving the different estimates for pressure in the system for 1<p<32 and 32≤p<6, it shows the a priori estimate for the corresponding linearized equation. This not only improves the previous results for 3D inhomogeneous incompressible Navier-Stokes equations with density-dependent viscosity, but also obtains the new results on the micropolar system without smallness for density in critical Besov spaces.

Global Well-Posedness for the 2-D Inhomogeneous Incompressible Navier-Stokes System with Large Initial Data in Critical Spaces

Without any smallness assumption, we prove the global unique solvability of the 2-D incompressible inhomogeneous Navier-Stokes equations with initial data in the critical Besov space, which is almost the energy space in the sense that they have the same scaling in terms of this 2-D system.

Global weak solutions to the inhomogeneous incompressible Navier–Stokes–Vlasov–Boltzmann equations

In this paper, the initial–boundary value problem of the coupled inhomogeneous incompressible Navier–Stokes equations and Vlasov–Boltzmann equation for the moderately thick spray is considered in three-dimensional space. The global existence of weak solutions is established by an approximation scheme, a fixed point argument and the weak convergence method.

Global well-posedness of 3D incompressible inhomogeneous Navier–Stokes equations

Global well-posedness of 3D incompressible inhomogeneous Navier–Stokes equations

The global regularity for 3D inhomogeneous incompressible fluids with vacuum

In this paper, we investigate the global regularity for 3D inhomogeneous incompressible Navier–Stokes equations with vacuum. More precisely, we establish several Prodi–Serrin type regularity criteria in the Besov space in terms of only velocity or gradient of velocity which allow the initial density to contain vacuum. By using the embedding theory of Besov space, it is shown that our results improve the Prodi–Serrin type regularity condition which is scaling invariant in the time–space Lebesgue space.