Applications of Bourgain-Brezis inequalities to Fluid Mechanics and Magnetism

TL;DR

This paper applies Bourgain-Brezis inequalities to analyze the vorticity and velocity in 2D fluid mechanics and to estimate magnetic fields in electromagnetism, demonstrating new bounds and regularity results.

Contribution

It extends Bourgain-Brezis inequalities to the vorticity equation and Maxwell equations, providing novel bounds and regularity results for these physical systems.

Findings

Vorticity remains in BV space for small time with BV initial data.

Velocity vector remains uniformly bounded for small time.

Magnetic field size estimated using only L^1 norm of current density gradient.

Abstract

We apply the borderline Sobolev inequalities of Bourgain-Brezis to the vorticity equation and Navier-Stokes equation in 2D. We take the initial vorticity to be in the space of functions of Bounded variation(BV). We obtain the subsequent vorticity to be in the space of functions of bounded variation, uniformly for small time, and the velocity vector to be uniformly bounded for small time. Such a conclusion cannot follow for initial vorticity taken to be just a measure or in L^1 from the Lamb-Oseen vortex example. Secondly we apply an improved Strichartz inequality obtained earlier by the first and third authors to the Maxwell equations of Electromagnetism. In particular we estimate the size of the magnetic field vector in terms of the gradient of the current density vector. The main point is that in this inequality only the L^1 norm in space appears for the gradient of the current density vector. Such a result is only possible because of a vanishing divergence inhomogeneity in the wave equation for the Magnetic field vector stemming from the Maxwell equations. A key ingredient in the proof of the improved Strichartz inequality is the Bourgain-Brezis borderline Sobolev inequalities.