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 $C^{3/5}$ 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.

• The paper demonstrates that the rSV equations induce a finite-time gradient blow-up with a sharp C^(3/5) Hölder profile, contrasting the classical C^(1/3) singularity.
• Methodology employs a nonlinear self-similar bootstrap in renormalized coordinates, leveraging advanced energy estimates and nonlocal operator analysis.
• Results reveal that Hamiltonian energy conservation in the rSV system enforces smoother, weakly singular shock structures, impacting shallow water modeling.

Asymptotic Self-Similar Blow-Up in the Regularized Saint-Venant Equations

Introduction and Motivation

The paper investigates the formation of singularities in the regularized Saint-Venant (rSV) equations, which serve as a conservative, non-dispersive regularization of the isentropic Euler (or equivalently, classical Saint-Venant) equations. The rSV equations are designed to preserve the Hamiltonian structure, maintaining not only mass and momentum, but also a higher-order ($H^1$-type) energy conservation, while removing the non-uniqueness and unphysical shocks associated with the hyperbolic cSV system. However, despite this regularization, the rSV equations admit finite-time singularity formation – gradient blow-up of smooth solutions.

A central question addressed is the structure of singularities in the rSV system. While in classical compressible Euler and inviscid Burgers equations gradient blow-up profiles display a universal $C^{1/3}$ Hölder regularity at the singularity, it is not a priori clear what regularity emerges in the Hamiltonian-regularized case. This paper rigorously establishes that, for a large class of initial data evolving under the rSV equations, the gradient blow-up is sharply characterized by a $C^{3/5}$ Hölder profile, fundamentally different from the classical $C^{1/3}$ case. Furthermore, the scalar regularized Burgers (rB) equation is shown to possess identical self-similar blow-up behaviour, demonstrating that this phenomenon is intrinsic to non-dispersive Hamiltonian regularizations of the Eulerian type.

Structural Setup and Main Results

The rSV equations are written as

$$\begin{aligned} & h_t + (hu)_x = 0, \ & (hu)_t + \left( hu^2 + \frac{1}{2}gh^2 + \gamma S \right)_x = 0, \end{aligned}$$

with the precise nonlocal structure of the regularization term $\mathcal{S}$ ensuring conservation laws and Hamiltonian structure. For gradient blow-up analysis, the authors work with Riemann variables $w = u + 2\sqrt{h}$ and $z = u - 2\sqrt{h}$, and exploit a nonlocal operator formulation tailored to the modulation of the blow-up profile.

The central theorem rigorously constructs solutions with finite-time blow-up in the $C^1$ norm, proves their existence and uniqueness in $H^5$, and establishes sharp spatial regularity of $C^{1/3}$0 at the singularity. Specifically, for a non-empty open set of smooth initial data (precisely described in Sobolev and weighted norms compatible with the blow-up scenario), the solution develops, at a single point $C^{1/3}$1, a blow-up with the following properties:

• The Riemann variable $C^{1/3}$2 (and equivalently, $C^{1/3}$3 and $C^{1/3}$4) is uniformly $C^{1/3}$5 Hölder everywhere up to $C^{1/3}$6.
• For any $C^{1/3}$7, the $C^{1/3}$8 norm grows asymptotically as $C^{1/3}$9 near the singularity; away from $C^{3/5}$0, the solution remains $C^{3/5}$1 for all time.
• The companion variable $C^{3/5}$2 remains globally bounded in $C^{3/5}$3.

A fundamental aspect of the analysis is a nonlinear bootstrap in self-similar coordinates, allowing precise control of the nonlinear and nonlocal perturbations to the leading-order dynamics. The reference profile is the self-similar solution to the Hunter–Saxton (HS) equation, known to saturate the $C^{3/5}$4 Hölder exponent. The regularized Burgers equation is treated analogously and shown to fall within the same universality class.

