Incompressible Inhomogeneous Navier Stokes Equations Research Articles

Inhomogeneous Incompressible Navier–Stokes Equations on Thin Domains

We consider the inhomogeneous incompressible Navier–Stokes equation on thin domains $${\mathbb {T}}^2 \times \epsilon {\mathbb {T}}$$, $$\epsilon \rightarrow 0$$. It is shown that the weak solutions on $${\mathbb {T}}^2 \times \epsilon {\mathbb {T}}$$ converge to the strong/weak solutions of the 2D inhomogeneous incompressible Navier–Stokes equations on $${\mathbb {T}}^2$$ as $$\epsilon \rightarrow 0$$ on arbitrary time interval.

Global regularity of density patch for the 3D inhomogeneous Navier–Stokes equations

This paper is concerning with the regularity propagation of density patches for the 3D inhomogeneous incompressible Navier–Stokes equations. By careful time-weighted energy estimates, Stokes estimates and the singular integral operators, we prove that the 3D density patches preserve the $$C^{k,\gamma }(k=1,2)$$ regularity for the initial interface given by $$\rho _0(x)=\eta _11_{\Omega _0}+\eta _21_{\Omega ^c_0}$$. In particular, we do not need the smallness assumption on $$|\eta _1-\eta _2|$$.

Incompressible inhomogeneous fluids in bounded domains of [formula omitted] with bounded density

In this paper, we study the incompressible inhomogeneous Navier-Stokes equations in bounded domains of R3 involving bounded density functions ρ=1+a. Based on the corresponding theory of Besov spaces on domains, we first obtain the global existence of weak solutions (ρ,u) with initial data a0∈L∞(Ω), u0∈Bq,s−1+3/q(Ω) for 1<q<3, 1<s<∞. Furthermore, with additional regularity assumptions on the initial velocity, we also prove the uniqueness of such a solution. It is a generalization of a result established by Huang et al. (2013) [20] for the whole space R3.

Global solution to inhomogeneous incompressible Navier-Stokes equations on thin domain

We consider the inhomogeneous incompressible Navier-Stokes equations on thin domains R2×εT. It is shown that when the vertical size ε is sufficiently small, there exists a unique global strong solution to the initial value problem with relatively large data.

Striated Regularity of 2-D Inhomogeneous Incompressible Navier–Stokes System with Variable Viscosity

In this paper, we investigate the global existence and uniqueness of strong solutions to 2D incompressible inhomogeneous Navier-Stokes equations with viscous coefficient depending on the density and with initial density being discontinuous across some smooth interface. Compared with the previous results for the inhomogeneous Navier-Stokes equations with constant viscosity, the main difficulty here lies in the fact that the $L^1$ in time Lipschitz estimate of the velocity field can not be obtained by energy method (see \cite{DM17,LZ1, LZ2} for instance). Motivated by the key idea of Chemin to solve 2-D vortex patch of ideal fluid (\cite{Chemin91, Chemin93}), namely, striated regularity can help to get the $L^\infty$ boundedness of the double Riesz transform, we derive the {\it a priori} $L^1$ in time Lipschitz estimate of the velocity field under the assumption that the viscous coefficient is close enough to a positive constant in the bounded function space. As an application, we shall prove the propagation of $H^3$ regularity of the interface between fluids with different densities.

From compressible to incompressible inhomogeneous flows in the case of large data

This paper is concerned with the mathematical derivation of the inhomoge-neous incompressible Navier-Stokes equations (INS) from the compressible Navier-Stokes equations (CNS) in the large volume viscosity limit. We first prove a result of large time existence of regular solutions for (CNS). Next, as a consequence, we establish that the solutions of (CNS) converge to those of (INS) when the volume viscosity tends to infinity. Analysis is performed in the two dimensional torus, for general initial data. In particular, we are able to handle large variations of density.

The inviscid limit in the Cauchy problem of the inhomogeneous incompressible Navier–Stokes equations

Applying similar arguments as introduced in Bona–Smith [4], we mainly study the inviscid limit of the inhomogeneous incompressible Navier–Stokes equations in Rd. Some results of local uniform estimates on solutions of the Navier–Stokes equations, independent of the viscosity, are also obtained.

