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Asymptotic self-similar blow-up for the regularized Saint-Venant equations

Yunjoo Kim, Bongsuk Kwon, Wanyong Shim

Abstract

We investigate singularity formation in the regularized Saint--Venant (rSV) equations, a conservative, non-dispersive shallow water system that is formally regarded as a Hamiltonian regularization of the isentropic Euler equations. While it is known that smooth solutions to the rSV system can develop gradient blow-up in finite time, the precise structure of such singularities has not been rigorously characterized. In this work, we establish stability of self-similar blow-up profiles of the Hunter--Saxton equation within the rSV framework, using a nonlinear bootstrap argument in dynamically rescaled coordinates. Our analysis captures the detailed space-time dynamics of solutions near the singularity, and proves their sharp $C^{3/5}$ Hölder regularity at the singular time. This regularity differs from the $C^{1/3}$ Hölder regularity of the cubic-root singularities found in the compressible Euler and inviscid Burgers equations. This contrast highlights the structural influence of the Hamiltonian regularization on singularity formation. To illuminate this effect, we also show that the same $C^{3/5}$ blow-up profile emerges in the regularized Burgers equation, a scalar analogue of the rSV system.

Asymptotic self-similar blow-up for the regularized Saint-Venant equations

Abstract

We investigate singularity formation in the regularized Saint--Venant (rSV) equations, a conservative, non-dispersive shallow water system that is formally regarded as a Hamiltonian regularization of the isentropic Euler equations. While it is known that smooth solutions to the rSV system can develop gradient blow-up in finite time, the precise structure of such singularities has not been rigorously characterized. In this work, we establish stability of self-similar blow-up profiles of the Hunter--Saxton equation within the rSV framework, using a nonlinear bootstrap argument in dynamically rescaled coordinates. Our analysis captures the detailed space-time dynamics of solutions near the singularity, and proves their sharp Hölder regularity at the singular time. This regularity differs from the Hölder regularity of the cubic-root singularities found in the compressible Euler and inviscid Burgers equations. This contrast highlights the structural influence of the Hamiltonian regularization on singularity formation. To illuminate this effect, we also show that the same blow-up profile emerges in the regularized Burgers equation, a scalar analogue of the rSV system.

Paper Structure

This paper contains 19 sections, 31 theorems, 419 equations.

Table of Contents

  1. Introduction
  2. Reformulation using Riemann functions
  3. Self-similar Hunter--Saxton profiles
  4. Main result
  5. Blow-up for the regularized Burgers equation
  6. Discussion
  7. Stability estimates and Proof of Theorem \ref{['mainthm']}
  8. Self-similar variables and modulations
  9. Bootstrap argument
  10. Proof of Theorem \ref{['mainthm']}
  11. Closure of bootstrap assumptions
  12. Preliminary estimates
  13. Proof of (\ref{['dottau_close']}) and (\ref{['lem_Zy_close']})--(\ref{['Zyweight_close']}) in Proposition \ref{['Boot']}
  14. Proof of (\ref{['Utildey_close']})--(\ref{['Wy4_close']}) in Proposition \ref{['Boot']}
  15. Outline of the proof of Theorem \ref{['mainthm2']}
  16. ...and 4 more sections

Key Result

Proposition 1.1

The ODE problem Weq--decay-infty admits a one-parameter family of smooth solutions $\{ \overline{W}_\beta(y) : \beta>0 \}$ such that for each $\beta>0$, $\overline{W}_\beta(y)$ is monotonically decreasing on $\mathbb{R}$ and is odd, i.e., $\overline{W}_\beta(-y) = - \overline{W}_\beta(y)$ for all $y and In particular, and, as $|y|\rightarrow \infty$, $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (60)

  • Proposition 1.1: KKY, Proposition 2.1
  • Theorem 1.2
  • Remark 1
  • Remark 2
  • Theorem 1.3
  • Lemma 2.1: Local existence and blow-up criterion
  • Remark 3: Initial conditions for $(W,Z)$
  • Remark 4
  • Proposition 2.2
  • proof
  • ...and 50 more