Nonlinear transport equations and quasiconformal maps

TL;DR

This paper proves the existence of solutions to a nonlinear transport equation in the plane where the velocity field is derived from a convolution with the Cauchy Kernel, utilizing quasiconformal mapping properties.

Contribution

It introduces a novel approach using quasiconformal mappings to establish existence results for a class of nonlinear transport equations with unbounded divergence.

Findings

Existence of solutions for the nonlinear transport equation in the plane.

Velocity field derived from convolution with the Cauchy Kernel can have unbounded divergence.

Quasiconformal mappings provide the key compactness property for the proof.

Abstract

We prove existence of solutions to a nonlinear transport equation in the plane, for which the velocity field is obtained as the convolution of the classical Cauchy Kernel with the unknown. Even though the initial datum is bounded and compactly supported, the velocity field may have unbounded divergence. The proof is based on the compactness property of quasiconformal mappings.