Incompressible Navier Stokes System Research Articles

Incompressible Navier–Stokes limit from nonlinear Vlasov–Fokker–Planck equation

The aim of this paper is to justify the rigorous derivation of the incompressible Navier–Stokes equations from the nonlinear Vlasov–Fokker–Planck (VFP) equation with a constant temperature. Under the incompressible Navier–Stokes scaling, we first establish the global existence of regular solutions to the rescaled nonlinear VFP equation near the Maxwellian, obtaining some uniform bound estimates. We then show the strong convergence of solution to the nonlinear VFP equation towards the incompressible Navier–Stokes system.

The forward self-similar solution of fractional incompressible Navier-Stokes system: The critical case

In this paper, we study the regularity and pointwise estimates of forward self-similar solutions of fractional Navier-Stokes system under the critical case. By employing a Caffarelli,Kohn and Nirenberg-type iteration, L∞ estimates of the self-similar solution's profile are established, which is a key ingredient to ensure that the global weighted energy estimate procedure used in [28] is performed under the critical case. As a product, its natural pointwise bounds are recovered. Moreover, to obtain the optimal spatial decay estimate of self-similar solution's profile, a new technique is required due to lack of the related regularity theory.

On the Structure of Axisymmetric Helical Solutions to the Incompressible Navier–Stokes System

On the Structure of Axisymmetric Helical Solutions to the Incompressible Navier–Stokes System

Approximate numerical procedures for the Navier–Stokes system through the generalized method of lines

Abstract This article develops approximate numerical solutions through the generalized method of lines for the time-independent, incompressible Navier–Stokes system in fluid mechanics. More specifically, we highlight the main objective of this article is the development of new approximate procedures for solving numerically the equation systems originated from a domain discretization related to a finite differences scheme. We recall that for such a method, the domain of the partial differential equation in question is discretized in lines (or more generally in curves), and the concerning solutions are written on these lines as functions of the boundary conditions and the domain boundary shape. Finally, it is worth emphasizing that in this text, we have presented softwares and results for a concerning approximate proximal approach, as well as results based on the original conception of the generalized method of lines.

Existence of global weak solutions of inhomogeneous incompressible Navier–Stokes system with mass diffusion

Existence of global weak solutions of inhomogeneous incompressible Navier–Stokes system with mass diffusion

Stability of Time-Dependent Motions for Fluid–Rigid Ball Interaction

We aim at the stability of time-dependent motions, such as time-periodic ones, of a rigid body in a viscous fluid filling the exterior to it in 3D. The fluid motion obeys the incompressible Navier–Stokes system, whereas the motion of the body is governed by the balance for linear and angular momentum. Both motions are affected by each other at the boundary. Assuming that the rigid body is a ball, we adopt a monolithic approach to deduce Lq\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^q$$\\end{document}–Lr\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^r$$\\end{document} decay estimates of solutions to a non-autonomous linearized system. We then apply those estimates to the full nonlinear initial value problem to find temporal decay properties of the disturbance. Although the shape of the body is not allowed to be arbitrary, the present contribution is the first attempt at analysis of the large time behavior of solutions around nontrivial basic states, that can be time-dependent, for the fluid–structure interaction problem and provides us with a stability theorem which is indeed new even for steady motions under the self-propelling condition or with wake structure.

Energy conservation for weak solutions to the pressureless Euler–Navier–Stokes system in three‐dimensional space

We study the energy conservation for the weak solution of the 3D Euler–Navier–Stokes system. we provide two sufficient conditions on the regularity of weak solutions to guarantee the conservation of total energy, which also including the initial energy. Due to the couple structure between inhomogeneous Euler equation and incompressible Navier–Stokes system, we adopt a variant of the method from R. Chen and C. Yu (J. Math. Pures Appl. 131 (2019), 1–16.) and J. Lions (Rend. Semin. Mat. Univ. Padova 30 (1960), 16–23.) to deal with the density of Euler flows and velocity of Navier–Stokes fluid.

