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About this paper
Ulisse Iotti
This paper presents a method for demonstrating the uniform boundedness of solutions to the three-dimensional incompressible Navier-Stokes and Euler equations. It achieves this by introducing a geometric decomposition of the flow domain based on the orthogonality between the velocity field and the pressure gradient, which, if valid, implies global existence of smooth solutions.
Key insights
- The paper introduces a geometric decomposition of the flow domain into regions where vorticity vanishes (irrotational) and its complement (rotational).
- In the rotational region, the velocity field is claimed to be orthogonal to the pressure gradient ($u ext{ extperiodcentered} abla p = 0$), allowing a simplified analysis of the nonlinear term.
- This orthogonality condition leads to a representation of the pressure gradient as $ abla p = u imes abla heta$, which is used to derive bounds on the velocity field.
Summary by alphaXiv.
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