Maximal regularity of the Stokes operator in an exterior domain with moving boundary and application to the Navier–Stokes equations

2018 ◽  
Vol 375 (3-4) ◽  
pp. 949-972
Author(s):  
Reinhard Farwig ◽  
Hideo Kozono ◽  
David Wegmann
Keyword(s):  
Exterior Domain ◽  
Moving Boundary ◽  
Maximal Regularity ◽  
Stokes Equations ◽  
Stokes Operator ◽  
Navier Stokes ◽  
Navier Stokes Equations

Some Elements of Functional Analysis

2006 ◽  
Author(s):  
Jean-Yves Chemin ◽  
Benoit Desjardins ◽  
Isabelle Gallagher ◽  
Emmanuel Grenier
Keyword(s):  
Functional Analysis ◽  
Weak Solutions ◽  
Vector Fields ◽  
Stokes Equations ◽  
Stokes Operator ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Dual Spaces ◽  
Type Spaces ◽  
Definition Of

Before introducing the concept of Leray’s weak solutions to the incompressible Navier–Stokes equations, classical definitions of Sobolev spaces are required. In particular, when it comes to the analysis of the Stokes operator, suitable functional spaces of incompressible vector fields have to be defined. Several issues regarding the associated dual spaces, embedding properties, and the mathematical way of considering the pressure field are also discussed. Let us first recall the definition of some functional spaces that we shall use throughout this book. In the framework of weak solutions of the Navier– Stokes equations, incompressible vector fields with finite viscous dissipation and the no-slip property on the boundary are considered. Such H1-type spaces of incompressible vector fields, and the corresponding dual spaces, are important ingredients in the analysis of the Stokes operator.

The engine behind (wall) turbulence: perspectives on scale interactions

2017 ◽  
Vol 817 ◽  
Author(s):  
B. J. McKeon
Keyword(s):  
Coherent States ◽  
Systems Approach ◽  
Stokes Equations ◽  
Stokes Operator ◽  
Wall Turbulence ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Scale Interactions ◽  
High Reynolds ◽  
Data Informed

Known structures and self-sustaining mechanisms of wall turbulence are reviewed and explored in the context of the scale interactions implied by the nonlinear advective term in the Navier–Stokes equations. The viewpoint is shaped by the systems approach provided by the resolvent framework for wall turbulence proposed by McKeon & Sharma (J. Fluid Mech., vol. 658, 2010, pp. 336–382), in which the nonlinearity is interpreted as providing the forcing to the linear Navier–Stokes operator (the resolvent). Elements of the structure of wall turbulence that can be uncovered as the treatment of the nonlinearity ranges from data-informed approximation to analysis of exact solutions of the Navier–Stokes equations (so-called exact coherent states) are discussed. The article concludes with an outline of the feasibility of extending this kind of approach to high-Reynolds-number wall turbulence in canonical flows and beyond.

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