Γ–convergence for nearly incompressible fluids

We consider the time-dependent compressible Navier–Stokes equations in the low Mach number regime in a family of domains Ωɛ ⊂ Rd converging in the sense of Mosco to a domain Ω ⊂ Rd, d ∈ {2, 3}. We show the limit is the incompressible Navier–Stokes system in Ω.

A posteriori error estimates for the Large Eddy Simulation applied to incompressible fluids

We study the two dimensional time dependent Large Eddy Simulation method applied to the incompressible Navier–Stokes system with Smagorinsky’s eddy viscosity model and a filter width that depends on the local mesh size. The discrete model is based on the implicit Euler scheme and a conforming finite element method for the time and space discretizations, respectively. We establish a reliable and efficient a posteriori error estimation between the numerical LES solution and the exact solution of the original Navier–Stokes system, which involves three types of error indicators respectively related to the filter and to the discretizations in time and space. Numerical results show the effectiveness of adaptive simulations

Traveling wave solutions to the inclined or periodic free boundary incompressible Navier-Stokes equations

This paper concerns the construction of traveling wave solutions to the free boundary incompressible Navier-Stokes system. We study a single layer of viscous fluid in a strip-like domain that is bounded below by a flat rigid surface and above by a moving surface. The fluid is acted upon by a bulk force and a surface stress that are stationary in a coordinate system moving parallel to the fluid bottom. We also assume that the fluid is subject to a uniform gravitational force that can be resolved into a sum of a vertical component and a component lying in the direction of the traveling wave velocity. This configuration arises, for instance, in the modeling of fluid flow down an inclined plane. We also study the effect of periodicity by allowing the fluid cross section to be periodic in various directions. The horizontal component of the gravitational field gives rise to stationary solutions that are pure shear flows, and we construct our solutions as perturbations of these by means of an implicit function argument. An essential component of our analysis is the development of some new functional analytic properties of a scale of anisotropic Sobolev spaces, including that these spaces are an algebra in the supercritical regime, which may be of independent interest.

On a Class of Exact Solutions to the Incompressible Navier–Stokes System in a Ball and a Spherical Layer

On a Class of Exact Solutions to the Incompressible Navier–Stokes System in a Ball and a Spherical Layer

On a Class of Exact Solutions to the Incompressible Navier–Stokes System in a Ball and a Spherical Layer

A class of exact solutions to the Navier–Stokes equations for rotational incompressible flows is obtained. A three-parameter family of solutions in a ball, spherical layers, and the entire spaceis constructed.

Incompressible limit for the free surface Navier-Stokes system

We establish uniform regularity estimates with respect to the Mach number for the three-dimensional free surface compressible Navier-Stokes system in the case of slightly well-prepared initial data in the sense that the acoustic components like the divergence of the velocity field are of size $$\sqrt{\varepsilon }$$ , $$\varepsilon $$ being the Mach number. These estimates allow us to justify the convergence towards the free surface incompressible Navier-Stokes system in the low Mach number limit. One of the main difficulties is the control of the regularity of the surface in presence of boundary layers with fast oscillations.

Stability of the density patches problem with vacuum for incompressible inhomogeneous viscous flows

We consider the inhomogeneous incompressible Navier–Stokes system in a smooth twoor three-dimensional bounded domain, in the case where the initial density is only bounded. Existence and uniqueness for such initial data was shown recently in Danchin and Mucha [Comm. Pure Appl. Math. 72 (2019)], but the stability issue was left open. After having shown that the solutions constructed therein have exponential decay, a result of independent interest, we prove the stability with respect to initial data, first in Lagrangian coordinates, and then in the Eulerian frame. We actually obtain stability in the energy space for the velocity and in a Sobolev space with negative regularity for the density. Let us underline that, as opposed to prior works, our stability estimates are valid even in the case of a vacuum. In particular, our result applies to the classical density patches problem, where the density is a characteristic function.

Sharper dispersive estimates and asymptotics for a Boussinesq-type system with larger ill-prepared initial data

The aim of this article is to extend previous works about the asymptotics of an ill-prepared fast rotating, highly stratified incompressible Navier–Stokes system. Thanks to improved Strichartz and a priori estimates, we are able not only to cover a case which was unanswered in our previous work (allowing bigger ill-prepared initial data) but also to improve the convergence rates and reduce some assumptions on the initial data. In passing we also widen the range of some parameters and compare two methods to obtain dispersive estimates.