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.

Analyticity of the inhomogeneous incompressible Navier–Stokes equations

In this paper, we obtain analyticity of the inhomogeneous Navier–Stokes equations. The main idea is to use the exponential operator e ϕ ( t ) | D | , where ϕ ( t ) = δ − θ ( t ) , δ > 0 is the analyticity radius of ( ρ 0 − 1 , u 0 ) , and | D | is the differential operator whose symbol is given by ‖ ξ ‖ l 1 . We will show that for sufficiently small initial data, solutions are analytic globally in time in critical Besov spaces.

Global Regularity of 2D Density Patches for Inhomogeneous Navier–Stokes

This paper is about Lions’ open problem on density patches (Lions in Mathematical topics in fluid mechanics. Vol. 1, volume 3 of Oxford Lecture series in mathematics and its applications, Clarendon Press, Oxford University Press, New York, 1996): whether or not inhomogeneous incompressible Navier–Stokes equations preserve the initial regularity of the free boundary given by density patches. Using classical Sobolev spaces for the velocity, we first establish the propagation of $${C^{1+\gamma}}$$ regularity with $${0 < \gamma < 1}$$ in the case of positive density. Furthermore, we go beyond this to show the persistence of a geometrical quantity such as the curvature. In addition, we obtain a proof for $${C^{2+\gamma}}$$ regularity.

On the well-posedness of 3-D inhomogeneous incompressible Navier–Stokes equations with variable viscosity

In this paper, we mainly study the well-posedness for the 3-D inhomogeneous incompressible Navier–Stokes equations with variable viscosity. With some smallness assumption on the BMO-norm of the initial density, we first get the local well-posedness of (1.1) in the critical Besov spaces. Moreover, if the viscosity coefficient is a constant, we can extend this local solution to be a global one. Our theorem implies that we have successfully extended the integrability index p of the initial velocity which has been obtained by Abidi, Gui and Zhang in [3], Burtea in [8] and Zhai and Yin in [32] to approach the ideal one i.e. 1<p<6. The main novelty of this work is to apply the CRW theorem obtained by Coifman, Rochberg, Weiss in [11] to get a new a priori estimate for an elliptic equation with variable coefficients. The uniqueness of the solution also relies on a Lagrangian approach as in [16–18].

On the decay and stability of global solutions to the 3D inhomogeneous MHD system

In this paper, we investigative the large time decay and stability to any given global smooth solutions of the $3$D incompressible inhomogeneous MHD systems. We proved that given a solution $(a, u, B)$ of (\ref{mhd_a}), the velocity field and magnetic field decay to $0$ with an explicit rate, for $u$ which coincide with incompressible inhomogeneous Navier-Stokes equations \cite{zhangping}. In particular, we give the decay rate of higher order derivatives of $u$ and $B$ which is useful to prove our main stability result. For a large solutions of (\ref{mhd_a}) denoted by $(a, u, B)$, we proved that a small perturbation to the initial data still generates a unique global smooth solution and the smooth solution keeps close to the reference solution $(a, u, B)$. Due to the coupling between $u$ and $B$, we used elliptic estimates to get $\|(u, B)\|_{L^{1}(\mathbb{R}^{+};\dot{B}_{2,1}^{5/2})} < C$, which is different to Navier-Stokes equations.

Global well-posedness for the 3D incompressible inhomogeneous Navier–Stokes equations and MHD equations

The present paper is dedicated to the global well-posedness for the 3D inhomogeneous incompressible Navier–Stokes equations, in critical Besov spaces without smallness assumption on the variation of the density. We aim at extending the work by Abidi, Gui and Zhang (2012) [2], and (2013) [3] to a lower regularity index about the initial velocity. The key to that improvement is a new a priori estimate for an elliptic equation with nonconstant coefficients in Besov spaces which have the same degree as L2 in R3. Finally, we also generalize our well-posedness result to the inhomogeneous incompressible MHD equations.

