Book contents
- Frontmatter
- Contents
- Contributors
- Preface
- 1 Remarks on recent advances concerning boundary effects and the vanishing viscosity limit of the Navier–Stokes equations
- 2 Time-periodic flow of a viscous liquid past a body
- 3 The Rayleigh–Taylor instability in buoyancy-driven variable density turbulence
- 4 On localization and quantitative uniqueness for elliptic partial differential equations
- 5 Quasi-invariance for the Navier–Stokes equations
- 6 Leray’s fundamental work on the Navier–Stokes equations: a modern review of “Sur le mouvement d’un liquide visqueux emplissant l’espace”
- 7 Stable mild Navier–Stokes solutions by iteration of linear singular Volterra integral equations
- 8 Energy conservation in the 3D Euler equations on T2 × R+
- 9 Regularity of Navier–Stokes flows with bounds for the velocity gradient along streamlines and an effective pressure
- 10 A direct approach to Gevrey regularity on the half-space
- 11 Weak-Strong Uniqueness in Fluid Dynamics
7 - Stable mild Navier–Stokes solutions by iteration of linear singular Volterra integral equations
Published online by Cambridge University Press: 15 August 2019
- Frontmatter
- Contents
- Contributors
- Preface
- 1 Remarks on recent advances concerning boundary effects and the vanishing viscosity limit of the Navier–Stokes equations
- 2 Time-periodic flow of a viscous liquid past a body
- 3 The Rayleigh–Taylor instability in buoyancy-driven variable density turbulence
- 4 On localization and quantitative uniqueness for elliptic partial differential equations
- 5 Quasi-invariance for the Navier–Stokes equations
- 6 Leray’s fundamental work on the Navier–Stokes equations: a modern review of “Sur le mouvement d’un liquide visqueux emplissant l’espace”
- 7 Stable mild Navier–Stokes solutions by iteration of linear singular Volterra integral equations
- 8 Energy conservation in the 3D Euler equations on T2 × R+
- 9 Regularity of Navier–Stokes flows with bounds for the velocity gradient along streamlines and an effective pressure
- 10 A direct approach to Gevrey regularity on the half-space
- 11 Weak-Strong Uniqueness in Fluid Dynamics
Summary
By their use of mild solutions, Fujita-Kato and later on Giga-Miyakawa opened the way to solving the initial-boundary value problem for the Navier-Stokes equations with the help of the contracting mapping principle in suitable Banach spaces, on any smoothly bounded domain $$\Omega \subset \R^n, n \ge 2$$, globally in time in case of sufficiently small data. We will consider a variant of these classical approximation schemes: by iterative solution of linear singular Volterra integral equations, on any compact time interval J, again we find the existence of a unique mild Navier-Stokes solution under smallness conditions, but moreover we get the stability of each (possibly large) mild solution, inside a scale of Banach spaces which are imbedded in some $$C^0 (J, L^r (\Omega))$$, $$1 < r < \infty$$.
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- Partial Differential Equations in Fluid Mechanics , pp. 204 - 223Publisher: Cambridge University PressPrint publication year: 2018