A new class of fractional differential hemivariational inequalities with application to an incompressible Navier-Stokes system coupled with a fractional diffusion equation

This paper is devoted to the study of a new and complicated dynamical system, called a fractional differential hemivariational inequality, which consists of a quasilinear evolution equation involving the fractional Caputo derivative operator and a coupled generalized parabolic hemivariational inequality. Under certain general assumptions, existence and regularity of a mild solution to the dynamical system are established by employing a surjectivity result for weakly-weakly upper semicontinuous multivalued mappings, and a feedback iterative technique together with a temporally semi-discrete approach through the backward Euler difference scheme with quasi-uniform time-steps. To illustrate the applicability of the abstract results, we consider a nonstationary and incompressible Navier-Stokes system supplemented by a fractional reaction-diffusion equation, which is studied as a fractional hemivariational inequality. Bibliography: 57 titles.

A new class of fractional differential hemivariational inequalities with application to an incompressible Navier-Stokes system coupled with a fractional diffusion equation

This paper is devoted to the study of a new and complicated dynamical system, called a fractional differential hemivariational inequality, which consists of a quasilinear evolution equation involving the fractional Caputo derivative operator and a coupled generalized parabolic hemivariational inequality. Under certain general assumptions, existence and regularity of a mild solution to the dynamical system are established by employing a surjectivity result for weakly-weakly upper semicontinuous multivalued mappings, and a feedback iterative technique together with a temporally semi-discrete approach through the backward Euler difference scheme with quasi-uniform time-steps. To illustrate the applicability of the abstract results, we consider a nonstationary and incompressible Navier-Stokes system supplemented by a fractional reaction-diffusion equation, which is studied as a fractional hemivariational inequality.

Global Unique Solutions for the Inhomogeneous Navier-Stokes equations with only Bounded Density, in Critical Regularity Spaces

We here aim at proving the global existence and uniqueness of solutions to the inhomogeneous incompressible Navier-Stokes system in the case where the initial density \(\rho _0\) is discontinuous and the initial velocity \(u_0\) has critical regularity. Assuming that \(\rho _0\) is close to a positive constant, we obtain global existence and uniqueness in the two-dimensional case whenever the initial velocity \(u_0\) belongs to the critical homogeneous Besov space \(\dot{B}^{-1+2/p}_{p,1}({\mathbb {R}}^{2})\) \((1<p<2)\) and, in the three-dimensional case, if \(u_0\) is small in \(\dot{B}^{-1+3/p}_{p,1}({\mathbb {R}}^{3})\ (1<p<3)\). Next, still in a critical functional framework, we establish a uniqueness statement that is valid in the case of large variations of density with, possibly, vacuum. Interestingly, our result implies that the Fujita-Kato type solutions constructed by Zhang (Adv Math 363:107007, 2020) are unique. Our work relies on interpolation results, time weighted estimates and maximal regularity estimates in Lorentz spaces (with respect to the time variable) for the evolutionary Stokes system.

Asymptotic profiles and concentration–diffusion effects in fractional incompressible flows

In this paper, we study the behavior of solutions to the fractional dissipative incompressible Navier–Stokes equations at large distances in Rd,d=2,3. The solvability of the fractional Navier–Stokes system in weighted L∞ spaces and the asymptotic profile of solutions are investigated. Based on the asymptotic behavior of the local solution, the concentration–diffusion phenomenon is discussed by constructing suitable initial data for the fractional dissipative incompressible Navier–Stokes system.

Homogenization of Evolutionary Incompressible Navier–Stokes System in Perforated Domains

In this paper, we consider the homogenization problems for evolutionary incompressible Navier-Stokes system in three dimensional domains perforated with a large number of small holes which are periodically located. We first establish certain uniform estimates for the weak solutions. To overcome the extra difficulties coming from the time derivative, we use the idea of Temam and consider the equations by integrating in time variable. After suitably extending the weak solutions to the whole domain, we employ the generalized cell problem to study the limit process.