Global Solution to the Incompressible Inhomogeneous Navier–Stokes Equations with Some Large Initial Data

In this paper, we prove that the incompressible inhomogeneous Navier–Stokes equations have a unique global solution with initial data \({(a_0,u_0)}\) in critical Besov spaces \({\dot{B}_{q,1}^{\frac{n}{q}}(\mathbb{R}^{n})\times\dot{B}_{p,1}^{\frac{n}{p}-1}(\mathbb{R}^{n})}\) satisfying a nonlinear smallness condition for all \({(p,q)\in[1,2n)\times[1,\infty)}\), \({-\frac1n\leq\frac1p-\frac1q\leq\frac1n}\) and \({\frac1p+\frac1q > \frac1n}\). We also construct an initial data satisfying that nonlinear smallness condition, but the norm of each component of the initial velocity field can be arbitrarily large in \({\dot{B}_{p,1}^{\frac{n}{p}-1}(\mathbb{R}^{n})}\) with \({n < p < 2n}\).

The energy balance relation for weak solutions of the density-dependent Navier–Stokes equations

We consider the incompressible inhomogeneous Navier–Stokes equations with constant viscosity coefficient and density which is bounded and bounded away from zero. We show that the energy balance relation for this system holds for weak solutions if the velocity, density, and pressure belong to a range of Besov spaces of smoothness 1/3. A density-dependent version of the classical Kármán–Howarth–Monin relation is derived.

On the strong solutions of the inhomogeneous incompressible Navier-Stokes equations in a thin domain

In this work, we will show the global existence of the strong solutions of the inhomogeneous incompressible Navier-Stokes equations in a thin three dimensional domain $\Omega=\mathbb R^2\times [0,\epsilon]$, with Dirichlet boundary condition on the top and bottom boundary: the global well-posedness may hold for large initial data when the vertical size $\epsilon$ is sufficiently small. Furthermore, when $\epsilon\rightarrow 0$ the velocity tends to vanish away from the initial time. The analysis relies on the a priori $H^2$-estimate for the solutions (similar as in [4, 5, 21]) and one pays attention to the dependence on the vertical size $\epsilon$.

Global well-posedness to the 3-D incompressible inhomogeneous Navier-Stokes equations with a class of large velocity

In this article, we consider the global well-posedness to the 3-D incompressible inhomogeneous Navier-Stokes equations with a class of large velocity. More precisely, assuming a0∈Ḃq,13q(R3) and u0=(u0h,u03)∈Ḃp,1−1+3p(R3) for p, q ∈ (1, 6) with sup(1p,1q)≤13+inf(1p,1q), we prove that if Ca0Ḃq,13qα(u03Ḃp,1−1+3p/μ+1)≤1, Cμ(u0hḂp,1−1+3p+u03Ḃp,1−1+3p1−αu0hḂp,1−1+3pα)≤1, then the system has a unique global solution a∈C˜([0,∞);Ḃq,13q(R3)), u∈C˜([0,∞);Ḃp,1−1+3p(R3))∩L1(R+;Ḃp,11+3p(R3)). It improves the recent result of M. Paicu and P. Zhang [J. Funct. Anal. 262, 3556–3584 (2012)], where the exponent form of the initial smallness condition is replaced by a polynomial form.

A Note on Blow-up Criterion of Strong Solutions for the 3D Inhomogeneous Incompressible Navier-Stokes Equations with Vacuum

In this paper, we study the three-dimensional inhomogeneous incompressible Navier-Stokes equations, and establish several regularity criteria in terms of only velocity which allow the initial density to contain vacuum. Therefore, our results can be considered as further improvement to the previous results.

Decay estimates of global solution to 2D incompressible Navier-Stokes equations with variable viscosity

In this paper, we investigate the time decay behavior to weak solution of 2D incompressible inhomogeneous Navier-Stokes equations. Granted the decay estimates, we gain a global well-posed result of these solutions.

On some large global solutions to 3-D density-dependent Navier–Stokes system with slow variable: Well-prepared data

In this paper, we consider the global wellposedness of 3-D incompressible inhomogeneous Navier–Stokes equations with initial data slowly varying in the vertical variable, that is, initial data of the form (1+εσa0(xh,εx3),(εu0h(xh,εx3),u03(xh,εx3))) for some σ>0 and ε being sufficiently small. We remark that initial data of this type does not satisfy the smallness conditions in [11,18] no matter how small ε is.