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
4 - On localization and quantitative uniqueness for elliptic partial differential equations
Published online by Cambridge University Press: 15 August 2019
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
Summary
We address the decay and the quantitative uniqueness properties for solutions of the elliptic equation with a gradient term, $$\Delta u=W\cdot \nabla u$$. We prove that there exists a solution in a complement of the unit ball which satisfies $$|u(x)|\le C\exp (-C^{-1}|x|^2)$$ where $$W$$ is a certain function bounded by a constant. Next, we revisit the quantitative uniquenessfor the equation$$-\Delta u= W \cdot \nabla u$$ and provide an example of a solution vanishing at a point with the rate$${\rm const}\Vert W\Vert_{L^\infty}^2$$.We also review decay and vanishing results for the equation $$\Delta u= V u$$.
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- Partial Differential Equations in Fluid Mechanics , pp. 68 - 96Publisher: Cambridge University PressPrint publication year: 2018