Exact self-similar finite-time blowup of the Hou-Luo model with smooth profiles

TL;DR

This paper proves the existence of exact self-similar finite-time blowup solutions with smooth profiles for the 1D Hou-Luo model, using an analytic fixed-point approach and providing detailed properties of these solutions.

Contribution

It introduces a purely analytic method to establish smooth self-similar blowup profiles and characterizes their monotonicity, convexity, and decay properties, advancing understanding of the model's singularity formation.

Findings

Existence of smooth self-similar blowup profiles proven analytically.

Profiles exhibit specific monotonicity and convexity properties.

Rigorous estimates on decay rates of the profiles in the far field.

Abstract

We show that the 1D Hou-Luo model on the real line admits exact self-similar finite-time blowup solutions with smooth self-similar profiles. The existence of these profiles is established via a fixed-point method that is purely analytic. We also prove that the profiles satisfy some monotonicity and convexity properties that were unknown before, and we give rigorous estimates on the algebraic decay rates of the profiles in the far field. Our result supplements the previous computer-assisted proof of self-similar finite-time blowup for the Hou-Luo model with finer characterizations of the profiles.