Stokes System Research Articles

On the Neumann problem for the nonstationary Stokes system in angles and cones

AbstractThe authors consider the Neumann problem for the nonstationary Stokes system in a two‐dimensional angle or a three‐dimensional cone. They obtain existence and uniqueness results for solutions in weighted Sobolev spaces and prove a regularity assertion for the solutions.

Robustness of 3D Navier–Stokes System with Increasing Damping

The principal objective of the paper is the study of the three-dimensional Navier–Stokes system with non-autonomous perturbation force term and increasing damping term, which often appears in the fluid system within saturated porous media and other complex media. With some suitable assumptions on the system parameters and external force term, based on the known result on global well-posedness, the existence of pullback attractors is educed, and the system robustness is shown via the upper semicontinuity of system attractors as the perturbation parameter approaches a certain value.

Convergence and Error Estimates of a Mixed Discontinuous Galerkin-Finite Element Method for the Semi-stationary Compressible Stokes System

Convergence and Error Estimates of a Mixed Discontinuous Galerkin-Finite Element Method for the Semi-stationary Compressible Stokes System

Singular Limit for the Compressible Navier–Stokes Equations with the Hard Sphere Pressure Law on Expanding Domains

The article is devoted to the asymptotic limit of the compressible Navier–Stokes system with a pressure obeying a hard–sphere equation of state on a domain expanding to the whole physical space $$\textbf{R}^3$$ . Under the assumptions that acoustic waves generated in the case of ill-prepared data do not reach the boundary of the expanding domain in the given time interval and a certain relation between the Reynolds and Mach numbers and the radius of the expanding domain we prove that the target system is the incompressible Euler system on $$\textbf{R}^3$$ . We also provide an estimate of the rate of convergence expressed in terms of characteristic numbers and the radius of domains.

Optimal control of the Navier— Stokes system with a space variable in a network-like domain

The research of the problem of optimal control of the Navier—Stokes evolutionary differential system, considered in Sobolev spaces, the elements of which are functions with carriers in an n-dimensional network-like domain, is presented. Such domain consists of a finite number of subdomains, mutually adjacent to certain parts of the surfaces of their boundaries according to the graph type. For functions that are elements of these spaces, the conditions for the existence of traces on the surfaces of the joining are presented and the conditions of adjacency subdomains to which these functions satisfy are described. In applied questions of the analysis of the processes of transport of continuous media, the conditions of adjacency describe the regularities of the flow of fluid through the boundaries of the adjacent domains. The paper presents the results of following main research questions: 1) weak solvability of the initial boundary value problem for the Navier—Stokes system and obtaining the conditions for the existence of a weak solution to this problem; 2) the formation and solution of optimal control problems of various types of Navier—Stokes system. The fundamental approach to the analysis of the weak solvability of the initial boundary value problem is its reduction to the differential-difference problem (semi-digitization of the original system by a time variable) and subsequent use of a priori estimates for weak solutions of the obtained boundary value problems. The obtained a priori estimates are used to prove the theorem of the existence of a weak solution of the original differential system and indicate the way of the actual construction of this solution. A universal approach to solving the problems of optimal distributed and starting control of the Navier—Stokes evolutionary system is presented. The latter essentially expands the possibilities of analyzing non-stationary network-like processes of applied hydrodynamics (for example, processes of transporting various types of liquids through network or main line pipelines) and optimal control of these processes.

Geometric singularities and regularity of solution of the Stokes system in nonsmooth domains

This paper deals with the geometrical singularities of the weak solution of the mixed boundary value problem governed by the stationary Stokes system in two-dimensional nonsmooth domains with corner points and points at which the type of boundary conditions changes. The presence 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. Moreover, the asymptotic singular representations for the solution which inherently depend on the zeros of certain transcendental functions are presented.

