Global Kato’s Solutions for Inhomogeneous Navier–Stokes System in \({L^{3}(\mathbb{R}^{3})}\)

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 a0≜1ρ0−1∈Bp,13p(R3) and the initial velocity u0=(u0h,u03)∈Bp,1−1+2p,1p(R3), with u0h belonging to H1(R3) , then the inhomogeneous NS system has a unique global solution provided that (∥a0∥Bp,13p+∥Λh−1∂3u0∥Bp,1−1+2p,1p)⋅f(u0) being sufficiently small for some bounded function f depending on ∥u0∥p,1−1+2p,1p and ∥u0h∥H1 . 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).

This study typically emphasizes analyzing the geometrical singularities of weak solutions of the mixed boundary value problem for the stationary Stokes and Navier-Stokes system in two-dimensional non-smooth domains with corner points and points at which the type of boundary conditions change. The existence of these points on the boundary generally generates local singularities in the solution. We will see the impact of the geometrical singularities of the boundary or the mixed boundary conditions on the qualitative properties of the solution including its regularity. The solvability of the underlying boundary value problem is analyzed in weighted Sobolev spaces and the regularity theorems are formulated in the context of these spaces. To compute the singular terms for various boundary conditions, the generalized form of the boundary eigenvalue problem for the stationary Stokes system is derived. The emerging eigenvalues and eigenfunctions produce singular terms, which permits us to evaluate the optimal regularity of the corresponding weak solution of the Stokes system. Additionally, the obtained results for the Stokes system are further extended for the non-linear Navier-Stokes system.

On some mixed-transmission problems for the anisotropic Stokes and Navier-Stokes systems in Lipschitz domains with transversal interfaces

The purpose of this paper is to show well-posedness results for Dirichlet problems for the Stokes and Navier–Stokes systems with $$L^{\infty }$$ -variable coefficients in $$L^2$$ -based Sobolev spaces in Lipschitz domains on compact Riemannian manifolds. First, we refer to the Dirichlet problem for the nonsmooth coefficient Stokes system on Lipschitz domains in compact Riemannian manifolds and show its well-posedness by employing a variational approach that reduces the boundary value problem of Dirichlet type to a variational problem defined in terms of two bilinear continuous forms, one of them satisfying a coercivity condition and another one the inf-sup condition. We show also the equivalence between some transmission problems for the nonsmooth coefficient Stokes system in complementary Lipschitz domains on compact Riemannian manifolds and their mixed variational counterparts, and then their well-posedness in $$L^2$$ -based Sobolev spaces by using the remarkable Necas–Babuska–Brezzi technique (see Babuska in Numer Math 20:179–192, 1973; Brezzi in RAIRO Anal Numer R2:129–151, 1974; Necas in Rev Roum Math Pures Appl 9:47–69, 1964). As a consequence of these well-posedness results we define the layer potential operators for the nonsmooth coefficient Stokes system on Lipschitz surfaces in compact Riemannian manifolds, and provide their main mapping properties. These properties are used to construct explicitly the solution of the Dirichlet problem for the Stokes system. Further, we combine the well-posedness of the Dirichlet problem for the nonsmooth coefficient Stokes system with a fixed point theorem to show the existence of a weak solution to the Dirichlet problem for the nonsmooth variable coefficient Navier–Stokes system in $$L^2$$ -based Sobolev spaces in Lipschitz domains on compact Riemannian manifolds. The well developed potential theory for the smooth coefficient Stokes system on compact Riemannian manifolds (cf. Dindos and Mitrea in Arch Ration Mech Anal 174:1–47, 2004; Mitrea and Taylor in Math Ann 321:955–987, 2001) is also discussed in the context of the potential theory developed in this paper.

ABSTRACTWe obtain well-posedness results in -based weighted Sobolev spaces for a transmission problem for anisotropic Stokes and Navier–Stokes systems with strongly elliptic coefficient tensor, in complementary Lipschitz domains of , . The strong ellipticity allows to explore the associated pseudostress setting. First, we use a variational approach that reduces the anisotropic Stokes system in the whole to an equivalent mixed variational formulation with data in -based weighted Sobolev spaces. We show that such mixed variational formulation is well-posed in the space , , for any p in an open interval containing 2. Then similar well-posedness results are obtained for two linear transmission problems. These results are used to define the Newtonian and layer potential operators for the considered anisotropic Stokes system and to obtain mapping properties of these operators. The potentials are employed to show the well-posedness of some linear transmission problems, which then is combined with a fixed point theorem in order to show the well-posedness of a nonlinear transmission problem for the anisotropic Stokes and Navier–Stokes systems in -based weighted Sobolev spaces, whenever the given data are small enough.

