Smooth global solutions to the Navier-Stokes equations always exist
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CONTESTED
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refutedsupported
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While some peer-reviewed studies claim to establish global smooth solutions under specific frameworks or conditions, reference sources note that the general existence of smooth solutions remains an open Millennium Prize Problem.
This paper investigates the globally dynamical stabilizing effects of the geometry of the domain at which the flow locates and of the geometry structure of the solutions with the finite energy to the three-dimensional (3D) incompressible Navier–Stokes (NS) and Euler systems. The global well-posedness for large amplitude smooth solutions to the Cauchy problem for 3D incompressible NS and Euler equations based on a class of variant spherical coordinates is obtained, where smooth initial data is not axi-symmetric with respect to any coordinate axis in Cartesian coordinate system. Furthermore, we establish the existence, uniqueness and exponentially decay rate in time of the global strong solution to the initial boundary value problem for 3D incompressible NS equations for a class of the smooth large initial data and a class of the special bounded domain described by variant spherical coordinates.
The Navier-Stokes equation on the Euclidean space R 3 \mathbb {R}^3 can be expressed in the form ∂ t u = Δ u + B ( u , u ) \partial _t u = \Delta u + B(u,u) , where B B is a certain bilinear operator on divergence-free vector fields u u obeying the cancellation property ⟨ B ( u , u ) , u ⟩ = 0 \langle B(u,u), u\rangle =0 (which is equivalent to the energy identity for the Navier-Stokes equation). In this paper, we consider a modification ∂ t u = Δ u + B ~ ( u , u ) \partial _t u = \Delta u + \tilde B(u,u) of this equation, where B ~ \tilde B is an averaged version of the bilinear operator B B (where the average involves rotations, dilations, and Fourier multipliers of order zero), and which also obeys the cancellation condition ⟨ B ~ ( u , u ) , u ⟩ = 0 \langle \tilde B(u,u), u \rangle = 0 (so that it obeys the usual energy identity). By analyzing a system of ordinary differential equations related to (but more complicated than) a dyadic Navier-Stokes model of Katz and Pavlovic, we construct an example of a smooth solution to such an averaged Navier-Stokes equation which blows up in finite time. This demonstrates that any attempt to positively resolve the Navier-Stokes global regularity problem in three dimensions has to use a finer structure on the nonlinear portion B ( u , u ) B(u,u) of the equation than is provided by harmonic analysis estimates and the energy identity. We also propose a program for adapting these blowup results to the true Navier-Stokes equations.
This paper establishes a unified mathematical framework independent of strong regularity constraints on initial data and external forces, and rigorously proves the existence, uniqueness, and stability of global smooth solutions for the 3D incompressible Navier-Stokes equations. The framework covers three classes of initial data: $H^{s}$-bounded, purely $L^{2}$-bounded, and locally weakly singular, with external forces restricted only to $L^{2}([0, \infty) ; L^{2}(\mathbb{R}^{3}))$. The core innovation lies in the trinity framework of compactly supported mollifier regularization, uniform double limit energy estimates, and Galerkin iteration, which seamlessly adapts to both weakly regular practical scenarios and highly regular ideal scenarios without structural reconstruction. Key conclusions include: (1) Local weak singularities of initial data vanish instantaneously for $t>0$, and solutions are globally smooth in $C^{\infty}((0, \infty) ; H^{\infty}(\mathbb{R}^{3}))$; (2) High-regularity initial data and external forces yield solutions with arbitrary-order smoothness at $t=0$ and for all subsequent time, excluding finite-time blow-up; (3) Turbulent "apparent singularities" are interpreted as spatiotemporal high-frequency oscillations of smooth solutions, without relying on physical assumptions. This work fills the gap in weakly regular well-posedness theory and provides rigorous mathematical support for ideal scenario analysis.
In this paper, we investigate the global well-posedness of the equilibrium diffusion model in radiation hydrodynamics. The model consists of the compressible Navier-Stokes equations coupled with radiation effect terms described by the fourth power of temperature. The global existence of classical solutions to the Cauchy problem in the whole space is established when initial data is a small smooth perturbation of a constant equilibrium state: moreover, an algebraic rate of convergence of solutions toward equilibrium is obtained under additional conditions on initial data. The proof is based on the refined energy method and Fourier’s analysis.
We give a rather short and self-contained presentation of the global existence for Leray-Hopf weak solutions to the three dimensional incompressible Navier-Stokes equations, with constant density. We give a unified treatment in terms of the domains and the relative boundary conditions and in terms of the approximation methods. More precisely, we consider the case of the whole space, the flat torus, and the case of a general bounded domain with a smooth boundary (the latter supplemented with homogeneous Dirichlet conditions). We consider as approximation schemes the Leray approximation method, the Faedo-Galerkin method, the semi-discretization in time and the approximation by adding a Smagorinsky-Ladyžhenskaya term. We mainly focus on developing a unified treatment especially in the compactness argument needed to show that approximations converge to the weak solutions.
Global Behavior of Spherically Symmetric Navier-Stokes Equations with Density-Dependent Viscosity
In this paper, we study a free boundary problem for compressible spherically symmetric Navier-Stokes equations without a solid core. Under certain assumptions imposed on the initial data, we obtain the global existence and uniqueness of the weak solution, give some uniform bounds (with respect to time) of the solution and show that it converges to a stationary one as time tends to infinity. Moreover, we obtain the stabilization rate estimates of exponential type in $L^\infty$-norm and weighted $H^1$-norm of the solution by constructing some Lyapunov functionals. The results show that such system is stable under the small perturbations, and could be applied to the astrophysics.
