Relaxed solutions for incompressible inviscid flows: a variational and gravitational approximation to the initial
1CNRS, Département de Mathématiques et Applications, Ecole Normale Supérieure, Université PSL, 45 rue d'Ulm 75005, Paris, France.
Summary
Researchers explored relaxed Euler equations for fluid dynamics. They propose a new model, the Euler-Poisson system, to address issues with initial value problems and turbulence theory, showing promise for recovering smooth solutions.
Area of Science:
- Fluid Dynamics
- Mathematical Physics
- Turbulence Theory
Background:
- Arnold's geometric interpretation links Euler equations to geodesic problems on diffeomorphism groups.
- Existing convex relaxations of Euler equations lack well-posedness for initial value problems, limiting their use in turbulence.
- The challenge lies in developing relaxed fluid equations suitable for initial value problems and turbulence.
Purpose of the Study:
- To develop a more relevant set of relaxed Euler equations.
- To investigate the initial value problem for a novel approximate model.
- To explore connections between fluid dynamics, gravitational systems, and turbulence.
Main Methods:
- Utilized Arnold's geometric framework for fluid dynamics.
- Investigated a convex relaxation of the Euler equations.
- Formulated and analyzed the multi-stream pressure-less gravitational Euler-Poisson system as an approximate model.
Main Results:
- Demonstrated that the initial value problem for the Euler-Poisson system can be framed as a concave maximization problem.
- Showed that this formulation allows for the recovery of a significant class of smooth solutions for limited time intervals.
- Identified limitations in existing relaxed Euler equations for initial value problems and turbulence.
Conclusions:
- The proposed multi-stream pressure-less gravitational Euler-Poisson system offers a more appropriate approximate model for relaxed fluid dynamics.
- This model provides a pathway to address the well-posedness issues of relaxed Euler equations in the context of initial value problems.
- Further research into this system could advance the understanding of turbulence and fluid dynamics.
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