Partial regularity of suitable weak solutions of the navier‐stokes equations

Caffarelli, Luis A.; Kohn, Robert V.; Nirenberg, Louis

doi:10.1002/cpa.3160350604

Cited by 1,384 publications

(1,858 citation statements)

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“…A weak solution does not satisfy the standard energy balance valid for a smooth solution of the Navier-Stokes equations, instead only an energy inequality can be shown to hold. Partial regularity of a weak solution was proven under the assumption that a generalized energy inequality holds [30,31], in which case the weak solution is referred to as a suitable weak solution. It can be shown that certain discrete approximations, including finite element approximations, converge to suitable solutions [32].…”

Section: Dissipative Weak Solutionsmentioning

confidence: 99%

Towards a parameter-free method for high Reynolds number turbulent flow simulation based on adaptive finite element approximation

Hoffman

Jansson

et al. 2015

Computer Methods in Applied Mechanics and Engineering

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Highlights• We present a parameter-free method for high Reynolds number turbulent flow.• The method is based on adaptive mesh optimization using adjoint techniques.• We connect the numerical approximation to dissipative weak solutions.• Turbulent boundary layers are left unresolved, no boundary layer mesh is needed.• We highlight benchmark problems for which the method is validated. AbstractThis article is a review of our work towards a parameter-free method for simulation of turbulent flow at high Reynolds numbers. In a series of papers we have developed a model for turbulent flow in the form of weak solutions of the Navier-Stokes equations, approximated by an adaptive finite element method, where: (i) viscous dissipation is assumed to be dominated by turbulent dissipation proportional to the residual of the equations, and (ii) skin friction at solid walls is assumed to be negligible compared to inertial effects. The result is a computational model without empirical data, where the only model parameter is the local size of the finite element mesh. Under adaptive refinement of the mesh based on a posteriori error estimation, output quantities of interest in the form of functionals of the finite element solution converge to become independent of the mesh resolution, and thus the resulting method has no adjustable parameters. No ad hoc design of the mesh is needed, instead the mesh is optimized based on solution features, in particular no boundary layer mesh is needed. We connect the computational method to the mathematical concept of a dissipative weak solution of the Euler equations, as a model of high Reynolds number turbulent flow, and we highlight a number of benchmark problems for which the method is validated. The purpose of the article is to present the computational framework in a concise form, to report on recent progress, and to discuss open problems that are subject to ongoing research.

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Section: Dissipative Weak Solutionsmentioning

confidence: 99%

Towards a parameter-free method for high Reynolds number turbulent flow simulation based on adaptive finite element approximation

Hoffman

Jansson

et al. 2015

Computer Methods in Applied Mechanics and Engineering

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“…We will derive the one-dimensional model for the three-dimensional axisymmetric Navier-Stokes equations. By the well-known Caffarelli-Kohn-Nirenberg theory [3], the singularity set of any suitable weak solution of the three-dimensional Navier-Stokes equations has one-dimensional Hausdorff measure 0. Thus, in the case of axisymmetric three-dimensional Navier-Stokes equations with swirl, if there is any singularity, it must be along the symmetry axis, i.e., the´-axis.…”

Section: Derivation Of the One-dimensional Modelmentioning

confidence: 99%

Dynamic stability of the three‐dimensional axisymmetric Navier‐Stokes equations with swirl

Hou

2007

Comm Pure Appl Math

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In this paper, we study the dynamic stability of the three-dimensional axisymmetric Navier-Stokes Equations with swirl. To this purpose, we propose a new one-dimensional model that approximates the Navier-Stokes equations along the symmetry axis. An important property of this one-dimensional model is that one can construct from its solutions a family of exact solutions of the threedimensionaFinal Navier-Stokes equations. The nonlinear structure of the onedimensional model has some very interesting properties. On one hand, it can lead to tremendous dynamic growth of the solution within a short time. On the other hand, it has a surprising dynamic depletion mechanism that prevents the solution from blowing up in finite time. By exploiting this special nonlinear structure, we prove the global regularity of the three-dimensional Navier-Stokes equations for a family of initial data, whose solutions can lead to large dynamic growth, but yet have global smooth solutions.

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“…To reflect this invariance one assigns scaling dimensions as given in [9], [21]. For example we assign dimension 2 to time variable and dimension 1 to spatial variable.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

An Approach to Rotating Boundary Layers Based on Vector Radon Measures

Giga

Saal

2012

J. Math. Fluid Mech.

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ABSTRACT. In this paper we develop a new approach to rotating boundary layers via Fourier transformed finite vector Radon measures. As an application we consider the Ekman boundary layer. By our methods we can derive very explicit bounds for existence intervals and solutions of the linearized and the nonlinear Ekman system. For example, we can prove these bounds to be uniform with respect to the angular velocity of rotation which has proved to be relevant for several aspects (see introduction). Another advantage of our approach is that we obtain well-posedness in classes containing nondecaying vector fields such as almost periodic functions. These outcomes give respect to the nature of boundary layer problems and cannot be obtained by approaches in standard function spaces such as Lebesgue, Bessel-potential, Hölder or Besov spaces.

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Partial regularity of suitable weak solutions of the navier‐stokes equations

Cited by 1,384 publications

References 18 publications

Towards a parameter-free method for high Reynolds number turbulent flow simulation based on adaptive finite element approximation

Towards a parameter-free method for high Reynolds number turbulent flow simulation based on adaptive finite element approximation

Dynamic stability of the three‐dimensional axisymmetric Navier‐Stokes equations with swirl

An Approach to Rotating Boundary Layers Based on Vector Radon Measures

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