Published as: J. Differential Equations, 236 (2007), no. 1, 293--341
DOI: 10.1016/j.jde.2007.01.025
arXiv categories: math.AP math-ph math.MP
Helical flows are flows which are covariant with respect to translation along helices. In recent work, B. Ettinger and E. Titi established global existence and uniqueness of helical flow solutions of the incompressible 3D Euler equations with bounded vorticity and no 'helical swirl' (the component of velocity along helices). In earlier work, A. Mahalov, E. Titi and S. Leibovich had established well-posedness, in H1, of viscous helical flows (solutions of the Navier-Stokes equations) with no restriction on helical swirl. Absence of helical swirl prevents vortex stretching, but, although a conse
The Navier-Stokes equation can be written in a form of Poisson equation. For laminar flow in a channel (plane Poiseuille flow), the Navier-Stokes equation has a non-zero source term (∇2u(x, y, z) = Fx (x, y, z, t) and a non-zero solution within the domain. For transitional flow, the velocity profile is distorted, and an inflection point or kink appears on the velocity profile, at a sufficiently high Reynolds number and large disturbance. In the vicinity of the inflection point or kink on the distorted velocity profile, we can always find a point where ∇2u(x, y, z) = 0. At this point, the Poisson equation is singular, due to the zero source term, and has no solution at this point due to singularity. It is concluded that there exists no smooth orphysically reasonable solutions of the Navier-Stokes equation for transitional flow and turbulence in the global domain due to singularity.
proven that smooth solutions always exist. This is called the Navier–Stokes existence and smoothness problem. The problem, restricted to the case of an
The Millennium Prize Problems are seven well-known complex mathematical problems selected by the Clay Mathematics Institute in 2000. The Clay Institute has pledged to pay one million US dollars for the first correct solution to each problem.
The Clay Mathematics Institute officially designated the title Millennium Problem for the seven unsolved mathematical problems, the Birch and Swinnerton-Dyer
The Navier–Stokes equations describe the motion of fluids, and are one of the pillars of fluid mechanics. However, theoretical understanding of their solutions is incomplete, despite its importance in science and engineering. For the three-dimensional system of equations, and given some initial conditions, mathematicians have not yet proven that smooth solutions always exist. This is called the Navier–Stokes existence and smoothness problem.
The problem, restricted to the case of an incompressible flow, is to prove either that smooth, globally defined solutions exist that meet certain conditions, or that they do not always exist and the equations break down. The official statement of the problem was given by Charles Fefferman.
Finite time blow up for a Navier-Stokes like equation
We consider an equation similar to the Navier-Stokes equation. We show that there is initial data that exists in every Triebel-Lizorkin or Besov space (and hence in every Lebesgue and Sobolev space), such that after a finite time, the solution is in no Triebel-Lizorkin or Besov space (and hence in no Lebesgue or Sobolev space). The purpose is to show the limitations of the so called semigroup method for the Navier-Stokes equation. We also consider the possibility of existence of solutions with initial data in the Besov space $\dot B^{-1,\infty}_\infty$. We give initial data in this space for which there is no reasonable solution for the Navier-Stokes like equation.
Published as: Proc. A.M.S., 129, (2001), 3017-3023
arXiv categories: math.AP math-ph math.CA math.FA math.MP
Vanishing of vacuum states and blow-up phenomena of the compressible Navier-Stokes equations
The Navier-Stokes systems for compressible fluids with density-dependent viscosities are considered in the present paper. These equations, in particular, include the ones which are rigorously derived recently as the Saint-Venant system for the motion of shallow water, from the Navier-Stokes system for incompressible flows with a moving free surface [14]. These compressible systems are degenerate when vacuum state appears. We study initial-boundary-value problems for such systems for both bounded spatial domains or periodic domains. The dynamics of weak solutions and vacuum states are investigated rigorously. First, it is proved that the entropy weak solutions for general large initial data satisfying finite initial entropy exist globally in time. Next, for more regular initial data, there is a global entropy weak solution which is unique and regular with well-defined velocity field for short time, and the interface of initial vacuum propagates along particle path during this time period.
The Navier–Stokes existence and smoothness problem concerns the mathematical properties of solutions to the Navier–Stokes equations, a system of partial
The Navier–Stokes existence and smoothness problem concerns the mathematical properties of solutions to the Navier–Stokes equations, a system of partial differential equations that describe the motion of a fluid in space. Solutions to the Navier–Stokes equations are used in many practical applications. However, theoretical understanding of the solutions to these equations is incomplete. In part
The Navier–Stokes existence and smoothness problem concerns the mathematical properties of solutions to the Navier–Stokes equations, a system of partial differential equations that describe the motion of a fluid in space. Solutions to the Navier–Stokes equations are used in many practical applications. However, theoretical understanding of the solutions to these equations is incomplete. In particular, solutions of the Navier–Stokes equations often include turbulence, which remains one of the greatest unsolved problems in physics, despite its immense importance in science and engineering.
Even more basic (and seemingly intuitive) properties of the solutions to Navier–Stokes have never been proven. For the three-dimensional system of equations, and given some initial conditions, mathematicians have neither proved that smooth solutions always exist, nor found any counter-examples. This is called the Navier–Stokes existence and smoothness problem.
Since understanding the Navier–Stokes equations is considered to be the first step to understanding the elusive phenomenon of turbulence, the Clay Mathematics Institute in May 2000 made this problem one of its seven Millennium Prize problems in mathematics. It offered a US$1,000,000 prize to the first person providing a solution for a specific statement of the problem:
There exists an initial condition
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