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Published on: July 19, 2016
Divergent expansion, Borel summability and three-dimensional Navier-Stokes equation
Ovidiu Costin1, Guo Luo, Saleh Tanveer
1Department of Mathematics, Ohio State University, OH 43210, USA.
Summary
This study introduces a novel integral equation approach for nonlinear partial differential equations (PDEs), extending Borel summation to handle non-analytic data and proving local existence for Navier-Stokes equations.
Area of Science:
- Applied Mathematics
- Fluid Dynamics
- Mathematical Analysis
Background:
- Borel summation is a technique for assigning values to divergent asymptotic series.
- Traditional Borel summation has limitations with non-analytic initial data in partial differential equations (PDEs).
- The Navier-Stokes (NS) system describes fluid motion but poses significant challenges in proving global existence of solutions.
Purpose of the Study:
- To extend Borel summation techniques to nonlinear PDEs.
- To develop an integral equation (IE) approach applicable to more general initial data.
- To investigate local existence proofs for the three-dimensional Navier-Stokes (NS) system.
Main Methods:
- Applying Borel summability of divergent asymptotic expansions to nonlinear PDEs.
- Formulating an integral equation (IE) applicable to non-analytic initial data.
- Analyzing the three-dimensional Navier-Stokes (NS) system using the IE approach.
Main Results:
- The IE approach accommodates more general initial data than standard Borel summation.
- Local existence proofs for the 3D NS system are derived using the IE method.
- Global existence for 3D NS is reframed as an asymptotic problem in a specific variable.
- Rigorous error control in numerical computations of the IE is achieved.
Conclusions:
- The developed IE method offers a robust framework for analyzing nonlinear PDEs, including the NS system.
- This approach provides a pathway to rigorously control errors in numerical solutions.
- The study demonstrates the potential of asymptotic analysis in addressing long-standing problems like global existence in fluid dynamics.
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