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Onsager's Mean Field Theory of Vortex Flows with Singular Sources: Blow-Up and Concentration without Quantization

Daniele Bartolucci, Paolo Cosentino, Lina Wu

TL;DR

This work extends Onsager's mean-field theory to include a fixed singular vortex of strength $\sigma$ at the origin, formulating canonical and microcanonical variational principles with a singular weight $H_{\lambda}(x)$ and establishing a sharp blow-up threshold $\lambda_{\sigma}$. It proves the equivalence of the CVP and MVP for a class of domains, develops a minimal mass framework that yields a lower bound on the concentration mass and introduces a novel blow-up regime—blow-up with concentration without quantization—where the concentrated mass varies in a real interval and depends on the singularity strength. The authors derive refined non-quantized bubbling profiles, prove existence of entropy-maximizing densities, and analyze high-energy limits, including a detailed Type I disk example that yields explicit entropy asymptotics and confirms the domain-type classification. Together, these results provide a comprehensive variational and asymptotic description of vortex flows with singular sources, bridging classical quantized blow-up with non-quantized concentration phenomena and suggesting geophysical applications related to principal vortex structures.

Abstract

Motivated by the Onsager statistical mechanics description of turbulent Euler flows with point singularities, we make a first step in the generalization of the mean field theory in [Caglioti, Lions, Marchioro, Pulvirenti; Comm. Math. Phys. (1995)]. On one side we prove the equivalence of statistical ensembles, on the other side we are bound to the analysis of a new blow up phenomenon, which we call "blow up and concentration without quantization", where the mass associated with the concentration is allowed to take values in a full interval of real numbers. This singular behavior may be regarded as lying between the classical blow up-concentration-quantization and the blow up without concentration phenomenon first proposed in [Lin, Tarantello; C.R. Math. Acad. Sci. Paris (2016)]. A careful analysis is needed to generalize known pointwise estimates in this non standard context, resulting in a complete description of the allowed asymptotic profiles.

Onsager's Mean Field Theory of Vortex Flows with Singular Sources: Blow-Up and Concentration without Quantization

TL;DR

This work extends Onsager's mean-field theory to include a fixed singular vortex of strength at the origin, formulating canonical and microcanonical variational principles with a singular weight and establishing a sharp blow-up threshold . It proves the equivalence of the CVP and MVP for a class of domains, develops a minimal mass framework that yields a lower bound on the concentration mass and introduces a novel blow-up regime—blow-up with concentration without quantization—where the concentrated mass varies in a real interval and depends on the singularity strength. The authors derive refined non-quantized bubbling profiles, prove existence of entropy-maximizing densities, and analyze high-energy limits, including a detailed Type I disk example that yields explicit entropy asymptotics and confirms the domain-type classification. Together, these results provide a comprehensive variational and asymptotic description of vortex flows with singular sources, bridging classical quantized blow-up with non-quantized concentration phenomena and suggesting geophysical applications related to principal vortex structures.

Abstract

Motivated by the Onsager statistical mechanics description of turbulent Euler flows with point singularities, we make a first step in the generalization of the mean field theory in [Caglioti, Lions, Marchioro, Pulvirenti; Comm. Math. Phys. (1995)]. On one side we prove the equivalence of statistical ensembles, on the other side we are bound to the analysis of a new blow up phenomenon, which we call "blow up and concentration without quantization", where the mass associated with the concentration is allowed to take values in a full interval of real numbers. This singular behavior may be regarded as lying between the classical blow up-concentration-quantization and the blow up without concentration phenomenon first proposed in [Lin, Tarantello; C.R. Math. Acad. Sci. Paris (2016)]. A careful analysis is needed to generalize known pointwise estimates in this non standard context, resulting in a complete description of the allowed asymptotic profiles.
Paper Structure (12 sections, 22 theorems, 473 equations)

This paper contains 12 sections, 22 theorems, 473 equations.

Table of Contents

  1. Introduction
  2. Blow up analysis: minimal mass Lemma and first consequences
  3. Blow up analysis: profile estimates without quantization
  4. Free energy and existence of solutions: the Canonical Variational Principle
  5. Entropy and existence of solutions: the Microcanonical Variational Principle, domains of type I/II.
  6. An example of a domain of Type I with $\sigma<0$.
  7. High energy limit of solutions $\rho_n$ of \ref{['mf:n.intro']}.
  8. Appendix
  9. Known results about problem \ref{['profile:tildew']}
  10. Proof of the estimate \ref{['profilew:H']}
  11. Initial estimate for \ref{['profilew:H']}
  12. Integral representation for $w_n$ and refined estimate

Key Result

Theorem 1.2

For any $\lambda \in [0,\lambda _\sigma)$ there exists one and only one solution $\rho_{\lambda }$ of cvp:intro and $\psi_{\lambda }=G[\rho_\lambda ]$ is a solution of the Singular Mean Field Equation,

Theorems & Definitions (41)

  • Remark 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Definition 1.1
  • Lemma 2.1: Minimal Mass Lemma
  • proof
  • Theorem 2.2: Concentration
  • proof
  • Remark 2.3
  • ...and 31 more