Controllability results for cascade systems of m coupled N-dimensional stokes and Navier-stokes systems by N – 1 scalar controls

In this paper we deal with the controllability properties of a system of m coupled Stokes systems or m coupled Navier-Stokes systems. We show the null-controllability of such systems in the case where the coupling is in a cascade form and when the control acts only on one of the systems. Moreover, we impose that this control has a vanishing component so that we control a m × N state (corresponding to the velocities of the fluids) by N — 1 distributed scalar controls. The proof of the controllability of the coupled Stokes systems is based on a Carleman estimate for the adjoint system. The local null-controllability of the coupled Navier-Stokes systems is then obtained by means of the source term method and a Banach fixed point.

The method of penalty functions in the analysis of optimal control problems of Navier — Stokes evolutionary systems with a spatial variable in a network-like domain

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.

The global generalized solution of the chemotaxis-Navier–Stokes system with logistic source

The global generalized solution of the chemotaxis-Navier–Stokes system with logistic source

Low-Mach type approximation of the Navier–Stokes system with temperature and salinity for free surface flows

We are interested in free surface flows where density variations coming e.g. from temperature or salinity differences play a significant role in the hydrody-namic regime. In water, acoustic waves travel much faster than gravity and internal waves, hence the study of models arising from compressible fluid mechanics often requires a decoupling between these waves. Starting from the compressible Navier-Stokes system, we derive the so-called Navier-Stokes-Fourier system in an incompressible regime using the low-Mach scaling, hence filtering the acoustic waves, neglecting the density dependency on the fluid pressure but keeping its variations in terms of temperature and salinity. A slightly modified low-Mach asymptotics is proposed to obtain a model with thermo-mechanical compatibility. The case when the density depends only on the temperature is studied first. Then the variations of the fluid density with respect to temperature and salinity are considered, and it seems to be the first time that salinity dependency is considered in this low Mach limit. We give a layer-averaged formulation of the obtained models in an hydrostatic context, allowing to derive numerical schemes endowed with strong stability properties that are presented in a companion paper. Several stability properties of the layer-averaged Navier-Stokes-Fourier system are proved.

Gradient estimates for the non-stationary Stokes system with the Navier boundary condition

Gradient estimates for the non-stationary Stokes system with the Navier boundary condition

Global strong solutions to the compressible Navier–Stokes system with potential temperature transport

Global strong solutions to the compressible Navier–Stokes system with potential temperature transport

Consistent splitting schemes for incompressible viscoelastic flow problems

AbstractViscoelastic fluids are highly challenging from the rheological standpoint, and their discretization demands robust, efficient numerical solvers. Simulating viscoelastic flows requires combining the Navier–Stokes system with a dynamic tensorial equation, increasing mathematical and computational demands. Hence, fractional‐step methods decoupling the calculation of the flow quantities are an attractive option. In consistent fractional‐step schemes, the splitting of the equations is derived from the continuous level, so that neither mass nor momentum balance is sacrificed. Thus, no corrections or velocity projections are needed, resulting in fewer algorithmic steps than other classical approaches. Moreover, consistency guarantees the absence of both numerical boundary layers and splitting errors, enabling high‐order accuracy in space and time. This article introduces the first consistent splitting methods for incompressible flows of viscoelastic fluids, for which arbitrary constitutive laws are allowed. We present the formulation and the algorithm, along with various numerical examples testing their accuracy. First‐, second‐ and third‐order backward‐differentiation schemes are numerically tested, and optimal convergence is confirmed for several spatial and temporal discretizations. Furthermore, good numerical stability is verified in challenging benchmark problems, including steady and time‐dependent solutions.

