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On well-posedness of a mildly dissipative family of active scalar equations in borderline Sobolev spaces

Anuj Kumar, Vincent R. Martinez

TL;DR

The paper develops a unified framework to study local well-posedness for a mildly dissipative family of active scalar equations (gSQG generalization) in borderline Sobolev spaces, where dissipation and constitutive-law modifications are only logarithmic. A central construct, the protean system, acts as a linear surrogate capturing the delicate commutator structure and guiding the derivation of a priori estimates and instantaneous smoothing in weighted Sobolev and log-Sobolev spaces. Under precise interrelations between the constitutive multiplier, the dissipation, and the frequency weights, the authors establish Hadamard well-posedness for $\beta\in(0,2]$ and global regularity for the Euler endpoint $\beta=0$, including smoothing effects and data-to-solution continuity. The results are complemented by sharp product and commutator estimates in logarithmic-type multiplier settings, and an application showing global regularity for the mildly dissipative 2D Euler equation, thus connecting subcritical borderline behavior with novel dissipative mechanisms.

Abstract

This paper considers a family of active scalar equations which modify the generalized surface quasi-geostrophic (gSQG) equations through its constitutive law and a dissipative perturbation. These modifications are characteristically mild in the sense that they are logarithmic. The problem of well posedness, in the sense of Hadamard, is then studied in a borderline setting of regularity in analogy to the scaling-critical spaces of the gSQG equations. A novelty of the system considered is the nuanced form of smoothing provided by the proposed mild form of dissipation, which is able to support global well-posedness at the Euler endpoint, but in a setting where the inviscid counterpart is known to be ill-posed. A novelty of the analysis lies in the simultaneous treatment of modifications in the constitutive law, dissipative mechanism, and functional setting, which the existing literature has typically treated separately. A putatively sharp relation is identified between each of the distinct system-modifiers that is consistent with previous studies that considered these modifications in isolation. This unified perspective is afforded by the introduction of a linear model equation, referred to as the protean system, that successfully incorporates the more delicate commutator structure collectively possessed by the gSQG family and upon which each facet of well-posedness can effectively be reduced to its study.

On well-posedness of a mildly dissipative family of active scalar equations in borderline Sobolev spaces

TL;DR

The paper develops a unified framework to study local well-posedness for a mildly dissipative family of active scalar equations (gSQG generalization) in borderline Sobolev spaces, where dissipation and constitutive-law modifications are only logarithmic. A central construct, the protean system, acts as a linear surrogate capturing the delicate commutator structure and guiding the derivation of a priori estimates and instantaneous smoothing in weighted Sobolev and log-Sobolev spaces. Under precise interrelations between the constitutive multiplier, the dissipation, and the frequency weights, the authors establish Hadamard well-posedness for $\beta\in(0,2]$ and global regularity for the Euler endpoint $\beta=0$, including smoothing effects and data-to-solution continuity. The results are complemented by sharp product and commutator estimates in logarithmic-type multiplier settings, and an application showing global regularity for the mildly dissipative 2D Euler equation, thus connecting subcritical borderline behavior with novel dissipative mechanisms.

Abstract

This paper considers a family of active scalar equations which modify the generalized surface quasi-geostrophic (gSQG) equations through its constitutive law and a dissipative perturbation. These modifications are characteristically mild in the sense that they are logarithmic. The problem of well posedness, in the sense of Hadamard, is then studied in a borderline setting of regularity in analogy to the scaling-critical spaces of the gSQG equations. A novelty of the system considered is the nuanced form of smoothing provided by the proposed mild form of dissipation, which is able to support global well-posedness at the Euler endpoint, but in a setting where the inviscid counterpart is known to be ill-posed. A novelty of the analysis lies in the simultaneous treatment of modifications in the constitutive law, dissipative mechanism, and functional setting, which the existing literature has typically treated separately. A putatively sharp relation is identified between each of the distinct system-modifiers that is consistent with previous studies that considered these modifications in isolation. This unified perspective is afforded by the introduction of a linear model equation, referred to as the protean system, that successfully incorporates the more delicate commutator structure collectively possessed by the gSQG family and upon which each facet of well-posedness can effectively be reduced to its study.
Paper Structure (28 sections, 25 theorems, 362 equations)

This paper contains 28 sections, 25 theorems, 362 equations.

Table of Contents

  1. Introduction
  2. The Protean System
  3. Mathematical Preliminaries
  4. Frequency-weighted Sobolev spaces
  5. Littlewood-Paley Decomposition
  6. Multiplier Classes
  7. Frequency weights associated to the regularity of the phase space
  8. Multipliers associated to the constitutive law
  9. Multipliers associated to the dissipation and its smoothing effect
  10. Statements of Main Results
  11. Application to the Mildly Dissipative 2D Euler Equation: Global regularity
  12. Product and Commutator estimates
  13. Product estimates
  14. Commutator estimates for logarithmic-type multipliers
  15. Analysis of the Protean System
  16. ...and 13 more sections

Key Result

Lemma 2.1

Let $\sigma\in\mathbb{R}$ and $1\le p \le q\le \infty$. Then there exists $C>0$ such that for all $j\in\mathbb{Z}$ and $f\in\mathscr{S}'(\mathbb{R}^d)$.

Theorems & Definitions (47)

  • Remark 2.1
  • Lemma 2.1: Bernstein inequalities
  • Lemma 2.2
  • Remark 2.2
  • Theorem 3.1
  • Theorem 3.2
  • Remark 3.1
  • Corollary 3.3
  • Corollary 3.4
  • Remark 3.2
  • ...and 37 more