Analytical Framework and Techniques

The proof proceeds via rescaling in dynamically modulated self-similar variables, tracking the evolution in a renormalized frame at the blow-up location. The modulation equations ensure the correct alignment with the self-similar scaling of the HS profile. An essential tool is the rigorous closure of a high-order nonlinear bootstrap, relying on advanced weighted maximum principles, energy estimates, and sharp asymptotic analysis of nonlocal Green’s functions associated with the operator structure of the regularization.

Considerable attention is paid to the selection of initial data compatible with the desired singularity: the necessary weighted and derivative bounds are nontrivial and involve matching to the stable manifold of the self-similar profile. The stability of the $C^{3/5}$5 singularity class is thus addressed, and its openness is established in suitable function spaces.

The approach elegantly decouples the fast blow-up mode (driven by the leading-order, HS-type nonlinearity) from subordinate perturbative corrections caused by the Hamiltonian regularization. Nonlocal contributions are shown to be strictly subcritical in the asymptotic regime, justifying the universality of the self-similar HS behaviour.

Implications and Theoretical Significance

Key analytical consequences and claims of the paper include:

• The $C^{3/5}$6 exponent is not an artefact of the scalar Hunter–Saxton or Camassa–Holm equations but emerges as the prototype singularity in a broad class of Hamiltonian-conservative non-dispersive equations, including physically relevant shallow water models like the rSV system.
• The energy structure fundamentally determines the possible Hölder exponents at singularity: while $C^{3/5}$7-energy conservation allows $C^{3/5}$8 singularities (Burgers, Euler, Euler–Poisson), the conservation of $C^{3/5}$9-energy – in conjunction with the particular structure of nonlocal Hamiltonian regularization – enforces a stricter threshold, and solutions cannot be less regular than $C^{1/3}$0. The observed $C^{1/3}$1 is thus the sharp, dynamically achieved regularity.
• This regularization modifies the nature of shock formation: the rSV and rB equations support weakly singular “cusped” waves rather than classical discontinuities, and their traveling wave profiles propagate at speeds coinciding with the shocks of the underlying hyperbolic system, but with markedly different internal structure.

Contradictory to prior heuristic expectations, the rSV regularization does not prevent gradient blow-up or restore global smoothness, but rather replaces classical $C^{1/3}$2 singularity formation with a smoother, energetically compatible $C^{1/3}$3 blow-up.

Broader Impact and Future Developments

• The results provide strong evidence that the leading-order structure of singularity formation in Hamiltonian-regularized systems is determined by the conservation of higher-order (Sobolev) invariants rather than the precise nature of the dispersive or dissipative regularization.
• The identification of the universality class for singularity formation in the rSV and rB equations suggests potential for future classification of blow-up exponents in other non-dispersive, non-dissipative Hamiltonian systems, both in one and higher dimensions.
• Practically, understanding the nature of singularity formation has implications for the numerical simulation of shallow water flows, possible breakdown mechanisms in (regularized) fluid models, and the design of stable conservation schemes for their integration.
• The techniques developed in this work, particularly the nonlinear self-similar bootstrap and the careful control of nonlocal terms, are likely to find application in further studies of singularity formation in PDEs where Hamiltonian structure and energy constraints are central.

Conclusion

The paper provides a definitive answer to the question of singularity structure in the regularized Saint-Venant and scalar regularized Burgers equations. It proves, with complete technical precision, that the Hamiltonian energy structure of these systems dramatically alters the regularity and dynamics of gradient blow-up, enforcing a sharp $C^{1/3}$4 Hölder singularity in place of the classical $C^{1/3}$5 root. Furthermore, the work establishes a quantitative and robust methodology for blow-up analysis in non-dispersive Hamiltonian PDEs, opening the door to further theoretical and applied investigations in singularity dynamics of regularized fluid models.

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Reference: “Asymptotic self-similar blow-up for the regularized Saint-Venant equations” (2604.03188).