Optimal control of thermal and wave processes in composite materials

The paper indicates the approach and the corresponding to it method of penalty functions for analyzing the problems of optimal control of thermal and wave processes in structural elements made of composite materials (composites). An object that is quite common in the industrial sphere, the structure of which is a set of layers (phases) of unidirectional composites — layered composites, is considered. When solving problems related to the analysis and description of the states of composites, quantitative characteristics of layers that are not functions of the coordinates of the points of the medium are usually used in order not to solve the corresponding problems for an inhomogeneous medium. Such functions are elements of Sobolev spaces, first of all, functions summable with a square. The convenience lies in the fact that when finding the conditions for solvability of initial-boundary value problems of various types (in most cases, such problems are the basis of mathematical models of many physical processes), it is possible to reduce to operator-difference systems, for which it is easy to construct a priori estimates of weak solutions. The next step after establishing the weak solvability of the initial-boundary value problem of the thermal or wave process in composites is the formulation and solution of the problem of optimal control of these processes. The proposed method of penalty functions on the example of solving such problems is a general method. It is applicable with slight modifications also not only in the case of elliptic, parabolic and other problems (including nonlinear) for scalar functions, but also for vector functions. An example of the latter is the Navier–Stokes system, widely used in the description of network-like hydrodynamic processes, considered in Sobolev spaces, the elements of which are functions with carriers on n-dimensional network-like domains, n greater or equal to 2.

Global well-posedness of perturbed Navier–Stokes system around Landau solutions

In this paper, we consider the perturbed Navier–Stokes equations around the Landau solution and investigate the global well-posedness results of the perturbed system with the small initial data in the X−1 space, where X−1=f∈D′R3:∫R3|ξ|−1|f̂|dξ<∞.

Regularity results in 2D fluid–structure interaction

We study the interaction of an incompressible fluid in two dimensions with an elastic structure yielding the moving boundary of the physical domain. The displacement of the structure is described by a linear viscoelastic beam equation. Our main result is the existence of a unique global strong solution. Previously, only the ideal case of a flat reference geometry was considered such that the structure can only move in vertical direction. We allow for a general geometric set-up, where the structure can even occupy the complete boundary. Our main tool—being of independent interest—is a maximal regularity estimate for the steady Stokes system in domains with minimal boundary regularity. In particular, we can control the velocity field in W2,2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$W^{2,2}$$\\end{document} in terms of a forcing in L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2$$\\end{document} provided the boundary belongs roughly to W3/2,2.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$W^{3/2,2}.$$\\end{document} This is applied to the momentum equation in the moving domain (for a fixed time) with the material derivative as right-hand side. Since the moving boundary belongs a priori only to the class W2,2,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$W^{2,2},$$\\end{document} known results do not apply here as they require a C2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^2$$\\end{document}-boundary.

Asymptotic Behavior of the Solution to Compressible Navier–Stokes System with Temperature-Dependent Heat Conductivity in an Unbounded Domain

This paper concerns the one-dimensional compressible Navier–Stokes system with temperature-dependent heat conductivity in R with large initial data. We prove that velocity and temperature are uniformly bounded from below and above in time and space when the heat conductivity coefficient takes κ=κ¯(1+θb) for all b>52. In addition, we show that the global solution is asymptotically stable as time tends to infinity.

Fundamental solutions of the Stokes system in quaternion analysis

UDC 532.5 The method of quaternionic analysis in fluid mechanics was developed by several generations of mathematicians with numerous important results. We add a small result in this direction. Thus, we introduce a new reformulation of fundamental solutions of the Stokes system within the framework of quaternion analysis and construct integral representations for its solutions.

Compressible Fluids Interacting with Plates: Regularity and Weak-Strong Uniqueness

In this paper, we study a nonlinear interaction problem between compressible viscous fluids and plates. For this problem, we introduce relative entropy and relative energy inequality for the finite energy weak solutions (FEWS). First, we prove that for all FEWS, the structure displacement enjoys improved regularity by utilizing the dissipation effects of the fluid onto the structure and that all FEWS satisfy the relative energy inequality. Then, we show that all FEWS enjoy the weak-strong uniqueness property, thus extending the classical result for compressible Navier–Stokes system to this fluid-structure interaction problem.

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.