This paper is build around the stationary anisotropic Stokes and Navier-Stokes systems with an L^infty -tensor coefficient satisfying an ellipticity condition in terms of symmetric matrices in {mathbb {R}}^{ntimes n} with zero matrix traces. We analyze, in L^2-based Sobolev spaces, the non-homogeneous boundary value problems of Dirichlet-transmission type for the anisotropic Stokes and Navier-Stokes systems in a compressible framework in a bounded Lipschitz domain with a transversal Lipschitz interface in {mathbb {R}}^n, nge 2 (n=2,3 for the nonlinear problems). Thus, the interface intersects transversally the boundary of the Lipschitz domain and divides the domain into two Lipschitz sub-domains. First, we use a mixed variational approach to prove the well-posedness of linear problems related to the anisotropic Stokes system. Then we show the existence of a weak solution to the Dirichlet and Dirichlet-transmission problems for the nonlinear anisotropic Navier-Stokes system. This is done by implementing the Leray-Schauder fixed point theorem and using various results and estimates from the linear case, as well as the Leray-Hopf and some other norm inequalities. Explicit conditions for uniqueness of solutions to the nonlinear problems are also provided.

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

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

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

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

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.

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 .

The article considers the Navier — Stokes evolutionary differential system used in the mathematical description of the evolutionary processes of transportation of various types of liquids through network or main pipelines. The Navier—Stokes system is considered in Sobolev spaces, the elements of which are functions with carriers on n-dimensional networklike domains. These domains are a totality of a finite number of mutually non-intersecting subdomains connected to each other by parts of the surfaces of their boundaries like a graph (in applications these are the places of branching of pipelines). Two main questions of analysis are discussed: the weak solvability of the initial boundary value problem of the Navier — Stokes system and the optimal control of this system. The main method of research of weak solutions is the semidigitization of the input system by a time variable, that is the reduction of a differential system to a differential-difference system, and using a priori estimates for weak solutions of boundary value problems to prove the theorem of the existence of a solution of the input differential system. For the optimal control problem a minimizing functional (the penalty function) and a family of the approximate functional with parameters that characterize the “penalty” for failure to fulfill the equations of state of the system are introduced. At the same time, a special Hilbert space is created, the elements of which are pairs of functions that describe the state of the system and controlling actions. The convergence of the sequence of such functions to the optimal state of the system and its corresponding optimal control is proved. The latter essentially widen the possibilities of analysis of stationary and nonstationary network-like processes of hydrodynamics and optimal control of these processesd.

Consider a smooth bounded domain \Omega\subseteq\mathbb R^3 with boundary \partial\Omega , a time interval [0,T) , with T\in(0,\infty] , and the Navier–Stokes system in [0,T) \times \Omega , with initial value u_0 \in L^2_{\sigma} (\Omega) and external force f= {\mathrm{div}}\,F , F \in L^2 (0,T;L^2(\Omega)) . Our aim is to extend the well-known class of Leray-Hopf weak solutions u satisfying u_{\vert{\partial \Omega}}=0 , {\mathrm{div}}\,u=0 to the more general class of Leray-Hopf type weak solutions u with general data u_{\vert{\partial \Omega}} =g , {\mathrm{div}}\,u=k satisfying a certain energy inequality. Our method rests on a perturbation argument writing u in the form u=v+E with some vector field E in [0,T)\times \Omega satisfying the (linear) Stokes system with f=0 and nonhomogeneous data. This reduces the general system to a perturbed Navier–Stokes system with homogeneous data, containing an additional perturbation term. Using arguments as for the usual Navier–Stokes system we get the existence of global weak solutions for the more general system.

We study an asymptotic analysis of a coupled system of kinetic and fluid equations. More precisely, we deal with the nonlinear Vlasov–Fokker–Planck equation coupled with the compressible isentropic Navier–Stokes system through a drag force in a bounded domain with the specular reflection boundary condition for the kinetic equation and homogeneous Dirichlet boundary condition for the fluid system. We establish a rigorous hydrodynamic limit corresponding to strong noise and local alignment force. The limiting system is a type of two-phase fluid model consisting of the isothermal Euler system and the compressible Navier–Stokes system. Our main strategy relies on the relative entropy argument based on the weak–strong uniqueness principle. For this, we provide a global-in-time existence of weak solutions for the coupled kinetic-fluid system. We also show the existence and uniqueness of strong solutions to the limiting system in a bounded domain with the kinematic boundary condition for the Euler system and Dirichlet boundary condition for the Navier–Stokes system.