Universal structure of blow-up in 1D conservation laws

TL;DR

This paper demonstrates that solutions to 1D conservation laws exhibit a universal blow-up profile, independent of initial conditions and specific equations, explained via renormalization group theory, with examples including Burgers' equation and gas dynamics.

Contribution

It reveals a universal blow-up structure in 1D conservation laws and connects it to renormalization group theory, supported by explicit examples.

Findings

Universal wave profile independent of initial conditions

Renormalization group explains blow-up universality

Logarithmic Fourier space wave emerges as a universal feature

Abstract

We discuss universality properties of blow-up of a classical (smooth) solutions of conservation laws in one space dimension. It is shown that the renormalized wave profile tends to a universal function, which is independent both of initial conditions and of the form of a conservation law. This property is explained in terms of the renormalization group theory. A solitary wave appears in logarithmic coordinates of the Fourier space as a counterpart of this universality. Universality is demonstrated in two examples: Burgers equation and dynamics of ideal polytropic gas.