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The divergence and Stokes' theorems are a variation of Green's theorem in a higher dimension. They are also a generalization of the fundamental theorem of calculus. The divergence theorem and Stokes' theorem are in a way similar to each other; The divergence theorem relates to the dot product of a vector, while Stokes' theorem relates to the curl of a vector. Many applications in physics and engineering make use of the divergence and Stokes' theorems, enabling us to write numerous physical laws...
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The Diffusion of Passive Tracers in Laminar Shear Flow
08:01

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Published on: May 1, 2018

Hardy-Littlewood-Sobolev inequalities via fast diffusion flows.

Eric A Carlen1, José A Carrillo, Michael Loss

  • 1Department of Mathematics, Hill Center, Rutgers University, 110 Frelinghuysen Road, Piscataway, NJ 08854, USA. carlen@math.rutgers.edu

Proceedings of the National Academy of Sciences of the United States of America
|November 3, 2010
PubMed
Summary
This summary is machine-generated.

Researchers proved the sharp Hardy-Littlewood-Sobolev inequality (λ = d - 2) for d≥3 and the sharp Logarithmic Hardy-Littlewood-Sobolev inequality for d = 2 using a fast diffusion equation flow.

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Area of Science:

  • Analysis
  • Partial Differential Equations
  • Geometric Inequalities

Background:

  • The Hardy-Littlewood-Sobolev inequality is a fundamental result in analysis with broad applications.
  • Previous proofs for specific cases often involved complex techniques.
  • The Logarithmic Hardy-Littlewood-Sobolev inequality is a crucial variant for d=2.

Purpose of the Study:

  • To provide a simplified proof for specific cases of the sharp Hardy-Littlewood-Sobolev inequality.
  • To establish a unified approach using monotone flows.
  • To investigate the sharp Logarithmic Hardy-Littlewood-Sobolev inequality for d=2.

Main Methods:

  • A monotone flow governed by the fast diffusion equation.
  • Analysis of the behavior of solutions under this flow.
  • Derivation of sharp inequalities from the properties of the flow.

Main Results:

  • A simple proof for the λ = d - 2 cases of the sharp Hardy-Littlewood-Sobolev inequality for dimensions d≥3.
  • A proof for the sharp Logarithmic Hardy-Littlewood-Sobolev inequality for d = 2.
  • Demonstration of the effectiveness of the fast diffusion equation in proving these inequalities.

Conclusions:

  • The fast diffusion equation provides an elegant and powerful tool for proving sharp integral inequalities.
  • This work simplifies existing proofs and extends results to the logarithmic case.
  • The findings have implications for harmonic analysis and related fields.